· table of contents table of contents ii abstract 1 acknowledgements 2 introduction 3 1 soliton...
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Sapienza Universita di Roma
Dottorato di Ricerca in Fisica
Scuola di dottorato ”Vito Volterra”
Einstein-Maxwell equations:
soliton solutions, equilibrium configurations
and related aspects.
Thesis submitted to obtain the degree of
”Dottore di Ricerca” - Doctor PhilosophiæPhD in Physics - XX cycle - October 2008
by
Armando Paolino
Program Coordinator Thesis Advisors
Prof. Enzo Marinari Prof. Vladimir Belinski
Prof. Antonio Degasperis
to Ayumi and Lisa
i
Table of contents
Table of contents ii
Abstract 1
Acknowledgements 2
Introduction 3
1 Soliton solutions of Einstein equations: Belinski-Zakharov technique 81.1 The integrable ansatz in general relativity . . . . . . . . . . . . . . . . . 81.2 The integration scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Construction of n-soliton solution . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 The physical metric components gab . . . . . . . . . . . . . . . . 161.3.2 The physical metric components f . . . . . . . . . . . . . . . . . 18
1.4 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Solitonic solutions of Einstein-Maxwell equations: Alekseev technique 202.1 The Einstein-Maxwell field equations . . . . . . . . . . . . . . . . . . . . 202.2 The spectral problem for Einstein-Maxwell fields . . . . . . . . . . . . . . 242.3 The component gab and the potentials Aa . . . . . . . . . . . . . . . . . 29
2.3.1 The n-soliton solution of the spectral problem . . . . . . . . . . . 292.3.2 The matrix X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.3 Verifications of the constraints . . . . . . . . . . . . . . . . . . . . 35
2.4 The metric component f . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5 Summary of prescriptions . . . . . . . . . . . . . . . . . . . . . . . . . . 392.6 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Exact stationary axially symmetric one-soliton solution on Minkowskybackground 41
Index notation and form of the line elements. . . . . . . . . . . . . . . . 423.1 Application of the first nine steps of the generating procedure. . . . . . . 42
3.1.1 Step-1: Background Einstein-Maxwell solution. . . . . . . . . . . 423.1.2 Step-2: Background value of the complex electromagnetic potential
Φ(0)a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.3 Step-3: Calculus of X(0) and X(0)−1. . . . . . . . . . . . . . . . . 43
ii
3.1.4 Step-4: Calculus of U(0)µ . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.5 Step-5: Deduction of β(xµ). . . . . . . . . . . . . . . . . . . . . . 453.1.6 Step-6: Calculus of W (0). . . . . . . . . . . . . . . . . . . . . . . . 453.1.7 Step-7: Deduction of background generating matrix ϕ(0) (w, xµ)
and its normalization. . . . . . . . . . . . . . . . . . . . . . . . . 463.1.8 Step-8: Costruction of m(k)A and p
(k)A vectors. . . . . . . . . . . . 48
3.1.9 Step-9: Costruction of Tkl matrix. . . . . . . . . . . . . . . . . . . 493.2 Determination of the mathematical parameters in terms of the physical one. 49
3.2.1 The McGuire-Ruffini one-body solution. . . . . . . . . . . . . . . 513.2.2 Calculus and comparison of the f conformal factors. . . . . . . . 533.2.3 Step-10 & 11: Determination of S matrix and calculus of gab
components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2.4 Components of the electromagnetic potential. . . . . . . . . . . . 58
4 A perturbative approach for stationary axially symmetric soliton solu-tions 604.1 Some preliminary remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2 A proposal for a perturbative approach. . . . . . . . . . . . . . . . . . . . 624.3 Outline of the perturbative solitonic generating technique. . . . . . . . . 64
4.3.1 Expansion of Einstein-Maxwell Equations. . . . . . . . . . . . . . 644.3.2 Lax Pair expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.3 Expanded dressing procedure. . . . . . . . . . . . . . . . . . . . . 67
5 Generation of approximate soliton solutions over a flat background. 695.1 Perturbative building block quantities . . . . . . . . . . . . . . . . . . . . 705.2 One-soliton approximate solution. . . . . . . . . . . . . . . . . . . . . . . 755.3 Two-soliton lowest order approximate solution . . . . . . . . . . . . . . . 81
6 Electric force lines of the double Reissner-Nordstrom exact solution 856.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2 Some Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3 Summary of the Alekseev-Belinski formulas . . . . . . . . . . . . . . . . . 896.4 Some further details of the solution . . . . . . . . . . . . . . . . . . . . . 906.5 Electric force lines definition . . . . . . . . . . . . . . . . . . . . . . . . . 946.6 Plots of the force lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.6.1 Two charges of equal sign ( e1e2 > 0 ) . . . . . . . . . . . . . . . . 966.6.2 Two charges of opposite sign ( e1e2 < 0 ) . . . . . . . . . . . . . . 986.6.3 Cases with only one charge . . . . . . . . . . . . . . . . . . . . . . 100
6.7 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7 A stability analysis of the double Reissner-Nordstrom Alekseev-Belinskiexact solution 1057.1 Force of the strut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.2 Analysis of equilibrium in the AB solution . . . . . . . . . . . . . . . . . 1097.3 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
iii
Conclusions and prospects 113
Additional contributions 115
I. Charged membrane as a source for repulsive gravity 116
II. Intersection of self-gravitating charged shells in a Reissner-Nordstromfield 132
Personal works 153
Bibliography 154
iv
Abstract
This thesis includes two parts, both concerning the balance problem of two charged
compact objects in General Relativity.
In the first part, chapters 1-5, we review the Belinsky-Zakharov and Alekseev solitonic
dressing techniques to generate respectively exact solutions for the Einstein equations in
vacuum and the Einstein-Maxwell equations in electro-vacuum. We recall those prob-
lems that these generating techniques have in solving the equilibrium problem of interest.
Then we construct a procedure to calculate the approximate terms of solitonic solutions
in respect to a control parameter interpretable as the Newtonian constant. We apply
this method to re-derive the field of two charged and rotating sources on Minkowski
background, giving the reparameterization between the mathematical constant and the
physical one.
In the second section of our work, chapters 6-7, we analyze the configurations of the
Alekseev-Belinski solution [5] for two Ressner-Nordstrom like sources in static equilib-
rium. As regards a qualitative classification in respect to the physical parameters we
show: a) which configurations are forbidden in respect to the equilibrium conditions; b)
the pictures of the force lines of the electric field; c) that most of the permitted config-
urations are stable in respect to displacements from the distance of equilibrium; d) that
some results, found in the past by means of test particle approximation, agree with this
more general exact solution.
Two additional contributions are enclosed at the end as a separate body.
The first one deals with the construction of an everywhere regular membrane model for
a Reissner-Nordstrom naked singularity. It is shown that such model is stable and allows
an external repulsive-gravity region. This toy model contributes to give a more sensible
physical meaning also to the Alekseev-Belinski solution.
The second one regards the intersection of two charged spherical thin shells in a central
Reissner-Nordstrom field. We give the energy-exchange formula, and compare the ejec-
tion mechanism of one of the two shells for these charged configurations, with the one
studied in a previous work of Barkov et al [7] for neutral shells.
1
Acknowledgements
I am very grateful to my supervisor, Vladimir Belinski, for his continued guidance and
support; to Remo Ruffini and George Alekseev for the stimulating discussions; to Decio
Levi for the useful suggestions about the drafting of this thesis.
A special thank to Antonio Degasperis, for his teachings, his constant presence and very
kind helpful, but, overall, for his dear friendship.
I would like to thank Marco Pizzi, since to work with him has been a very interesting
and beautiful experience.
Thanks to Monica Rizzo, Patrizia Mezzabotta, Francesca Paolino, Chiara Milano and
Roberto Ricci for their useful, and necessary, linguistic hints.
I wish to remember all those friends with whom I spent lovely moments during this
period of study, in particular Antonino Marciano, Fabio Briscese, Calogero Tornese and
Riccardo Benini, and all my old friends, which I feel close to me in each moment of my
life and that gave me a great help during these last years: Stefano Granata, Giangiacomo
Gandolfi, Fabrizio Ferrante, Daniela Bonetti, Mario Rosati, Paolo Branchesi.
I am grateful to my mother, my father and my brothers, for their help and encouragement.
Finally to Ayumi and to Lisa chan: the light of our eyes.
2
Introduction
In the Einstein theory of gravitation, as in any nonlinear theory, the construction and
investigation of diverse families of exact solutions of the field equations is of considerable
interest in studying the complicated nonlinear character of interaction of the gravita-
tional fields of different sources with each other and with other forms of matter. The
characterizations of simple model configurations among these solutions and their investi-
gations yield explanations for many qualitative features of their fields and of the process
of their interaction. Having a clear physical content, these model configurations can
turn out to be very useful in discussing many questions in the theory of gravitation and
various applications of the theory in cosmology, the physics of compact cosmic objects,
and the theory of radiation, propagation, and interaction of waves.
The developments in the area of exact solutions have ran through various phases (See as
review work examples [3], [19], [20], [22], [83]). In a first period, the principal way of ob-
taining exact solutions to Einstein’s field equations was to impose sufficient symmetries
in order to reduce the integrability problem of the system to a rather easily tractable
set of equations. Later, the 60s saw an increase of the number of exact solutions thanks
to the development of techniques to treat geometries with particular algebraic prop-
erties. In 1968 Ernst [34, 35] introduced complex potentials which turned out to be
useful in treating stationary vacuum and electro-vacuum fields through a system of two
complex second-order equations. In the beginning of the 70s, Geroch [40, 41] showed
that for empty space-times admitting a two-parameter commutating group of motions it
appeared possible (Geroch conjecture) to generate new exact solutions from flat space-
time through the action of a hidden symmetry group associated with the field equations.
At the end of the same decade Belinski and Zakahrov [14, 15] discovered that Einstein’s
vacuum equations for space-times, possessing those symmetries as for the Geroch conjec-
ture, can be solved by means of the inverse scattering methods (ISM)1. These techniques
1It is dutiful to mention also other independent works of Maison [61, 62], Harrison [44, 45] andNeugebauer [69], which appeared at about the same time. A wide review of references and abstractsconcerning the subsequent developments of these area of Einstein’s field equations generating techniquescan be found in http://members.localnet.com/∼atheneum/bib/bib.html .
3
4
handle nonlinear equations as compatibility conditions of an over-determined system of
linear equations and enable to obtain new exact solutions, starting from exact back-
ground solutions, giving algebraic algorithms for their computation.
Despite the essential restrictions involved in the consideration of fields configurations
depending on only two coordinates, this class of two-dimensional fields is still very inter-
esting and rich in its physical content. It includes fields created by all sorts of stationary
axially symmetric sources; various kinds of wave fields (with planar, cylindrical, and
other symmetries); some fields having an explicitly expressed dynamical nature and de-
scribing a uniformly accelerated motion of different sources, with consideration of their
gravitational and electromagnetic radiation; sets of solutions of cosmological type (both
homogeneous and non homogeneous). The study of this class of fields can answer many
questions in gravitation theory often yielding new and unexpected results. Moreover it
can be the basis for the investigation of more general but also considerably more com-
plicated situations.
The first part (chapters 1-5) of this dissertation deals with the application of such ISM
generating techniques used to find solutions for fields of two compact electrically charged
objects in stationary equilibrium. In particular we will take into consideration only the
Alekseev soliton method [1, 3]. This is a direct generalization of Belinski-Zakahrov
technique (which generates solutions for Einstein equations in vacuum) and enables to
generate electro-vacuum exact solutions of the Einstein-Maxwell equations by means of
a dressing (Darboux) transformation performed on background known solutions. This
method enables to construct solutions for the fields of systems of a number of compact
sources aligned on a common axis. Anyway, as well as other solitonic generating tech-
niques, it presents some limitations concerning difficulties in finding easily a physical
interpretation for them (that is to give a representation of the mathematical parameters
in terms of physical one) and the possibility of yielding regular solutions. For example,
considering the equilibrium problem for two massive, non rotating sources, it produces
non elementary-flat solutions, i.e. with conic-like singularities on the axis. These irregu-
larities are a consequence of some limitations which are specific of the solitonic generating
tools. In Newtonian physics this two-body system can stay in equilibrium if (in geomet-
rical units) the product of the masses is equal to the product of the charges. The limit
to this classical condition shows that, in the relativistic regime, static equilibrium should
subsist only if this two-body system is composed either by a under-extreme (mass greater
than the charge) and a super-extreme (mass smaller than the charge) Reissner-Nordstom
like sources, or by two extreme (mass equal to the charge) sources. The experiences on
solitonic generating techniques tell us that they can generate only solutions relevant to
5
super-extreme sources; hence up to the present they are not able to yield those equilib-
rium conditions to remove the conic singularities from the axis.
In 2007, Alekseev and Belinski [5, 4] found the general solution (hereafter denoted with
AB) for the static balancing problem of two charged non rotating sources, giving the
general equilibrium conditions to remove each axial irregularity. They reached this re-
sult not by means of the Alekseev dressing solitonic technique, but using the Alekseev
“Monodromy Transform Approach”. Nevertheless, it should be possible to reach the
same result through the dressing technique; how to do this is still an open question, but
if this is the case, it could yield a more direct and simple way to obtain such solutions.
In regard to this, it is worth to recall that the problem of finding a similarly general
result concerning the regular solution for two charged and rotating sources, as well as
the relative explicit general equilibrium conditions, is still open2.
The original work, related to the Alekseev solitonic technique, presented in the first
part of this thesis (chapters 4-5), deals with the construction of the procedure to gener-
ate perturbations of the exact solitonic solutions respect to the Newtonian gravitational
constant. The five chapters relevant to this part are organized as follows.
In chapter 1, we recall the Belinski-Zakahrov dressing generating technique for vacuum
solutions.
In chapter 2, we describe the Alekseev technique which generalizes the previous one to
Einstein-Maxwell equations for electro-vacuum solutions.
The third chapter is devoted to show as the Alekseev dressing procedure works taking, as
an example, the one-soliton stationary solution. The results, presented in this chapter,
are not new; its utility is to give the reader a description of all basic aspects concerning
this generating technique such as, in particular, those concerning the difficulties incurred
by working on multi-soliton solutions.
In the chapter 4, the construction of the procedure to generate perturbative solutions
in respect to the Newtonian gravitational constant γ is presented. Its contextual task
is to investigate some aspects of the two-body problem mentioned above, as regards the
Alekseev exact generating technique in the framework of a weak field approximation.
Nevertheless, it posses independent interests; in fact this tool permits to generate weak
solutions for electromagnetic solitons on flat or fixed curved space-time, since it enables
to handle separately electromagnetic and gravitational fields.
The chapter 5, after the presentation of all the formulas by means of which the perturba-
tive terms can be calculated, describes a first applications of this perturbative approach.
We give: first the approximate solutions up to the first order in γ of one-soliton solution
2A review relevant to the particular results obtained up to 2003 can be found on [83].
6
with the aim to check the procedure; then the zero order approximate terms of two-
soliton solution, finding a first reparameterization between the mathematical parameters
and the physical one. The work relevant to the successive order of approximation of the
two-soliton solution, is not given here, since it is still in progress.
The second part of this thesis, constituted by chapters 6 and 7, regards respectively
two works, [71] and [74], relevant to a first analysis of the AB exact solution3.
In chapter 6, after a brief summary of the AB solution describing two Reissner-Nordstrom
sources in reciprocal equilibrium, we study in some detail the coordinate systems used
and the main features of the gravitational and electric fields, as well as the different con-
figurations permitted by it. Many of these configurations are forbidden by Newtonian
physics; we find that the equilibrium is possible for two opposite charged sources too.
We graph the plots of the electric force lines in three qualitatively different situations:
equal-signed charges, opposite charges and the case of a naked singularity near a neutral
black hole.
The chapter 7 is devoted to the analysis of the stability of the AB solution with respect
to displacements from the equilibrium distance between the two sources. To do this
we define the force of the conic singularity, and assume that the force between the two
bodies is equal and opposite. Then we analyze the stability of the equilibria in the three
qualitatively different situations: equal-signed charges, opposite-signed charges and the
case of a naked singularity near a neutral black hole. We show that most of such con-
figurations result to be stable and in agreement with the analog stability classification
given by Bonnor [21] for test charged particles in a Reissner-Nordstrom field.
At the end of the thesis, as a separate part, the preprint version of two additional
works ( [13], [75])4 are attached. They deal with some spherically symmetric solutions
of Einstein-Maxwell equations relevant to shell like configurations of matter.
The first one, [13], is about a membrane model (i.e. a thin shell with tension) for a
naked singularity. The link with the AB solution is rather natural. As specified above,
the equilibrium is allowed only by the presence of a naked singularity and of its repulsive
region near the center. Therefore, in order to give a more sensible physical meaning
to this solution, we construct a model, at least a toy model, for a Reissner-Nordstrom
naked source presenting a repulsive region around it. Such model consists in a charged
membrane with a dark-matter-like equation of state, ε = −p . Obviously this is a very
simple model and it does not resolve all the problems concerning the naked singularity
3All these works were produced in collaboration with Marco Pizzi.4Both these two works were produced, during the last two years in collaboration with Marco Pizzi;
the first one reported, together with Vladimir Belinski too.
7
(i.e. how to arrive to such configuration). Anyway, it gives al lest a hint that such a
configuration is physically possible (and we know that something like this should exist
because the electron has the parameters corresponding to a naked singularity). The
static configuration we present is stable with respect to the radial displacements. It is
important to recall that such kind of naked-singularity model already exist in literature;
however, we re-derive such results using a method more habitual for physicist, that is by
means the direct integration of the field equations with appropriate δ-shaped sources.
The last work, [75], is about the motion of two spherical intersecting thin charged shells
with positive tangential pressure in a Reissner-Nordstrom field. It is a direct general-
ization to the case of charged shells of the work of Barkov et al [7] dealing with neutral
shells. The motion of each shell is independent from the other until their intersect;
indeed, until the intersection, the outer shell feels the inside shell as a simple Reissner-
Nordstrom source. We obtain the exchanging energy formula between the two shells
due to their intersection. Finally, we describe the ejection mechanism, for which one of
the two bounded shell can acquire enough energy to be driven out to infinity. We show
that, because the energy transfer is larger due to the Coulomb interaction, the ejection
mechanism is more efficient in the charged case than in the neutral one if the charges
have opposite sign.
The original contributions presented in this thesis, which, as mentioned above, has been
subject matter of publications, were produced in collaboration with other authors. We
worked in parallel, comparing the results and discussing what had to be included or
excluded, how to explain it. Therefore, it is not possible to define clearly each different
personal contribution.
Chapter 1
Soliton solutions of Einsteinequations: Belinski-Zakharovtechnique
In this chapter we resume the scheme of the soliton technique of Belinski-Zakharov [14]
[15] to generate exact solutions of the Einstein Equations in vacuum for metric tensors
depending only on two coordinates. A more detailed and comprehensive description,
together with references to other different approaches, can be found in [9].
1.1 The integrable ansatz in general relativity
The Belinski-Zakharov soliton technique generates exact solutions for the Einstein Equa-
tions in vacuum,
Rij = 0 , (1.1)
where Rij is the Ricci tensor, for metric tensors g4ij depending on two variables only1.
These correspond to space-times that admit two Killing vectors fields, i.e an Abelian
two-parameter group of isometries. Moreover it is necessary to impose 2 the existence of
2-surfaces orthogonal to the group orbits. The metric tensors belonging to this class can
assume the following block diagonal form:
ds2 = gab(xρ) dxadxb + f(xρ) ηµν dxµdxν , (1.2)
where Latin letters from the first part of the alphabet take only 0, 1 values, while the
Greek ones 2, 3 ; f > 0 and ηµν = diag (−e, 1) , with e = ±1 . If e = 1 (1.2) describes
1Here g4 denotes the four-dimensional metric tensor; we denote the four-dimensional index withi, j, ... = 0, 1, 2, 3.
2It is unknown at present whether the Inverse Scattering Methods can be applied without thisfurther physical restriction.
8
9
non-stationary solutions and xi = (ρ, ϕ, t, z) ; while if e = −1 it refers to stationary
solutions and, in this case, xi = (t, ϕ, ρ, z) . More explicitly, this geometrical reduction
reads;
ds2 = g00(t, z)dρ2 + 2g01(t, z)dρ dϕ + g11(t, z)dϕ2 + f(t, z)(−dt2 + dz2
), if e = 1 ,
ds2 = g00(ρ, ϕ)dt2 + 2g01(ρ, ϕ)dt dz + g11(ρ, ϕ)dz2 + f(ρ, ϕ)(dρ2 + dϕ2
), if e = −1 ,
where signature(g) = (e, 1) . In what follows we shall always denote by g the two-
dimensional real and symmetric matrix with components gab . For the determinant of
this matrix it is convenient to introduce the notation:
det g = e α2 (1.3)
and we shall always consider that α is nonnegative. Hereafter, for the sake of simplicity,
we will consider the non-stationary case and hence we will assume e = 1 . It is convenient,
to write the field equations, to introduce a pair of null-coordinates defined by: ζ =
(√
e x2 + x3)/2 , η = −(√
e x2 − x3)/2 3. The system (1.1) for the metric tensor (1.2),
implies that Raµ ≡ 0 . The remaining equations can be decomposed into two sets. The
first one follows from the equations Rab ≡ 0 and can be written in the form of a single
matrix equation: (α g,ζg
−1)
,η+
(α g,ηg
−1)
,ζ= 0 . (1.4)
The second set follows from the equations R22 + R33 = 0 and R23 = 0 , and gives the
metric coefficient f(xρ) in terms of the matrix g , as obtained by the (1.4):
(ln f),ζ(ln α),ζ = (ln α),ζζ +1
4 α2Tr A2 , (1.5)
(ln f),η(ln α),η = (ln α),ηη +1
4 α2Tr B2 , (1.6)
where the matrices A and B are defined by
A = −α g,ζ g−1 , B = α g,η g−1 . (1.7)
The integrability condition for (1.5) and (1.6), with respect to f , is automatically sat-
isfied if g satisfies (1.4). The equation R22 −R33 = 0 can be written in the form:
(ln f),ζη =1
4 α2Tr AB − (ln α),ζη . (1.8)
This last equation does not add anything new, since it is a consequence of the system
(1.3)-(1.7) when α is not a constant. While the special case in which it is constant
3Note that in the case e = −1 , these coordinates are complex variables, namely ζ = (z + iρ)/2 andη = ζ = (z − iρ)/2 .
10
corresponds to flat Minkowsky spacetime.
The basic point on which the generating technique is constructed lies on the fact that the
principal set of the field equations, i.e. (1.4), is very similar to the field equations for some
integrable relativistic invariant model called principal chiral field [89], to which it reduces
if α is constant. The general idea of the method is based on the study of the analytic
structure of the eigenfunctions of two operators (as functions of a complex parameter λ ),
which are associated to the system (1.3)-(1.4). In particular, for the soliton solutions of
(1.3)-(1.4), the structure of the poles, λn , of the corresponding functions in the λ-plane
plays a fundamental role. As shown in the next section, if α is not constant, (1.3)-(1.4)
require the introduction of generalized differential operators entering into the Lax-Pair
(see below). These operators depend on the function α(ζ, η) and contain derivatives
with respect to the spectral parameter λ . For soliton solutions this leads, instead of
stationary poles as for the principal chiral field, to pole trajectories, since now, the poles
will depend on the coordinates: λn = λn(ζ, η) .
1.2 The integration scheme
From the trace of (1.4), it is immediate that
αζη = 0 , (1.9)
namely, α satisfies the two-dimensional wave equation which as the general solution
α = a(ζ) + b(η) , (1.10)
where a(ζ) and b(η) are arbitrary functions. For later use we define a second indepen-
dent solution of (1.9)4
β = a(ζ)− b(η) . (1.11)
The equation (1.4) is equivalent to a system consisting of the definitions (1.7) and two
first order matrix equations for the matrices A and B .
A,η −B,ζ = 0 (1.12)
A,η + B,ζ + α−1 [A,B]− α,η α−1A− α,ζ α−1B = 0 (1.13)
Where this last equation, in which the square brackets denote the commutator, repre-
sents an integrability condition of (1.7) with respect to g . The main step now consists
4It is to recall that the metric (1.2) admits arbitrary coordinate transformations x′ 2 = f1(ζ)+f2(η) ,x′ 3 = f1(ζ)− f2(η) , which leave unchanged its conformally flat block. By an appropriate choice of thefunctions f1 and f2 , it is possible to bring the functions a(ζ) and b(η) , in (1.10)-(1.11), into aprescribed form. When this freedom is used to write (α, β) as spacetime coordinates, we say that themetric (1.2) has the canonical form and (α, β) are called canonical coordinates.
11
in representing (1.12) and (1.13) in the form of compatibility conditions of a more gen-
eral overdetermined system of matrix equations related to an eigenvalue-eigenfunction
problem for some linear differential operators. Such a system will depend on a complex
spectral parameter λ , and the solutions of the original equations for the matrices g , A
and B will be determined by the possible types of analytic structure of the eigenfunctions
in the λ-plane. Therefore, introducing the differential operators5
D1 = ∂ζ − 2 α,ζ λ
λ− α∂λ , D2 = ∂η +
2 α,η λ
λ + α∂λ , (1.14)
we consider the linear system6
D1 ψ =A
λ− αψ , D2 ψ =
B
λ + αψ , (1.15)
for the complex matrix function ψ(λ, ζ, η) , which in this context is usually called the
generating matrix. It is easy to see that if α is solution of the wave equation (1.9), then:
[D1, D2] = 0 . (1.16)
By means of this property, it results that the compatibility conditions for (1.15) reproduce
exactly the (1.12)-(1.13) equations for the matrices A and B . Moreover, putting λ = 0
in (1.15), we obtain
∂ζψ = − 1
αAψ = g ζ g−1 ψ , ∂ηψ =
1
αB ψ = g η g−1 ψ . (1.17)
Hence we can choose ψ such that
ψ(0, ζ, η) ≡ ψ(0) = g(ζ, η) . (1.18)
The method we apply assumes the a priori knowledge of a particular solution, g0(ζ, η) ,
of the system (1.3)-(1.4). From it, by means of (1.7) one can calculate the correspond-
ing A0(ζ, η) and B0(ζ, η) , and integrating (1.15), obtain the corresponding generating
matrix ψ0(λ, ζ, η) 7.
We then introduce the dressing matrix χ(λ, ζ, η) according to the definition:
ψ = χ ψ0 , (1.19)
where ψ , is the solution of (1.15) corresponding to the new solution g we want to con-
struct. This way, through the restriction (1.18), we obtain a new solution g = χ(0, ζ, η)g0
5The symbol ∂ with a subscript denotes partial differentiation with respect to the correspondingvariable and λ is a complex parameter independent of the coordinates ζ and η .
6The matrices A and B together with α are real and independent on the spectral parameter λ .7It is worth noting that this is the only step of this generating procedure where it is necessary to
perform the integration of a differential system, which, however, is linear.
12
for the equations (1.4). Substituting (1.19) in (1.15) (and assuming that this transfor-
mation g0 → g does not change the determinant, detg = detg0 , see (1.3)), we have the
following differential constraints for the matrix χ :
D1 χ =1
λ− α(Aχ− χA0) , D2 χ =
1
λ + α(B χ− χB0) . (1.20)
Anyway, we recall that the matrix g(ζ, η) must be real and symmetric. Hence, it is
necessary to perform a reduction on the dressing and generating matrices. This can be
done through the imposition of two restrictions. The first consists of requiring the reality
of these matrices on the real axis of the λ-plane, that is:
χ(λ) = χ(λ) , ψ(λ) = ψ(λ) , (1.21)
where the overset bar denotes complex conjugation 8. The second condition is based on
the invariance of the solutions of the system (1.20) under the substitution λ → α2/λ .
Introducing the matrix
χ′(λ) = g χ−1(α2/λ) g−10 , (1.22)
where the tilde denotes transposition of the matrix, it follows that χ′ is solution of (1.20)
if g is symmetric. Then we can choose χ′(λ) = χ(λ) , and hence
g = χ(λ) g0 χ(α2/λ) . (1.23)
When λ →∞ , g = χ(∞) g0 χ(0) , which, because of (1.18) and (1.19), gives the asymp-
totic behavior
χ(∞) = I . (1.24)
Thus, the problem now consists of solving (1.20) and determining the dressing matrix χ
in accordance with these restrictions. In this way we have for the dressed solution:
g = χ(0) g0 . (1.25)
It is important to keep in mind that the dressing procedure must preserve the determinant
of g , that is det g = α2 = det g0 . Therefore it is necessary to impose on matrix χ , as
follows from (1.25), the further restriction: det χ(0) = 1 . However, it is more convenient
to renormalize the final results in order to obtain the correct functions. These (correct)
functions will be called physical functions and denoted with the small up set suffix (ph) .
8For the sake of brevity, when it is not necessary, we do not indicate the arguments ζ and η ofsome functions.
13
To legitimate this procedure it is sufficient to see that, if we obtain a solution of (1.4)
with det g 6= α2 , the trace of this equation implies that det g satisfies the equation
(α(ln det g),ζ),η + (α(ln det g),η),ζ = 0 . (1.26)
It is easy to see that the matrix
g(ph) = α (det g)−1/2g , (1.27)
satisfies both (1.4) and the condition det g(ph) = α2 . Correspondingly
A(ph) = A− α ln[ α (det g)−1/2 ] ,ζ I , (1.28)
B(ph) = B + α ln[ α (det g)−1/2 ] ,η I , (1.29)
where A and B are defined in terms of g according to (1.7) and A(ph) and B(ph) are
defined by the same formulas but in terms of the matrix g(ph) .
1.3 Construction of n-soliton solution
According to the Inverse Scattering Theory, the general solution for χ is given by the sum
of a solitonic and a nonsolitonic parts. Here only the solitonic part will be considered.
The existence of particular solutions of this kind is due to the presence in the λ-plane
of points at which the determinant of χ has simple poles. Thus it is representable as
a rational function of the parameter λ with a finite number of simple poles, in such a
way that it goes to the unit matrix when λ → ∞ , as required by (1.24). The reality
condition (1.21) for g implies that the poles can lie either on the real axis of the complex
λ-plane or come in complex conjugate pairs. From the symmetry condition (1.23), which
implies that det χ(λ) det χ(α2/λ) = 1 , it follows that for each pole λ = µ, there is a
corresponding point λ = α2/µ where det χ = 0 . The inverse matrix χ−1 has the same
properties, as can be seen from (1.21) and (1.23). Therefore the matrix χ has the form
χ = I +n∑
r=1
Rk
λ− µk
, (1.30)
where the matrices Rk and the functions µk do not depend on λ . With respect to
the expression (1.30), the reality conditions for χ , implies that to each real pole µk, it
correspond a real matrix Rk , while to each complex µk, there must be another function
µk+1 = µk , to which it corresponds Rk+1 = Rk . Thus from (1.30) and (1.25), the
solution of the equation (1.4) assumes the form
g(ζ, η) =
(I −
n∑
k=1
µ−1k Rk
)g0 . (1.31)
14
To determine the functions µk and the matrices Rk , it is necessary to substitute (1.30)
into (1.20) and impose that these equations be satisfied at the poles λ = µk(ζ, η) . Since
the right hand sides of (1.20) have in λ = µk first order poles, whereas the left hand sides
have second order ones, the requirement that the coefficients of the powers (λ − µk)−2
vanish yields the following equations for the pole trajectories µk(ζ, η) :
µk,ζ =2 α,ζ µk
α− µk
, µk,η =2 α,η µk
α + µk
. (1.32)
These equations have the following invariance: if µk is a solution of (1.32), then α2/µk
is also a solution. The solutions of (1.32) are the roots of the quadratic equation
µ2k + 2 (β − wk) µk + α2 = 0 , (1.33)
where β is the other independent solution (1.11) of the wave equation (1.9) for α , and
wk are arbitrary complex constants. For each given wk , (1.33) yields two roots, µk and
α2/µk . Hence these solutions can be written in the form
µink = (wk − β)
1− [1− α2(β − wk)
−2]1/2
(1.34)
µoutk = (wk − β)
1 + [1− α2(β − wk)
−2]1/2
(1.35)
since, in the λ-plane, respectively, they are never located outside or inside the circle of
radius λ = |α| .Rewriting (1.20) in the form
A
λ− α= (D1 χ) χ−1 + χ
A0
λ− αχ−1 , (1.36)
B
λ + α= (D2 χ) χ−1 + χ
B0
λ + αχ−1 , (1.37)
since the left hand sides of these equations are regular at the polee of χ , λ = µk , the
residues of these poles on the right hand sides must vanish. This leads to the equations
for the matrices Rk :
Rk,ζ χ−1(µk) + RkA0
µk − αχ−1(µk) = 0 , (1.38)
Rk,η χ−1(µk) + RkB0
µk + αχ−1(µk) = 0 . (1.39)
To obtain these equations use has been made of the relation
Rk χ−1(µk) = 0 , (1.40)
15
which follows from the identity χχ−1 = I , at the poles λ = µk . The relation (1.40)
implies that Rk and χ−1(µk) are degenerate matrices, therefore their matrix elements
can be written in the form
(Rk)ab = n(k)a m
(k)b , [χ−1(µk)]ab = q(k)
a p(k)b . (1.41)
According with these representations, we have that9
m(k)a q(k)
a = 0 . (1.42)
As a consequence of (1.38) and (1.39), we obtain
(n(k)a m(k)
c ),ζg(k)c p
(k)b +
1
µk − αn(k)
a m(k)d (A0)dc q(k)
c q(k)b = 0 , (1.43)
which, combined with (1.42), yields the differential system for the vector functions m(k)a :
[m
(k)a,ζ + m
(k)b
(A0)ba
µk − α
]q(k)a = 0 ,
[m(k)
a,η + m(k)b
(B0)ba
µk + α
]q(k)a = 0 . (1.44)
Anyway, these equations present four ( m(k)a and q
(k)a , for a = 0, 1 ) unknown quantities,
hence we need other relations. To find these, let us define the matrices
Mk = (ψ−10 )λ=µk
= ψ−10 (µk, ζ, η) . (1.45)
Being ψ0 a particular solution of (1.15), from these equations we obtain:
Mk,ζ + MkA0
µk − α= 0 , Mk,η + Mk
B0
µk + α= 0 . (1.46)
By comparing (1.46) with (1.44), it is found that the matrices Mk are proportional to
the vectors m(k)a ; that is, a solution of (1.44) will be
m(k)a = m
(k)0b (Mk)ba = m
(k)0b [ ψ−1
0 (µk, ζ, η) ]ba , (1.47)
where the m(k)0b are arbitrary complex constant vectors. The expression (1.47) for m
(k)a
could also present an arbitrary factor depending on k , and on the coordinates; since it
disappears in the final expressions of the residues Rk , it can be set equal to 1 without
loss of generality.
To complete the determination of Rk we need to find the vectors n(k)a , as prescribed by
the representation (1.41) . Substituting (1.30) into (1.23), and considering the relation
9Here and in the following, summation will be understood to be over repeated vector and tensorindices a , b , c , d , which, we recall, takes the values 0 and 1 .
16
obtained in such a way at the points λ = µk , since g does not depend on λ , we have
the following system of n algebraic matrix equations for the matrices Rk :
Rk g0
[I +
n∑
l=1
(α2 − µk µl)−1µk Rl
]= 0 . (1.48)
Substituting in this the representation (1.41) of Rk , we obtain a system of linear algebraic
equations for the vectors n(k)a :
n∑
l=1
Γkl n(l)a = µ−1
k m(k)c (g0)ca , (1.49)
where the n× n matrix Γkl is symmetric and its elements are
Γkl = −m(k)c m
(l)b (g0)cb(α
2 − µkµl)−1 . (1.50)
Introducing the symmetric matrix Πkl inverse to Γkl , and the vectors
L(l)a = m(l)
c (g0)ca , (1.51)
we obtain
n(k)a =
n∑
l=1
µ−1l ΠklL
(l)a . (1.52)
Now, using (1.41), (1.52) and (1.51) we get, from (1.31), the metric componets gab :
gab = (g0)ab −n∑
k,l=1
µ−1k µ−1
l Πkl L(k)a L
(l)b . (1.53)
This expression is obviously symmetric. To ensure that the matrix g is also real, as
deducible from the second of the conditions (1.21) and from (1.47), it is necessary to
choose the arbitrary constants m(k)0b so that the vectors m
(k)a corresponding to real poles
λ = µk are real. It results that all the complex poles have to appear only as conjugate
pairs: for each complex pole λ = µ , its complex conjugate λ = µ must also appear. We
can therefore denote, for each complex pole λ = µk , µk+1 = µk . In this case, the choice
of the constants m(k)0b , is constrained by the request that the vectors m
(k)a and m
(k+1)a
corresponding to such pairs of complex conjugate poles are complex conjugate to each
other.
1.3.1 The physical metric components gab
The matrix g of (1.53) does not satisfy (1.3), therefore, as explained at the end of para-
graph 1.2, it is necessary to perform a renormalization to obtain the physical solution.
17
Here, we will not describe all the steps to deduce this expression. We will limit ourselves
to describe the scheme which bring to the final formulas, remanding to [9] for the details.
The form (1.53) is not convenient for this calculation. It is better to dispose of a formula
for χ given by a product among factors, each one being specific of each single soliton.
This is feasible since the n-soliton solution can be obtained either as a simultaneous
dressing or as an iteration of n steps introducing one soliton at each step, in succession:
g(n) = χ(n) g0 =(χ(1)
n χ(1)n−1 ... χ
(1)1
)g0 = χ(1)
n
(χ
(1)n−1
(...
(χ
(1)1 g0
))). (1.54)
Therefore, it is better to start with one-soliton dressing. In this case the matrix χ1 can
be expressed in the following form
χ1 = I + µ−11 (λ− µ1)
−1(µ21 − α2) P1 , (1.55)
where the matrix P1 , given by
(P1)ab =m
(1)c (g0)ca m
(1)b
m(1)d (g0)df m
(1)f
, (1.56)
satisfies the properties
P 21 = P1 , Tr P1 = 1 , det P1 = 0 . (1.57)
Denoting with g1 the one-soliton solution, we have from (1.25) and (1.55)
g1 = χ1(0) g0 =[I − µ−2
1 (µ21 − α2) P1
]g0 , (1.58)
the determinant of which, using the general relation holding for any 2 × 2 matrix F ,
det(I + F ) = 1 + Tr F + det F , is:
det g1 = µ−21 α2 det g0 . (1.59)
This result is obviously independent by the soliton indices. That is, it can automatically
be generalized as
det gk+1 = µ−2k+1 α2 det gk . (1.60)
Therefore, the determinant of the final n-soliton matrix g will be
det g = α2n
(n∏
k=1
µ−2k
)det g0 = α2n+2
n∏
k=1
µ−2k . (1.61)
We can now write explicitly the equation (1.27)10:
g(ph) = α
(α2n+2
n∏
k=1
µ−2k
)−1/2
g = α−n
(n∏
k=1
µk
)g . (1.62)
10It is worth noting that both signs are allowed in front of the matrix g(ph) due the invariance ofthe Einstein Equations (1.3)-(1.7) with respect to the reflection g(ph) → −g(ph) . This sign should bechosen separately for each case in order to ensure the correct signature of the metric.
18
1.3.2 The physical metric components f
To complete the construction of the n-soliton solutions for the metric (1.2), we also need
to calculate the metric coefficient f from (1.5)-(1.6) using the matrix g already found.
It is possible to see that the coefficient f , in the general n-soliton case, can also be
calculated explicitly by algebraic operations only like the metric components gab . Here
we trace the general outline to get the final result. The first step consists in substituting
into (1.5)-(1.6) the nonphysical solution g given by (1.53), to obtain the nonphysical
factor f . It is convenient to proceed one soliton a time. Thus, the one-soliton case yields
f1 = C1 f0
(µ2
1 − α2)−1
α µ21 Γ11 , (1.63)
where C1 is an arbitrary constant, f0 is the particular background solution for f , which
corresponds to the solution g0 and Γ11 is the single component of the matrix (1.50) . It
is, in this case for which k = 1 and l = 1 , a 1× 1 matrix:
Γ11 =(µ2
1 − α2)−1
m(1)a m
(1)b (g0)ab . (1.64)
The repetition of this operation n-times yields:
f = Cn f0 αn
(n∏
k=1
µ2k
)[n∏
k=1
(µ2
k − α2)]−1
det Γkl , (1.65)
where Cn is an arbitrary constant. Substituting into (1.5)-(1.6) g(ph) , A(ph) and B(ph)
(calculated by means of (1.28) and (1.28)), we find that the physical coefficient f (ph) is
given by the formula
f (ph) = f α1/2F , (1.66)
where the function F is11:
F = CF α−(n2+2n+1)/2
(n∏
k=1
µk
)n−1 [n∏
k=1
(µ2
k − α2)] [
n∏
k>l=1
(µk − µl)−2
]. (1.67)
CF is an arbitrary constant, and∏n
k>l=1 (µk − µl)−2 is equal to 1 for n = 1. From
(1.65)-(1.66) we get the final expression for the physical value of the coefficient f :
f (ph) = Cf f0 α−n2/2
(n∏
k=1
µk
)n+1 [n∏
k>l=1
(µk − µl)−2
]det Γkl , (1.68)
where Cf is an arbitrary constant which should be taken with the appropriate sign in
order to ensure the correct sign for f (ph) .
11We remand to the book [9] for further details about the deduction of this function.
19
1.4 Some remarks
It is worth noting that the number of solitons is constrained in the case of stationary
solutions. In fact if e = −1 , using canonical coordinates12, (1.3) becomes:
det g = −α2 = −ρ2 , (1.69)
consequently, the expression (1.61) gives:
det g = (−1)nρ2n
(n∏
k=1
µ−2k
)det g0 . (1.70)
If we take the particular solution g0, which by definition satisfies det g0 = −ρ2 , it follows
from (1.70) that the number of solitons , n , must always be even, since an odd number
would change the sign of det g and lead to an unphysical metric signature. Therefore, in
contrast to the nonstationary case, on a physical background all stationary axisymmetric
solitons (even those which correspond to real poles λ = µk ) can only appear in pairs
forming bound two-soliton states. Nevertheless, we can obtain physical solutions with
an odd number of solitons, but for this it is necessary to take a background solution with
a nonphysical signature, det g0 = ρ2 . The first examples of solutions of this kind were
obtained and investigated in [86].
To end this chapter, it is important to recall that, from the physical point of view, the
metrics of the kind (1.2) have many applications in gravitational theory. A lot of well
known solutions belong to such a class as the Robinson-Bondi plane waves, the Einstein-
Rosen cylindrical wave solutions, the homogeneous cosmological models of Bianchi types
I-VII, the Schwarzschild and Kerr solutions, Weyl’s axisymmetric solutions, etc.. The
reader can find a wide review of the solutions belonging to this class in [83]. In particular,
for the solutions obtained by means of the Belinski-Zakharov technique, we indicate the
table on page 546 of this book.
12See footnote number 1.2 at page 10.
Chapter 2
Solitonic solutions ofEinstein-Maxwell equations:Alekseev technique
As we have seen in the previous chapter, the Belinski-Zakharov technique produces soli-
ton solutions for the equations (1.1), that is the Einstein equations in vacuum. The
generalization represented by the inclusion of matter, i.e. the appearance of a nonzero
right hand side in the Einstein equations, generally destroys the applicability of the
Inverse Spectral methods. This is because the stress-energy tensor produces a non-
vanishing right hand side in the basic equation (1.4) that prevents the application of
the technique. However, there are some exceptions. This chapter describes the case of
the coupled Einstein-Maxwell field equations. Some examples of the application of the
Inverse Spectral methods to other non-vacuum cases can be found in the book [9].
The main new step for the solution of the Einstein-Maxwell equations was made by G.
A. Alekseev in 1980 [1]; the most detailed presentation of his approach was given in
his 1988 paper [3]. In this chapter we will follow very near the exposition given in [9],
adopting the notation used therein.
It is worth to be mentioned that the integrability ansatz of the Einstein-Maxwell equa-
tions was also given by other authors using different approaches; we remand to the books
[9, 83] for the relevant references.
2.1 The Einstein-Maxwell field equations
The problem we are dealing with, is represented by the coupled differential equation
of Einstein and Maxwell; in the general case (with the usual notation and with an
20
21
appropriate choice of units), they read as
Rij − 1
2g4
ij R = Tij
(√−g4 F ik
),k
=√−g4 j i
,
where g4ij is the unknown four-dimensional metric tensor1, and g4 is its determinant; Rij
is the Ricci tensor, R is the Ricci scalar and Tij is the energy-momentum tensor. For
the electromagnetic fields it is:
Tij =1
2
(−FikF
kj +
1
4FlmF lm g4
ij
),
where, being Ai the covariant components of the electromagnetic vector potential,
Fij = Aj,i − Ai,j .
Anyway, since the solitonic technique works with metric tensors of the form (1.2),
ds2 = gab(xρ) dxadxb + f(xρ) ηµν dxµdxν , (2.1)
it is convenient to adapt the Einstein-Maxwell equations according with this symmetry
reduction. The notation will be the same as that adopted in the previous chapter.
Hereafter, for convenience of the reader, we will sometimes rewrite some of the formulas
already used. The two-dimensional matrix ηµν is2
ηµν =
(−e 0
0 1,
)(2.2)
where e = 1 and e = −1 for the non-stationary and stationary solutions, respectively.
The determinant of the two-dimensional matrix gab is
det g = e α2. (2.3)
In order to make the integrable ansatz compatible with the metric (2.1) one should
assume the following structure for the electromagnetic potentials:
Aµ = 0 , Aa = Aa(xρ) . (2.4)
1We put the index 4 only to distinguish the total metric tensor from the 2×2 block gab , a, b = 0, 1 ,as made in the previous chapter.
2In this chapter we will often use a matrix notation. Thus, for definiteness in any matrix(Mik, M ik, M i
k or M ki ) the first index, independent of its up or down position, will always enumerate
the rows, and the second index will enumerate the columns.
22
Then, the only nonvanishing components for the covariant and controvariant electromag-
netic tensor field are
Fµa = Aa,µ , F µa =1
fηµνgacAc,ν . (2.5)
As usual, gab are the components of the inverse matrix of gab and thus defined by
gacgcb = δab , being δa
b the Kronecker symbol.
The Einstein-Maxwell equations can be written (with an appropriate choice of units) in
the form
Rba = 2
(FλaF
λb − 1
2δbaFλcF
λc
), (2.6)
Rµν = 2
(FνcF
µc − 1
2δµν FλcF
λc
), (2.7)
(fαF µa),µ = 0 . (2.8)
Since the two-dimensional trace in the Greek indices on the right hand side of 2.7 vanishes
identically, these equations can be written in the following equivalent form:
Rµµ = 0 , Rµ
ν −1
2δµν Rλ
λ = 2
(FνcF
µc − 1
2δµν FλcF
λc
). (2.9)
Direct calculation shows that the first of these equations is
ηµν(ln f),µν + ηµν(ln α),µν +1
4gabgcdηµνgbc,µ gda,ν = 0 , (2.10)
and that the second does not contain the second derivatives of the metric coefficient f :
ηµρ
[1
2(ln f),µ(ln α),ρ +
1
2(ln f),ρ(ln α),µ
]− 1
2δνµ ηρσ(ln f),ρ(ln α),σ
− ηνρ(ln α),µρ +1
2δνµ ηρσ(ln α),ρσ
− 1
4gabgcd
[ηνρgbc,µgda,ρ − 1
2δνµ ηρσgbc,ρgda,σ
]
= gcdAd,ρ
[2 ηνρAc,µ − δν
µ ηρσAc,σ
].
(2.11)
The trace of this equation vanishes identically. Consequently it gives only two indepen-
dent relations from which the metric coefficient f can be found by quadratures if the
matrix gab and the potential Aa are known. As in the vacuum case, (2.10) will then
be satisfied due to the Bianchi identities, and we can forget about this equation from
now on. Also, as in the vacuum case, the integration of the coefficient f from (2.11)
does not present a major difficulty and will be carried out at the end of the procedure.
Thus we turn our attention to the problem of the matrix gab and the potentials Aa
from the system (2.6) and (2.8). It is easy to see that this system does not contain the
23
coefficient f and that it forms a closed self-consistent system of equation for gab and
Aa . Calculating Rab and using the definitions (2.5) we can write these equation in the
form:
ηµν 1
α(α gbcgac,µ),ν = −4 ηµνgbcAa,µAc,ν + 2 δb
a ηµνgcdAc,µAd,ν , (2.12)
ηµν(α gacAc,µ),ν = 0 . (2.13)
It is noteworthy that the trace of right hand side of (2.12) vanishes identically and that
the function α(xµ) , in accordance with its definition (2.3), should satisfy the vacuum
’wave’ equation,
ηµνα,µν = 0 . (2.14)
Let us introduce the two-dimensional antisymmetric matrices,
εµν = εµν =
(0 1
−1 0
), (2.15)
and the same for Latin indexes,
εab = εab =
(0 1
−1 0
). (2.16)
Now we are in the position to start the integration scheme. Following Alekseev’s sugges-
tion of exploiting the duality property of the electromagnetic field, we introduce some
auxiliary potentials Ba , which will only play an intermediary role and will not be present
in the final results. In terms of the original potentials Aa , these are defined by,
Ba,µ = − 1
αηµν ενλ gab εbcAc,λ . (2.17)
It is easy to verify that the integrability condition for this equation, εµνBa,µν = 0 ,
coincides with the Maxwell equation (2.13). Relation (2.17) can also be written in the
inverse form:
Aa,µ =1
αηµν ενλ gab εbcBc,λ . (2.18)
Let us now combine Aa and Ba into a single complex potential Φa , defined by
Φa = Aa + i Ba . (2.19)
Then (2.17)-(2.18) are, respectively, the imaginary ad real part of the equation for Φa :
Φa,µ = − i
αηµνε
νλ gabεbc Φc,λ , (2.20)
24
from which the Maxwell equations for Φa trivially follow:
ηµν(α gac Φc,µ),ν = 0 . (2.21)
By direct calculation one can show that Einstein equations (2.12) can be written as
ηµν 1
α
(α gbc gac,µ
),ν
= −2 gbcηµνΦa,µΦc,ν . (2.22)
The imaginary part of the right hand side of this equation vanishes because the left hand
side is real. This is indeed a consequence of (2.17). Also due to this relation and the
identity
εadεbc = δab δ
dc − δa
c δdb , (2.23)
the real part of the right hand side of (2.22) coincides exactly with the right hand side
of (2.12). It is thus clear that any solution of (2.19)-(2.22), gab and Aa = Re Φa , is also
a solution of the Einstein-Maxwell equations (2.12)-(2.13).
2.2 The spectral problem for Einstein-Maxwell fields
Now we want to represent the Einstein-Maxwell equations (2.19)-(2.21) as self-consistency
condition of a linear spectral problem. The five-dimensional generalization3 of the
Belinski-Zakahrov technique, suggests that it is reasonable to look for the solution in
the framework of the same spectral problem (1.15), but for three-dimensional matrices
A , B and ψ . This three-dimensional generalization is straightforward4 and can be
written as:
Πµψ =
(e λ
λ2 − e α2ηµρ ερσKσ − e α
λ2 − e α2Kµ
)ψ , (2.24)
where the operators Πµ are
Πµ = ∂µ +2 e (λ2 ηµρ ερσα,σ − λα α,µ)
λ2 − e α2∂λ , (2.25)
and ηµρ and ερσ are given in (2.2) and (2.15). The matrices Kµ and ψ are now three-
dimensional and the function α(xµ) is the same as before, i.e. it satisfies (2.14).
Let us note that the equations (2.24) and (2.25) are written in terms of the coordinates xµ
introduced at the beginning of the paragraph 1.1, while the Lax-Pair (1.15) is written in
terms of the null coordinates ζ, η . This is made in order to develop a universal approach
3See paragraph 1.5 in [9].4Namely, even if it is written in a different form, at least of the dimension of the matrices, it
is formally equal to Lax Pair of Belinski-Zakharov for Einstein equations in vacuum. See below theexpressions of this Lax Pair for the non-stationary and stationary cases.
25
both to the stationary and the nonstationary cases. Hereafter we open a short aside to
give the link between the expressions of the Lax Pair (2.24) with its expression as given
by the (1.15) in null coordinates, and that one given in chapter 8 of the book [9] for the
stationary case. In these transformations the dimension of the matrices is not important.
———————–
Written for each component the system (2.24) becomes:
Π2ψ = − 1
λ2 − e α2(λK3 + e αK2) ψ , Π3ψ = − e
λ2 − e α2(λK2 + αK3) ψ ,
where
Π2 = ∂2 − 2 λ (λα,3 + e α,2 α)
λ2 − e α2∂λ , Π3 = ∂3 − 2 e λ (λ α,2 + α,3 α)
λ2 − e α2∂λ .
We recall that the null coordinates ζ, η are defined in terms of the coordinates xµ by:
ζ = (√
e x2 + x3)/2 , η = −(√
e x2 − x3)/2 .
• For nonstationary case we have: x2 = t , x3 = z , e = 1 , thus, the system (2.24),
expressed in terms of the null coordinates, reduces to the system (1.15)
D1 ψ =A
λ− αψ
D2 ψ =B
λ + αψ
, where D1 = ∂ζ − 2 α,ζ λ
λ− α∂λ , D2 = ∂η +
2 α,η λ
λ + α∂λ ,
if Π3 + Π2 = D1 , Π3 − Π2 = D2 , and A = −(K3 + K2) , B = K3 −K2 .
• For stationary case it results: x2 = ρ , x3 = z , e = −1 and α = ρ . Now
the system (2.24) becomes (see chapter 8 of the book [9]):
D1ψ =ρ V − λU
λ2 + ρ2ψ
D2ψ =ρU + λV
λ2 + ρ2ψ
, where D1 = ∂z − 2λ2
λ2 + ρ2∂λ , D2 = ∂ρ +
2λρ
λ2 + ρ2∂λ ,
if Π2 = D1 , Π3 = D2 , and U = K3 , V = K2 . In this case, U = ρ g,ρ g−1 and
V = ρ g,z g−1 .
———————–
26
Due to the ’wave’ equation (2.14) for the function α the operators Πµ commute:
ΠµΠν − ΠνΠµ = 0 . (2.26)
The self-consistency conditions for (2.24) are:
ηµν Kµ,ν = 0 , (2.27)
εµν
(Kµ,ν +
1
αKµ Kν − 1
αα,νKµ
)= 0 . (2.28)
The second of these equations implies that the two matrices Kµ can be written in terms
of a single matrix X in the form
Kµ = αX,µX−1 . (2.29)
Note that the matrix X plays the same role as g in the two-dimensional vacuum case.
Then the (2.28) is just the integrability condition of (2.29) for X , and (2.27) gives
a really nontrivial condition in the form of the following differential equation for the
matrix X
ηµν(αX,µ X−1),ν = 0 . (2.30)
In general this equation does not reproduce the Einstein-Maxwell system. We should
perform a reduction on X introducing some additional constraints that are not a con-
sequence of the self-consistency conditions of the spectral problem (2.24), but that are
compatible with them. To formulate these constraints, let us first introduce the following
matrix Ω :
Ω =
0 1 0
1 0 0
0 0 0
. (2.31)
In what follows we shall use the small Latin indices from the first part of the alphabet
for the first and the second rows and columns of the three-dimensional matrices and, we
recall, that they can take the values 0 and 1; for the third rows and columns we shall use
the star symbol. Capital letters of the first part of the Latin alphabet will be used to
enumerate the matrix components, thus A = (a, ∗) , B = (b, ∗) , ... . With this convention
the matrix Ω , for example, can be written in the following form:
Ω = ΩAB =
(Ωab Ωa∗
Ω∗b Ω∗∗
)=
(εab 0
0 0
). (2.32)
It is now convenient to introduce two special combinations made up of the matrix X and
its derivatives. Thus, we define Uµ by
Uµ = i e α ηµρ ερσX−1X,σ + 4 e (α2X−1),µ Ω . (2.33)
27
The additional constraints we need to impose on the matrix X can now be written as
X = X† , (2.34)
XUµ = −4 i e α ηµν ενσ Ω Uσ , (2.35)
where † means Hermitian conjugation. These are the two fundamental constraints.
However, we can still impose three new constraints on the matrix X . These new con-
straint are weaker, are easily imposed and do not represent a loss of generality. It is easy
to see from (2.31)-(2.35) that X33,µ = 0 , so that X33 = constant . Due to the invari-
ance of equations with respect to the rescaling X → cX , α → c α ( c is an arbitrary
constant), the value of X33 can be chosen at will. We chose to put
X33 = 2 . (2.36)
The second of these new constraints is the requirement that the determinants of the
two-dimensional blocks, constructed from the first two rows and columns of the matrices
Uµ , are not zero. As follows from definition (2.33), the upper and left blocks of matrices
Uµ are
(Uµ)ab = i e α ηµν ενσ
[(X−1)a3 X3b
,σ + (X−1)ac Xcb,σ
]+ 4 e
[α2(X−1)ac
],µ
εcb . (2.37)
Because det X 6= 0 and det Ω = 0 , it follows from (2.35) that det Uµ = 0 . Thus our
second new constraint can be formulated as
rank(Uµ) = 2 , det[(Uµ)a
b] 6= 0 . (2.38)
These properties will be satisfied automatically by the construction of the solutions and,
in practice, do not mean a loss of generality.
The third of these new constraints is that at least one of the diagonal elements of the
two-dimensional matrix
Xab − 1
2Xa3X3b (2.39)
does not vanish. This condition can also be easily imposed and does not represent a loss
of generality for the Einstein-Maxwell fields. It is easy to show that the unique structure
for the matrix X that follows from the constraints (2.34)-(2.39) is
X =
−4εacgcdε
db + 8εacΦcεbdΦd 4εacΦc
4εacΦc 2
,
X−1 =
−1
4εacgcdεdb −1
2εacg
cdΦd
−14εbcgcdΦd −1
2+ gcdΦcΦd
.
(2.40)
28
It is useful to notice that the matrix X is similar to the matrix X ′ ,
X ′ = 4
(gab + 2 ΦaΦb Φa
Φb12
).
That is X ′ = R−1XR if R = diag (εab, 1) . Moreover, it is worth underlining that
det X = 32 det g = 32 e α2 5.
Equation (2.20), which is satisfied by the complex electromagnetic potentials Φa , is now
a consequence of of the (a∗)-components of (2.35). Now we can see that the substitution
of this form of the matrix X into the self-consistency equations (2.30) exactly reproduces
the Einstein-Maxwell equations (2.21)-(2.22) and nothing else.
It is now worth to emphasizing that it is not possible to construct a dressing method
based on the spectral equation (2.24) and producing dressed solutions satisfying the con-
straints (2.34)-(2.35), at least following the standard approach described in chapter 16.
For this reason, Alekseev introduces, mapping the λ-spectral plane on a new w-spectral
plane, a different Lax Pair.
We introduce a new generating matrix φ ,which is related to the generating matrix ψ of
the vacuum case by
ψ = (X − 4iΩ) ϕ . (2.41)
The substitution of this expression into (2.24) shows that, due to the additional con-
straints (2.35), this new generating matrix ϕ satisfies the following spectral equation:
Πµϕ =i λ
(λ2 − e α2)2
[ (λ2 + e α2
)Uµ − 2 e α λ ηµρ ερσUσ
]ϕ , (2.42)
where Uµ are the matrices (2.33). The advantage of this representation of our spectral
problem is that it consists of rational functions not only with respect to the original
parameter λ , but also with respect to a new parameter w defined by
w = −1
2
(λ + 2β +
e α2
λ
). (2.43)
5The factor 32 is just a consequence of the choice X33 = 2 .6To preserve the structure (2.40) of the X matrices under the dressing Darboux transformation
(1.25), it is necessary to generalize the condition (1.23) substituting the transposition with the Hermitianconjugation: X = χ (λ)X0 χ†
(α2/λ
). As a consequence the 3 × 3 generalization of (1.53) becomes
XAB = (X0)AB − ∑k,l µ
−1k µ−1
l ΠklL(l)A L
(k)B , where Πkl =
[− (
α2 − µkµl
)−1m
(k)A m
(k)B
(X0
)AB
]−1
and
L(l)A = m
(l)B (X0)BA . A direct calculus shows that these formulas do not allow to generate new electro-
magnetic fields; namely the dressed components of the electromagnetic potential result to be triviallynull.
29
Here β is the second independent solution of the ‘wave’ equation (2.14), which has the
following connection to the function α :
β,µ = −e ηµρ ερσ α,σ . (2.44)
Due to this connection, the parameter w(xµ, λ) satisfies the identity
Πµw = 0 . (2.45)
The relation (2.43) can be understood as a transformation λ = λ(α, β, w) from the
parameter λ to new spectral parameter w . After this transformation is applied to any
generating matrix ϕ(xµ, λ) it must be understood that such a matrix becomes a function
of xµ and w only (more precisely as ϕ [xµ, λ(α, β, w)] ). In this sense and due to identity
(2.45), for any matrix φ we have
Πµφ = (∂µφ)w , (2.46)
where the right hand side is the usual partial derivatives with respect to xµ performed
under the assumption that w is some free parameter independent from xµ . The key
point now is that the application of this transformation to (2.42) shows its rational
dependence on w together with a simple structure of differential operators:
∂ ϕ
∂xµ=
1
2 i [(w + β)2 − eα2][(w + β) Uµ + e α ηµρ ερσUσ] ϕ. (2.47)
The analyticity of this equation with respect to the spectral parameter w is important,
because it allow us to apply to the construction of its solitonic solutions the dressing
procedure used in the vacuum case, but with the meromorphic structure of the dressing
matrices in the complex w-plane. At the same time the simplicity of the differential
operators allow us to impose the additional constraints (2.34)-(2.35) in a simple way.
2.3 The component gab and the potentials Aa
The construction for the metric components gab and the electromagnetic potentials Aa
needs, as first thing, the building of the n-soliton solution of the spectral problem (2.47);
it will be resolved at first in general, i.e. without assuming any additional structure for
the matrices Uµ . After that, we will impose all the necessary additional constraints (i.e.
the conditions which follow from (2.33)-(2.39).
2.3.1 The n-soliton solution of the spectral problem
Let us start this first stage with the introduction of a new matrix Λ νµ :
Λ νµ =
1
2 i
(w + β)δνµ + e α ηµρ ερν
(w + β)2 − e α2, (2.48)
30
and then the spectral equation (2.49) takes the form
ϕ,µ = Λ νµ Uνϕ . (2.49)
Let ϕ(0) and U(0)µ be some background solution of (2.49) with some given function
α and β . Then we search for the new ’dressed’ solution, corresponding to the same
functions α and β , of the form
ϕ = χϕ(0) . (2.50)
Because ϕ(0) is a solution, from (2.49) we obtain the following equation for the dressing
matrix χ :
χ,µ = Λ νµ ( Uν χ− χU (0)
ν ) . (2.51)
Now we will use the Latin indices of the last part of the alphabet (i.e. letters i, j, k... )
to enumerate quantities related to the poles of matrix χ . We assume that χ and χ−1
have n simple poles,
χ = I +n∑
k=1
Rk
w − wk
, χ−1 = I +n∑
k=1
Sk
w − wk
. (2.52)
Here and in the following we do not assume summation on indices i, j, k, ... ; such a
summation will be always indicated by the symbol∑
. At this stage wk and wk can
be arbitrary functions of the coordinates xµ . Also, in what follow, we consider that all
the 2n functions wk and wk are different. From the identity χχ−1 = I we have the
following condition for the matrices Rk(xµ) and Sk(x
µ) :
Rk χ−1(wk) = 0 , χ(wk) Sk = 0 , (2.53)
where the expression of the type F (wr) means the value of the function F (w, xµ) at
w = wk . The dependence of the coordinates is omitted for simplicity. Equation (2.53)
imply that we can look for matrices Rk and Sk of the form
(Rk)AB = n
(k)A m(r)B , (Sk)A
B = p(k)A q(k)B . (2.54)
It is worth noting that the construction of the solution of the spectral problem (2.47)
that we are carrying out is valid for matrices of any dimension.
The substitution of (2.54) into (2.53) gives two systems of algebraic equations from which
one can express all vectors n(k)A and q(k)A in terms of vectors m(k)A and p
(k)A as
n∑
l=1
p(k)B m(l)B
wl − wk
n(l)A = p
(k)A , (2.55)
n∑
l=1
m(k)Bp(l)B
wk − wl
q(l)A = −m(k)A . (2.56)
31
If we now introduce the n× n matrix Tkl and its inverse (Tkl)−1 ,
Tkl =p
(k)B m(l)B
wl − wk
,
n∑
l=1
Til(T−1)lk = δik , (2.57)
we obtain
q(k)A = −n∑
l=1
(T−1)lk m(l)A , n(k)A =
n∑
l=1
(T−1)kl p(k)A , (2.58)
for the vectors q(k)A and n(k)A .
To obtain the vectors m(k)A and p(k)A we use (2.51). It can be written in the form
Λ νµ Uν = χ,µ χ−1 + Λ ν
µ χU (0)µ χ−1 , (2.59)
or, equivalently, as
Λ νµ Uν = −χ(χ−1),µ + Λ ν
µ χU (0)µ χ−1 . (2.60)
All the terms in that functions are meromorphic functions of w that vanish at w →∞ .
Thus, to satisfy these equations it suffices to eliminate the residues of all their poles.
The first terms on the right hand side generate the second order poles at these points if
wk and wk depend on the coordinates xµ . Consequently, the first result we have from
(2.59) and (2.60) is that
wk = constant , wk = constant . (2.61)
Note that, due to the simplicity of the differential operators in the spectral equation
(2.49), the poles and the zeros of matrices χ and χ−1 in the w-plane are stationary
points and not trajectory as in the vacuum case.
Now the right hand side of (2.59) contain only simple poles at the points w = wk and
w = wk . The elimination of their residues gives the following equations for the matrices
Rk and Sk :
Rk,µ χ−1(wk) + Λ νµ (wk) Rk U (0)
ν χ−1(wk) = 0 , (2.62)
χ−1(wk) Sk,µ − Λ νµ (wk) χ(wk) U (0)
ν Sk = 0 . (2.63)
The solution of these equations can be expressed in terms of the background matrix
ϕ(0) . It is easy to check that if we substitute the matrices Rk and Sk into (2.62)-(2.63),
taking into account the conditions (2.53) and the fact that ϕ(0) and U (0) are solutions
of (2.49), then the system (2.62)-(2.63) is a set of differential equations for the vectors
m(k)A and p(k)A . The general solution of which is:
m(k)A = k(k)B [ (ϕ)(0)−1(wk) ]BA , (2.64)
p(k)A = l(k)B [ ϕ(0)(wk) ]AB . (2.65)
32
where k(k)B and l(k)B are 2n arbitrary constant vectors.
The structure of the coefficients Λ νµ , see (2.48), shows that in (2.59) we still have poles
with nonzero residues at the two points where (w + β)2 − e α2 = 0 . If we define√
e as
√e = 1 if e = 1 ,
√e = i if e = −1 , (2.66)
these poles can be written as w = w+ and w− , where
w+ = −β + e√
e α , w− = −β − e√
e α . (2.67)
The elimination of the residues of (2.59), at these poles does not produce any new
constraints on the matrices Rk and Sk , but gives the value of the matrices Uµ in terms
of Rk , Sk and the background matrices U(0)µ :
Uµ =1
2
[χ(w+) U (0)
µ χ−1(w+) + χ(w−) U (0)µ χ−1(w−)
]
+1
2e√
e ηµρ ερσ[χ(w+) U (0)
σ χ−1(w+)− χ(w−) U (0)σ χ−1(w−)
].
(2.68)
With this formula we have finished the construction, in general, of the n-soliton solution
of the spectral equation (2.49). This means that we can now express the matrices Uµ ,
ϕ and χ in terms of the background solution U(0)µ , ϕ(0) up to the freedom of choosing
arbitrary constants wk , wk and the arbitrary constants k(k)B and l(k)B in the vectors
m(k)A and p(k)A .
It is clear that one can use such a freedom to further specify the solution when necessary.
This is indeed necessary because the solution we have constructed for the matrices Uµ
does not guarantee that these are the same matrices that can be expressed in terms of X
matrix in (2.33), and that such a matrix X satisfies (2.30) and the additional constraints
(2.34)-(2.39).
It is remarkable and nontrivial that all these additional requirements can be satisfied
due to the freedom of the parameters. This is a consequence of the fact that our spectral
equations have ‘conserved integrals’ (some authors call them ‘involutions’), i.e. some
expressions quadratic in the generating matrix that give zero under the action of the
operators Πµ .
2.3.2 The matrix X
Let us now return to the Einstein-Maxwell three-dimensional problem. Here the analogue
of the two-dimensional metric tensor is the three-dimensional matrix X . However this
matrix needs to be Hermitian, not symmetric. Furthermore the dressing matrix χ(w) is
rational on w , which means that the replacement λ → e α2/λ is irrelevant in this case.
33
This suggests that we impose the basic additional constraints (2.34)-(2.35) assuming the
existence of a ‘conserved integral’ of the following form:
∂µ
[ϕ†(w, xµ) W (w, xµ) ϕ(w, xµ)
]= 0 (2.69)
with some and, as yet unknown, matrix W 7.
The existence of the integral (2.69) means that ϕ† W ϕ = Q(w) , where Q does not
depend on xµ ; we impose that matrix Q(w) be hermitian: Q† = Q . In this case the
freedom of the transformation ϕ(w, xµ) → ϕ(w, xµ)γ(w) , which obviously exist for the
spectral equation (2.49), allows to normalize each solution in such a way that the matrix
Q transforms as Q → γ†Qγ . Since Q is hermitian its transformed form can be made
universal, i.e. the same for all solutions, by choosing an appropriate transformation
matrix γ(w) for each solution. Moreover, this universal form can be made diagonal,
real and independent of w . Thus, without loss of generality, and within the class of
Hermitian matrices Q , the integral (2.69) can be written as
ϕ†(w, xµ) W (w, xµ) ϕ(w, xµ) = C , (2.70)
where
C = diag (C1, C2, C3) , C1, C2, C3 = constant , (2.71)
and where the three constants, C1, C2 and C3 , are real. This implies that
C = C† . (2.72)
Even if the constants C1, C2, C3 can be eliminated from the solutions by making their
modulus equal to 1 , we will keep the matrix C in the more general form (2.71) in order
to leave open the possibility for more convenient choices of arbitrary parameters in the
final form of the solution.
Since (2.70) is universal, it is also valid for the background solution
ϕ(0)† W (0) ϕ(0) = C , (2.73)
where W (0) is the matrix W calculated for the background solution. From (2.70) and
(2.73) we have
W−1 = χ(W (0))−1 χ†. (2.74)
Now let us assume that the matrix W−1 has no singularities at the points where the
matrices χ and χ−1 have poles. Then in order to satisfy (2.74) one needs first to
7Note: The definition of the Hermitian conjugation of matrix functions that depend on the complexparameter w is the following: to obtain the hermitian conjugation M†(w, xµ) of any matrix M(w, xµ)as function of w, one should first calculate the value of the matrix M at the complex conjugate point w,i.e. the value M(w, xµ), and then take the usual Hermitian conjugate of this value.
34
eliminate the residues on the right hand side of this relation, i.e. at w = wk and
w = wk . Let us consider first the set of points w = wk . The residues at these points
vanish if (I +
n∑
k=1
Rk
wl − wk
)(W (0))−1(wl)R
†l = 0 . (2.75)
or in components
n∑
k=1
m(k)D[(W (0))−1(wl)
]DB
m(l)B
wk − wl
n(k)A =
[(W (0))−1(wl)
]AD
m(l)D . (2.76)
It follows from (2.72)-(2.73) that we should construct any background solution in such
a way that the matrix W (0) is hermitian. Now it is easy to check that the equation
eliminating the residues on the right hand side of (2.74) at the second set of poles, i.e.
at w = wk , coincides exactly with (2.76). Therefore this is the only equation we need in
order to have regularity of (2.74) at the points where matrices χ and χ−1 are singular.
Equation (2.76) is an algebraic system where vectors n(k)A can be expressed in terms of
the vectors m(k)A . If we substitute into such a system (2.65) for the vectors p
(k)A , (2.55)
takes the form:
n∑
k=1
m(k)D[ϕ(0)(wl)
]DB
l(l)B
wk − wl
n(k)A =
[ϕ(0)(wl)
]AD
l(l)D . (2.77)
Of course, this equation should coincide with (2.75). The coincidence takes place when
wk = wk , (2.78)
and [(W (0))−1(wl)
]DB
m(l)B = [ ϕ(0)(wl) ]DB l(l)B . (2.79)
The first condition shows that the pole of the inverse matrix χ−1 should be located at
the points which are complex conjugate to the pole of the matrix χ . To discover the
second condition we should substitute (2.64) into (2.79) for the vectors m(k)A and the
expression for (W (0))−1 in terms of the ϕ(0) and C which follows from (2.73):
(W (0))−1 = ϕ(0)C−1ϕ(0)† . (2.80)
After this substitution we have to take in account the conditions (2.71)-(2.72) for matrix
C and the fact that now wk = wk . Then the resulting from 2.79 is very simple:
k(k)A = CAB l(k)B , (2.81)
where CAB are the components of the diagonal matrix C . This allows us to write all
the constants k(k)A in terms of l(k)A , or viceversa.
35
The same results can be obtained if we start our analysis from the ’conserved integral’
(2.74) written in its inverse form:
W = (χ−1)†W (0)χ−1 . (2.82)
2.3.3 Verifications of the constraints
Up to this point we do not need to know the explicit structure of the matrix W , apart
from their regularity at points w = wk and w = wk and the hermiticity of their
background values. Under these conditions the relations (2.78) and (2.80) between the
free constant parameters ensure the absence of poles at the points w = wk and w = wk
on the right hand side of (2.74) or (2.82). Now, we have to fix the exact structure of
matrix W in such a way that (2.82) is satisfied not only at the poles but everywhere in
the complex w-plane, and that it also satisfies the constraints (2.33)-(2.35). This goal
can be achieved if we choose the matrix W to be a linear function of w of the following
form:
W = X − 1
4XEX + 4 i (w + β) Ω , (2.83)
where
E =
0 0 0
0 0 0
0 0 1
. (2.84)
With this choice the matrix
W − (χ−1)†W (0)χ−1 (2.85)
clearly has no singularities at finite values in the w-plane. It has also no singularities at
infinity because the matrix χ−1 tends to unity as w → ∞ and W → W (0) since the
fixed constant matrix Ω has the same values for the background and dressed solutions.
This eliminates the poles in (2.85) at infinity in the w-plane. However, this expression
still has non-zero finite values at w → ∞ , which should vanish if we wish to satisfy
(2.82). Using (2.52) and (2.83) it is easy to calculate the first nonvanishing term of the
matrix (2.85) at w →∞ . Equating this term to zero we get
X − 1
4XEX = X(0) − 1
4X(0)EX(0) + 4 i (S†Ω + ΩS) , (2.86)
where X(0) is the background value of the matrix X and
S =n∑
k=1
Sk . (2.87)
36
The components of S follow from (2.54) and (2.58),
S BA = −
n∑
k,l=1
(T−1)kl p(l)A m(k)B . (2.88)
Now (2.82) is completely satisfied because (2.85) represents an analytic function at each
point on the w-plane which vanishes at infinity. Such a function is everywhere zero in
the w-plane.
Due to the special structure of the matrix E , it is easy to prove by direct computation
the hermiticity of the matrix X from the hermiticity of the matrix X− 14XEX . Then it
is also easy to see from (2.86) that the hermiticity of the dressed matrix X(0) implies the
hermiticity of the dressed matrix X . Another important property follows from (2.86),
namely, that X∗∗ = 2 if X(0)∗∗ = 2 . Because the background solution X(0) satisfies, by
definition, all the additional constraints (including (2.34) and (2.36) ), (2.86) guarantees
that all these constraints are satisfied for the dressed solution X .
Since (2.86) gives the matrix X− 14XEX , we need to know how to calculate the matrix
X from this. Let us introduce a new matrix
G = X − 1
4XEX . (2.89)
This expression can be easily inverted:
X = G + GEG , if X∗∗ = 2 . (2.90)
Due to this property and the trivial identity EΩ = 0 , it is easy to prove that we have
an equivalent form of condition (2.35), which is obtained by just replacing X on the left
hand side by G
GUµ = −4 i e α ηµρ ερσΩ Uσ . (2.91)
Another equivalent equation can be obtained by multiplying (2.91) by e√
e ηνλ ελµ , and
taking the sum and the difference of the new equation and the original one. The result
is
(G± 4 i e√
e α Ω)(Uµ ± e√
e ηµρ ερσ Uσ) = 0 . (2.92)
From (2.83), (2.89) and (2.67) we have,
G± 4 i e√
e α Ω = W (w±) . (2.93)
Using (2.82) we see that the dressing formulae for the first factors in (2.92) are
G± e√
e α Ω = (χ−1)†(w±)(G(0) ± e
√e α Ω
)χ−1(w±) . (2.94)
37
The dressing formulae for the second factor in (2.92) can be obtained by multiplying
(2.68) by e√
e ηνλ ελµ , and taking the sum and the difference of this new equation with
(2.68) itself. We thus obtain
Uµ ± e√
e ηµρ ερσUσ = χ(w±(U (0)
µ ± e√
e ηµρ ερσ U (0)σ
)χ−1(w±) . (2.95)
The product of these last two equations shows that if the left hand side of (2.92) is zero
for the background solution, it is also zero for the dressed solution. Thus we conclude
that condition (2.35) is valid because the background solution verifies it and because we
have already ensured that (2.49) and (2.69) are satisfied.
It follows from (2.95) that the traces of the matrices Uµ e√
e ηµρ ερσUσ are also conserved
under the dressing procedure. However, it is more convenient to deal directly with the
traces of the matrices Uµ , by taking trace of (2.68). From this equation we have simply
that Tr(Uµ) = Tr(U(0)µ ) . Since (2.33) is trivially valid for the background solution, we
have that Re[ Tr(U(0)µ ) ] = 0 , which one can easily verify using (2.40) for the background
matrix (X(0))−1 and the fact that the two-dimensional background matrix is real and
symmetric. As a consequence we have that for the dressed matrices Uµ ,
Re [ Tr(Uµ) ] = 0 . (2.96)
Finally we need to prove that our matrices Uµ and X are connected by (2.33). Again,
for the background solution such a relation is trivially satisfied because we started with
a given matrix X(0) and the new matrices U(0)µ were defined just using (2.33). In this
case (2.33) represents an additional constraint connecting Uµ and X .
To prove the validity of the constraint (2.33) one can start from the conserved integral,
[ϕ†(G + 4 i (w + β) Ω) ϕ
],µ
= 0 . (2.97)
After differentiation and by substitution into this formula of the expression (2.47) for
ϕ,µ (and for is hermitian conjugated ϕ†,µ ), we multiply the result by (w + β)2−e α2 and
obtain on the left hand side of (2.97) a quadratic polynomial in the spectral parameter
w , or more precisely, in w + β . Since our solution already ensures that condition (2.97)
is satisfied, all the coefficients in this polynomial vanish. The zero value of the coefficient
of the quadratic term gives the identity
(G + 4 i β Ω),µ = 2 (U †µ Ω− Ω Uµ) . (2.98)
The remaining coefficients of the polynomial give nothing new: the linear coefficient just
lead again to (2.98), and the free coefficient leads to (2.91).
It would be nice to prove the validity of (2.33) by the method we used before, i.e. by
38
proving that the dressing procedure preserves this relation. However, we have no suitable
dressing formula for the right hand side of (2.33). Instead, we can calculate the exact
structure of the matrices Uµ from those equation for which we have already proved the
validity. Then we can check the correctness of (2.33) by direct substitution. At this
stage we know that (2.34)-(2.39), (2.96) and (2.98) are valid and that matrix X has the
structure of (2.40). Detailed analysis in which a key rule is played by (2.35) in the form
(2.91) and by (2.96) and (2.98), shows that this system leads to the following unique
structure for the matrices Uµ :
(Uµ) ba = gac,µε
cb − i
αηµρ ερσ gac εcd gdf,σ εfb + 2 Φa,µΦc εcb
(Uµ) ∗a = −Φa,µ
(Uµ) b∗ = 2 Φc εcd (Uµ) b
d
(Uµ) ∗∗ = 2 εab Φa,µ Φb
. (2.99)
Direct substitution of this result together with the matrix X , see (2.40), into (2.33)
shows that this equation is identically verified. This is the final step in the proof that the
matrix X which first appeared in (2.29), has the structure (2.40) and satisfies (2.30). As
a consequence, the functions gab and Re(Φa) , which can be extracted from (2.86) using
(2.89), (2.90) and (2.40), indeed represent a solution of the Einstein-Maxwell equations.
2.4 The metric component f
To complete the construction of the n-soliton solution for the Einstein-Maxwell equa-
tions, we need to compute the metric coefficient f from (2.11). Without giving the
details (the reader can find them in [9]), we limit here to tell that the equation (2.11)
can be written as:
(ln f),µ = (ln |D|),µ − (ln α),µ + iD−1ελνα,ν Tr[(G−1 + E) U †
λ Ω Uµ
], (2.100)
where
D = ηµνα,µα,ν . (2.101)
The integration of (2.100) is rather long and will not be done here. It was performed by
Alekseev [3], and yields the very simple result:
f = C0f(0)T T , (2.102)
where, C0 = constant , f (0) is the background value of the metric coefficient f and T
is the determinant of the n× n matrix Tkl ,
T = det Tkl . (2.103)
39
2.5 Summary of prescriptions
Let us summarize now, step by step, the set of practical prescriptions for constructing
n-soliton solution of the Einstein-Maxwell equations starting with a given background
solution.
1. Take some background solution g(0)ab and A
(0)a of the Einstein-Maxwell equations
(2.12)-(2.13). Calculate the determinant of the matrix g(0)ab and find the function
α(xµ) from the relation α2 = e det g(0)ab , after choosing some definite root of this
quadratic equation, for example α > 0 .
2. Take the previous g(0)ab , A
(0)a and α , and find, using (2.17), the auxiliary potentials
B(0)a (up to two arbitrary real additive constants), and write the background value
of the complex electromagnetic potentials Φ(0)a = A
(0)a + i B
(0)a .
3. Substitute the values Φ(0)a and g
(0)ab into (2.40). This gives the background value
X(0) of the matrix X .
4. Calculate the background matrices U(0)µ by substituting into (2.33) the previous
values of X(0) and α .
5. Use (2.44) to find the function β(xµ) , up to some arbitrary real additive constant.
6. From (2.83) compute the background matrix W (0) in terms of X(0) and β .
7. Substitute α and β and U (0) into the spectral equation (2.47) and find the nor-
malized solution for the background generating matrix ϕ(0)(w, xµ) , i.e. the solution
that satisfies (2.73) with the matrix C defined in by (2.71)-(2.72).
8. Using the previous ϕ(0) , construct the vectors m(k)A and p(k)A according to (2.64)-
(2.65), where wk = wk , and where the constants k(k)A and l(k)A are related by
(2.81).
9. With these values for m(k)A and p(k)A construct the matrix Tkl using (2.57), and
again taking wk = wk .
10. Substitute the matrix Tkl and the vectors m(k)A and p(k)A into (2.88) to obtain
the matrix S .
11. Finally, from (2.86), with the help of (2.89) and (2.90) calculate the components of
the matrix X in terms of X(0) and S . The matrix X , thus obtained when written
40
in the form (2.40), gives the dressed solution gab and Aa of the Einstein-Maxwell
equations in terms of the Xab and Xa∗ components of X as
gab =1
4εca
(Xcd − 1
2Xc∗Xd∗
)εdb , (2.104)
Aa =1
4εcaReXc∗ . (2.105)
12. Takeing the coefficient f (0) from the background solution (2.1) and, after calculat-
ing the determinant T of the Tkl matrix defined by (2.57), use (2.102) to obtain
the coefficient f .
2.6 Some remarks
It is worth making some remarks on the relation between soliton solutions described
here for the particular case when Φa = 0 (vacuum) and the vacuum soliton solutions
which can be constructed using the Belinski-Zakharov technique described in the first
chapter. There is not a comprehensive analysis of this relation yet. However the results
obtained in [29, 30, 31, 37] show that to all appearances the n-soliton vacuum solution
corresponding to n complex coordinate-independent poles in the complex w-plane in
Alekseev’s approach is equivalent to the 2n-soliton solution corresponding to n pairs of
complex conjugate (coordinate-dependent) poles in the complex λ-plane in the framework
described in chapter 1. By equivalent we mean two solutions can be transformed into each
other by a coordinate transformation. Nevertheless, in the vacuum case, the Inverse Scat-
tering Method described in chapter 1 using the complex structure in the λ-plane, gives a
richer set of soliton solutions, since it also includes solutions which correspond to an odd
number of poles in the λ-plane. In the Alekseev approach for each single complex pole a
distinct complex conjugate pole must appear in the inverse of the generating matrix.
Hence, if a single pole of the Belinski-Zakharov approach must be real, such kind of poles
has no place in the Alekseev framework since, in this case, the generating matrix and its
inverse would have the same pole in such a way that the procedure becomes singular.
Thus there are no analogues of such solutions in the framework that uses the complex
structure in the w-plane, at least following all the prescriptions of Alekseev procedure.
At this regard, it is worth mentioning the work of Micciche and Griffiths [67], in which,
following a different prescription, they obtained such 1-soliton solutions in w-plane by
introducing distinct real poles in the inverse of the generating matrix. Anyway, they
found this possibility only for solutions in vacuum, that is for null electromagnetic fields.
Chapter 3
Exact stationary axially symmetricone-soliton solution on Minkowskybackground
In this chapter, we show how to apply the Alekseev solitonic technique to generate a
stationary axially symmetric one-soliton solution on a Minkowsky background. The aim
of this chapter is to illustrate practically the application of the procedure, since both the
solution that will be found [66] and the reparameterization to get it [3] are well known.
It is worth reminding here that a generating technique provides exact solutions of the
field equations in terms of a certain number of mathematical constants; after this primer
product is obtained, further work is necessary on it to find, when possible, its physical
meaning. This work generally consists, on the one side, in a reparameterization that
links the mathematical constants to the physical constants; on the other side, in finding
further conditions that have to be imposed on the parameters to eliminate some possible
spurious behaviour, or to characterize, or select, a particular solution of interest among
a more general family, naturally obtained by means of the generating technique.
This chapter is organized as follows: in the first part the generating procedure will
be applied, step by step, until the determination of the T matrix defined in (2.57).
In the second part a known solution will be introduced, possessing the same degree of
freedom (that is the number of independent constants) of the one-soliton solution; after
an appropriate comparison, the transformations to express the mathematical constants
in terms of the physical ones will be found1. The final part consists in the application
of the last steps of the procedure to calculate those components that are useful to the
1The same result can be obtained by an analysis of the asimptotic behaviour of the fields far fromtheir source.
41
42
determination of the one-soliton solution in the form of the introduced known solution.
The result, of interest here, is not the exact solution representing a one-body source
characterized by mass, angular momentum, electric and magnetic charge and NUT pa-
rameter. Rather, it is principally the reparameterization to get the one-soliton solution
in a physically readable form.
Index notation and form of the line elements.
Hereafter, the coordinates will be enumerated as (x0, x1, x2, x3) = ( t, ϕ, ρ, z) , the small
Latin letters take the two values 0, 1 while, Greek small indexes take the two values
2, 3 . The background line element and the dressed one-soliton solution are therefore of
the form:
ds2 = gab(xρ)dxadxb + f(xρ)ηµνdxµdxν , (3.1)
where, since we are dealing with an axially symmetric stationary metric, the sign indi-
cator e is set equal to −1, hence the ηµν matrix will be:
ηµν=
(1 0
0 1
),
while, since det g(0) = −ρ2 and α2= e det g(0) , then α = ρ .
3.1 Application of the first nine steps of the gener-
ating procedure.
3.1.1 Step-1: Background Einstein-Maxwell solution.
As mentioned above, we start by taking a flat background solution, that is, automatically
null values for A(0)a electromagnetic background potentials. In cylindrical coordinate,
the Minkowsky line element is:
ds2 = −dt2 + ρ2dϕ2 + dρ2 + dz2 . (3.2)
Therefore it results that, respectively for the background metric components and for the
background electromagnetic potentials, we have:
g(0)ab =
(−1 0
0 ρ2
), f (0) = g(0)
µµ = 1 (3.3)
and
A(0)a = 0 . (3.4)
43
3.1.2 Step-2: Background value of the complex electromagneticpotential Φ
(0)a .
Recalling that:
εµν = εµν=
(0 1
−1 0
), εab = εab=
(0 1
−1 0
),
the immaginary components B(0)a of the complex background electromagnetic fields Φ
(0)a
follows as the solution of the differential system (2.17)
Ba,µ = − 1
αηµν ενλ gab εbcAc,λ .
Being the imaginary part of the complex electromagnetic potential depending only on ρ
and z variable and being a constant potential inessential for the field value, then we put
B(0)a = 0
and therefore:
Φ(0)a = A(0)
a + iB(0)a = 0 .
3.1.3 Step-3: Calculus of X(0) and X(0)−1.
From the general structure of the X and X−1 matrices given in (2.40),
X =
−4 εacgcdε
db + 8 εacΦcεbdΦd 4 εacΦc
4 εacΦc 2
X−1 =
−1
4εacgcdεdb −1
2εacg
cdΦd
−14εbc gcdΦd −1
2+ gcdΦcΦd
.
Substituting the values found above for g(0)ab and Φ
(0)a , and since
εacg(0)cd εdb =
(−ρ2 0
0 1
),
we obtain:
X(0) =
4ρ2 0 0
0 −4 0
0 0 2
X(0)−1 =
1
4ρ20 0
0 −1
40
0 01
2
.
44
3.1.4 Step-4: Calculus of U(0)µ .
From the equation (2.33)
Uµ = i e α ηµρ ερσX−1X,σ + 4 e(α2X−1
),µ
Ω
putting
εµσ .
= ηµρερσ =
(0 1
−1 0
),
and recalling the definition (2.31)
Ω=
(εab 0
0 0
)=
0 1 0
−1 0 0
0 0 0
,
substituting the background values just found above, for the stationary case we have:
U (0)µ = −
[i ρ εµ
σ X(0)−1X(0),σ + 4
(ρ2X(0)−1
),µ
Ω]
.
Since
X(0),σ =
8 ρ δ2σ 0 0
0 0 0
0 0 0
,
where we have put ρ,σ = δ2σ , being δ the Kronecker symbol,
X(0)−1X(0),σ =
2
ρδ2σ
1 0 0
0 0 0
0 0 0
and
(ρ2X(0)−1
),µ
= 2 ρ δ2µ X(0)−1 + ρ2
(X(0)−1
),µ
= −1
2ρ δ2
µ
0 0 0
0 1 0
0 0 −2
,
than we have for each component of U(0)µ matrices that:
U(0)2 = −
[i ρ
(6ε2
2X(0)−1X(0),2 + ε2
3X(0)−1X(0),3
)+ 4
(ρ2X(0)−1
),2
Ω]
=
= −[i ρ X(0)−1X
(0),3 + 4
(ρ2X(0)−1
),2
Ω],
U(0)3 = −
[i ρ
(ε3
2X(0)−1X(0),2 + 6ε3
3X(0)−1X(0),3
)+ 4
(ρ2X(0)−1
),3
Ω]
=
= −[−i ρX(0)−1X
(0),2 + 4
(ρ2X(0)−1
),3
Ω].
45
Therefore, the final expressions are:
U(0)2 =
0 0 0
−2 ρ 0 0
0 0 0
U
(0)3 =
2i 0 0
0 0 0
0 0 0
.
3.1.5 Step-5: Deduction of β(xµ).
From the equation (2.44) β,µ = −e ηµρ ερσ α,σ , which now reduces to
β,2 = ρ,3
β,3 = −ρ,2
,
it follows immediately that
β = −z + z0 ,
where z0 is an arbitrary constant that gives information on the location of the source
on the axis of symmetry.
3.1.6 Step-6: Calculus of W (0).
From the equation (2.83) we have that:
W (0) = X(0) − 1
4X(0)EX(0) + 4 i (w + β) Ω ,
where w ∈ C and E = diag( 0, 0, 1 ) .
Since
X(0)EX(0) =
4ρ2 0 0
0 −4 0
0 0 2
0 0 0
0 0 0
0 0 1
4ρ2 0 0
0 −4 0
0 0 2
= 4E ,
then
W (0) =
4ρ2 0 0
0 −4 0
0 0 2
− 1
64 64 E + 4 i (w + β)
0 1 0
−1 0 0
0 0 0
.
Introducing2 λ = − (β + w) = z − w , where it has be choosen Re(w) = z0 , the W (0)
matrix assumes the form:
W (0) =
4ρ2 −4iλ 0
4iλ −4 0
0 0 1
.
2Here, the λ symbol has nothing to do with the same symbol used to indicate the spectral parameterin the context of the Belinski-Zakharov technique.
46
3.1.7 Step-7: Deduction of background generating matrixϕ(0) (w, xµ) and its normalization.
This is the only step in which we have to solve a differential problem; in fact, the great
advantage of the soliton method is that, now, we only have to integrate a linear system
instead of the nonlinear one of Einstein-Maxwell.
Integration of Lax Pair. To get the background generating matrix ϕ(0) we have to
integrate the spectral equation (2.47)
ϕ(0),µ =
1
2i
(w + β
(w + β)2 + ρ2Uµ − ρ
(w + β)2 + ρ2εµ
σUσ
)ϕ(0) .
Introducing the symbol Γ =√
λ2 + ρ2 , the Lax Pair assumes the form:
ϕ(0),µ =
1
2iΓ2(−λUµ − ρ εµ
σUσ) ϕ(0) ,
that is:
ϕ(0),2 =
1
2iΓ2(−λU2 − ρU3) ϕ(0)
ϕ(0),3 =
1
2iΓ2(−λU3 + ρU2) ϕ(0)
.
Since
(−λU2 − ρU3)=
−2iρ 0 0
2λρ 0 0
0 0 0
and (−λU3 + ρU2)=
−2iλ 0 0
−2ρ2 0 0
0 0 0
,
the spectral system is reduced to:
∂ρϕ(0) =
1
Γ2
−ρ 0 0
−iλρ 0 0
0 0 0
ϕ(0)
∂zϕ(0) =
1
Γ2
−λ 0 0
iρ2 0 0
0 0 0
ϕ(0)
;
or, introducing explicit value of Γ , and recalling that the capital Latin letters of the first
part of the alphabet ( i.e. from A to H ) can assume the values ( 0, 1, ∗ ) and since
47
∂z ≡ ∂λ ,
∂ρϕ(0)0A = − ρ
λ2 + ρ2ϕ
(0)0A
∂ρϕ(0)1A = − iλρ
λ2 + ρ2ϕ
(0)1A
∂ρϕ(0)∗A = 0
∂λϕ(0)0A = − λ
λ2 + ρ2ϕ
(0)0A
∂λϕ(0)1A = − iρ2
λ2 + ρ2ϕ
(0)1A
∂λϕ(0)∗A = 0
.
The integration of this system yields the following solution:
ϕ(0)0A =
aA
Γ
ϕ(0)1A =
iaAλ
Γ+ bA
ϕ(0)∗A = cA
where aA, bA, cA are nine real integration constants.
Normalization of generating matrix. Using the above general solution for the gen-
erating matrix, its hermitian conjugate and the matrix W (0) :
ϕ(0) =
1Γ
aA
bA + iΓ
λ aA
cA
, ϕ(0)† =
(1Γ
aA, bA − iΓ
λ aA, cA
), W (0) =
4ρ2 −4iλ 0
4iλ −4 0
0 0 1
,
from the conserved integral (2.70) ϕ(0)†W (0)ϕ(0) = C where the 3× 3 C matrix can be
choosen to be real and diagonal C = diag (C1, C2, C3) we have that:
(ϕ(0)†W (0)ϕ(0)
)= 4aAaB − 4bAbB + cAcB = C ,
that is
4 a 20 − 4 b 2
0 + c 20 = C1
4 a 21 − 4 b 2
1 + c 21 = C2
4 a 2∗ − 4 b 2
∗ + c 2∗ = C3
,
4 a0 a1 − 4 b0 b1 + c0 c1 = 0
4 a1 a∗ − 4 b1 b∗ + c1 c∗ = 0
4 a∗ a0 − 4 b∗ b0 + c∗ c0 = 0
.
Finally, choosing aA = (1, 0, 0) , bA = (0, 1, 0) , cA = (0, 0, 1) , it follows that C1 = 4 ,
48
C2 = −4 , C3 = 1 and:
[ϕ(0)
]AB
=
1
Γ0 0
iλ
Γ1 0
0 0 1
,[ϕ(0)−1
]AB=
Γ 0 0
−iλ 1 0
0 0 1
.
Rationalization of Γ function. It is convenient to replace the cylindrical coordinates
(ρ, z) with a system of ellipsoidal coordinates (r, θ) expressed by:
ρ =√
R2 − w2 s θ z = R c θ (3.5)
where s.= sin , c
.= cos and R = R (r) is at the moment an unknown function of the
radial coordinate r . In fact, it is easy to see that, in these new coordinates, Γ depends
on them in the following rational way
Γ = R− w c θ .
3.1.8 Step-8: Costruction of m(k)A and p(k)A vectors.
From the matrix ϕ(0) , and its inverse, it is now possible to evaluate the vectors m(k)A
and p(k)A through the formulas (2.64)-(2.65):
m(k)A= k(k)B
[(ϕ(0)
)−1(wk)
]BA
, p(k)A = l(k)B
[ϕ(0) (wk)
]AB
,
where we have replaced directly wk with wk in virtue of (2.78), and where wk are
the values of the fixed poles. For one-soliton, the values of the soliton index k is 1 .
Hence, since we are dealing with w1 , henceforth in this chapter, we replace it with w ,
remembering that, from now on, it will be no more a variable but a constant. The soliton
index will be removed from all the other symbols too. The vectors k(k)A depend on l(k)A
vectors through the equation (2.81), k(k)A = CAB l(k)B , where CAB = diag (C1, C2, C3) .
Therefore, as Γ(w) = Γ and λ(w) = λ , it follows that:
m0 = C1 Γ l0 − i C2 λ l1 p0 =1
Γl0
m1 = C2 l1 p1 = i1
Γλ l0 + l1
m∗ = C3 l∗ p∗ = l∗
;
49
or, substituting the values chosen for the diagonal components of constant C matrix:
m0 = 4 Γ l0 + i 4 λ l1 p0 =1
Γl0
m1 = −4 l1 p1 = i1
Γλ l0 + l1
m∗ = l∗ p∗ = l∗
. (3.6)
3.1.9 Step-9: Costruction of Tkl matrix.
Having at our disposal the vectors mA and pA , it is now possible to construct the unique
component (since the indexes of the matrix T enumerate the solitons) T11 of the matrix
Tkl . We denote it simply with T through the equation (2.57). We rewrite this equation
here, substituting directly w with w
T =pA mA
w − w. (3.7)
It is worth noting that z and the complex pole w appear in T , in the S BA matrix
and, therefore, in the final solution, always through the combination λ = z−w . Then ,
for the arbitrariness of the z0 constant, and since Re(w) is an arbitrary fixed constant
too, it follows that we have only one independent constant that we have already used
above when we put z0 = Re(w) . As the most convenient choice, we now take the pole
to be purely imaginary, that is w = i σ , where σ ∈ R ; this position is not a reduction
of the generality of the result since the arbitrariness in the choice of z0 reflects, in this
one-soliton case, the invariance under translation along the z-axis of the Minkowsky
background line element (3.2). From this position it follows that:
Γ = R− i σ c θ , (3.8)
λ = R c θ − i σ , (3.9)
T =1
2 i σΓ
[4 Γ | l0|2 + Γ
(−4 | l1|2 + | l∗|2 )+ 8 σ l0 l1
]. (3.10)
3.2 Determination of the mathematical parameters
in terms of the physical one.
In the previous paragraphs, we have constructed the m(1)A and p(1)A vectors and the
unique component of the T matrix. They all will be the ingredients by which the S
matrix is defined through the (2.88) equation. The mathematical constants present in
50
these objects are the six real constants of the three complex constant vectors lA and
the imaginary part σ of the pole w . Looking at the formulas (2.64)-(2.65) which give
the m(1)A and p(1)A vectors, it results that there may also be arbitrary complex factors
which can depend on the index k and the coordinates xµ , and that such factors are not
present in final expressions for the matrices Rk and Sk . This means that we dispose of
a rescaling freedom lA → ι lA for an arbitrary complex constant ι , which lets us fix one
of the three constant vectors lA according to our convenience, without loss of generality.
Therefore, the six real constants given by lA are reduced to four; adding σ , we have a
total of five arbitrary independent constants.
It is necessary to remind the unknown function R(r) present in the definition (3.5) of
the ellipsoidal coordinates (r, θ) , has not been defined yet.
Now, we are going to introduce the one-body solution of McGuire-Ruffini [66], which
depends on five free parameters representing: mass, angular momentum for unit mass3,
electric and magnetic charge4 and NUT parameter of one-body source. A direct compar-
ison between the respective f conformal factors and the g00 components will enable us
to express the five mathematical constants in terms of the above mentioned physical ones.
It is worth to recall that here the qualification of ”physical”, extended to the mag-
netic charge and the NUT parameter, is improper; we have used it for them just to
distinguish the five parameters of the McGuire-Ruffini from the purely mathematical
parameters present in the one-soliton solution. The magnetic charge parameter appears
in the McGuire-Ruffini as a consequence of the invariance of the Einstein-Maxwell equa-
tions respect to a duality rotation e = q c θ g = q s θ , where q is a constant and the
electric charge e and the magnetic charge g are described in terms of θ . The NUT
parameter5 appeared in the literature as a new constant present in the solution of New-
man, Unti and Tamburino [70]. It reduces to the Schwarzschild solution if this parameter
is equated to zero otherwise it presents singularities along the symmetry axis at θ = 0
and θ = π . If one of them can be removed by a coordinate transformation, this does
not get rid of the other one. Some literature was devoted to look for an interpretation
of this parameter (see among the most recent works for example [65] and the literature
cited therein). Anyway it carries some singularities. Because of this, it should be better
to refer to the magnetic charge and NUT terms as unphysical parameters and hence to
3Dealing with axially symmetric solution the angular momentum is parallel to the axis of symmetryand oriented along the θ = 0 direction.
4We call McGuire-Ruffini solution the generalization of the Kerr-Newmann-NUT solution [33] givenby the presence of this additional magnetic charge parameter.
5It can also be referred to as the dual mass, the magnetic mass, the gravitomagnetic monopolemomentum and so on.
51
remove them from the final result if a physical meaningful solution is desired.
3.2.1 The McGuire-Ruffini one-body solution.
The line element of McGuire-Ruffini one-body solution in ellipsoidal coordinates
(t, ϕ, r, θ) is:
ds2 = − 1
Σ
(∆− a2s2θ
)dt2 + 2
1
Σ
(∆ χ− a% s2θ
)dt dϕ +
1
Σ
(%2s2θ −∆ χ2
)dϕ2+
+Σ
(1
∆dr2 + dθ2
) (3.11)
where:
B = b + a c θ , Σ = r2 + B2 , ∆ = r2 − 2 mr − b2 + a2 + q2 ,
χ = a s2θ − 2 b c θ , % = r2 + b2 + a2 , q2 = e2 + g2
and m , b , a , e and g are respectively the Schwarzchild mass, the NUT parameter, the
angular momentum per unit mass, the electric and magnetic charge.
To perform a first comparison, we transform the conformally flat bedimensional block of
the line element (3.1) in ellipsoidal coordinates (3.5),
ρ =
√R2 − w2s θ
z = R c θ, where R = R (r) ,
and equate it to the corresponding block of (3.11). Under this transformation of coordi-
nates we have:
dρ2 + d z2 = R2,r
(R2 − w2c2θ
R2 − w2
)dr2 +
(R2 − w2c2θ
)d θ2 ,
where R,r= ∂rR . From the identification
f
[R2
,r
(R2 − w2c2θ
R2 − w2
)dr2 +
(R2 − w2c2θ
)d θ2
]= Σ
(1
∆dr2 + d θ2
),
we get the system
f R2r
(R2 − w2c2θ
R2 − w2
)=
r2 + B2
r2 − 2 mr − b2 + a2 + q2
f (R2 − w2c2θ) = r2 + B2
; (3.12)
52
from the second equation of (3.12), we get that
f =r2 + B2
(R2 − w2c2θ).
and, substituting this in the first of (3.12),
R2r
R2 − w2=
1
r2 − 2 mr − b2 + a2 + q2.
The solution (with positive determinations of the radicals) of these last relation gives:
R +√
R2 − w2 = (r −m) +
√(r −m)2 −m2 − b2 + a2 + q2 .
Since w = i σ, it follows that:
R = r −m, (3.13)
σ2 = −m2 − b2 + a2 + q2 . (3.14)
The line element of (3.11) is now rewritable in the form
ds2 = gab(r, θ)dxadxb + f(r, θ)(dρ2 + dz2
),
where the components, expanded and ordered in accordance with decreasing power of r
and c θ to facilitate future comparison, are:
g00 = − 1
Σ
[r2 − 2mr + a2c2θ +
(q2 − b2
)](3.15)
g01 =1
Σ
[(−2b c θ) r2 + 2m
(a c2θ + 2b c θ − a
)r
+ a(2b2 − q2
)c2θ + 2b
(b2 − a2 − q2
)c θ + a
(q2 − 2b2
)] (3.16)
g11 =1
Σ
s2θ r4 +
(−a2c4θ − 4ab c3θ − 6b2c2θ + 4ab c θ + a2 + 2b2)r2
+ 2m(a c2θ + 2b c θ − a
)2r
+ a2(b2 − q2 − a2
)c4θ + 4ab
(b2 − q2 − a2
)c3θ
+[2(a2 − 2b2
)q2 +
(a4 + 3b4 − 8a2b2
)]c2θ
− 4ab(b2 − a2 − q2
)c θ
+(b4 + 3a2b2 − a2q2
)
(3.17)
f =r2 + (b + a c θ)2
(r −m)2 + σ2c2θ(3.18)
It is easy to see that, equating some of the five parameters to zero, this solution contains,
as particular cases, well known one-body solutions as showed below in the table.
53
m b a e g
Schwarzschild 6= 0 = 0 = 0 = 0 = 0
Reissner-Nordstrom 6= 0 = 0 = 0 6= 0 = 0
NUT 6= 0 6= 0 = 0 = 0 = 0
Kerr 6= 0 = 0 6= 0 = 0 = 0
Kerr-NUT 6= 0 6= 0 6= 0 = 0 = 0
Kerr-Newman 6= 0 = 0 6= 0 6= 0 = 0
Kerr-Newman-NUT 6= 0 6= 0 6= 0 6= 0 = 0
3.2.2 Calculus and comparison of the f conformal factors.
Introducing the following terms:
p.= 4|l0|2
q.= −4|l1|2 + |l∗|2
,
ξ.= 8 σRe
(l0l1
)
η.= 8 σIm
(l0l1
) , ζ.= ξ + iη ,
the (3.10) expression for T becomes
T =1
2iσΓ
(Γ p + Γ q + ζ
).
Using this expression in the (2.102) for the f factor, and since f (0) = 0, we have that:
f = C(0)T T = C(0)1
4σ2|Γ|2(Γ p + Γ q + ζ
) (Γ p + Γ q + ζ
).
If F.= 4σ2|Γ|2T T , then from the relations
|Γ|2 = R2 + σ2c θ
Γ2 + Γ2 = 2 (R2 − σ2c2θ)(p ζ + q ζ
)γ +
(p ζ + q ζ
)Γ = 2 [(p + q) ξR− (p− q) η σc θ] ,
it follows that
F =[P 2R2 + 2PξR + Q2σ2c2θ − 2Qησ c θ + |ζ|2] ,
where P = p + q and Q = p − q . Using the expression for R , given by (3.13), we
have
f = C(0)1
4σ2|Γ|2[P 2r2 + 2P (ξ − Pm) r + Q2σ2c2θ − 2Qησc θ + |ζ|2 + Pm (Pm− 2ξ)
],
54
which, compared with (3.18), gives the system
C(0)P 2
4σ2= 1
C(0)2P (ξ − Pm)
4σ2= 0
C(0)Q2σ2
4σ2= a2
−2C(0)Qησ
4σ2= 2ab
C(0)|ζ|2 + Pm (Pm− 2ξ)
4σ2= b2
.
Therefore P =ε1 2 σ√
C(0)
, Q =ε2 2 a√
C(0)
, ξ =ε1 2 mσ√
C(0)
, η =−ε2 2 b σ√
C(0)
, where ε1 and ε2
are two undetermined indicators of sign. This leads to the following relations for the lA
vectors:
4|l0|2 − 4|l1|2 + |l∗|2 = ε12σ√C(0)
4|l0|2 + 4|l1|2 − |l∗|2 = ε22a√C(0)
Re(l0l1
)= ε1
m
4√
C(0)
Im(l0l1
)= −ε2
b
4√
C(0)
, (3.19)
from which
4|l0|2 =ε1σ + ε2a√
C(0)
(3.20)
−4|l1|2 + |l∗|2 =ε1σ − ε2a√
C(0)
. (3.21)
Hence we have:
T =1
i√
C(0)Γ[ ε1r − ε2i (b + a c θ) ] . (3.22)
Now, we can proceed with the final steps of the generating procedure to get all the
remaining components of the metric tensor and of the electromagnetic potentials. From
the comparison with the g00 component (3.15), we will be able to define, in a complete
manner, both the undetermined terms C(0) , ε1 , ε2 , and the lA vectors in terms of the
five physical constants.
55
3.2.3 Step-10 & 11: Determination of S matrix and calculusof gab components.
Before we calculate all the components of the S matrix, it is convenient to have the final
formulas for the dressed solution expressed only in terms of the background solution and
of the S matrix. In fact, not all its components appear in these formulas. We get this
explicit version of the (2.104) and (2.105) formulas from the (2.86) through the (2.89)
and (2.90). In the particular case, adapted to the Minkowsky background, we have:
g00 = −1 + 2 Im (S 10 )− 4|S ∗
0 |2
g01 = i[S 0
0 + S11
]− 4S ∗0 S∗1
g10 = −i[S 1
1 + S00
]− 4S ∗1 S∗0
g11 = ρ2 − 2 Im (S 01 )− 4|S ∗
1 |2
, (3.23)
A0 = −2 Im (S ∗0 )
A1 = −2 Im (S ∗1 )
. (3.24)
A direct calculation shows that the condition valid to assure, at the same time, the
reality and simmetry of gab matrix
Re(S 00 + S 1
1 )− 4 Im(S ∗0 S ∗
1 ) = 0 ,
reduces to
Im(8 σ p0 p1 − p0 m0 + p1 m1
)= 0 .
This can be verified to be identically satisfied if we substitute the expressions (3.6) into
it.
Comparison of g00 components.
To get the g00 component we have to calculate the S 10 and S ∗
0 .
———————–
• Calculus of S 10 component:
S 10 = − 1
Tp0m
1 = − 1
T
(− 4
Γl0 l1
)= i
(ε1m− ε2i b
ε1r − ε2iB
)=
1
Σ(ε2b + ε1im) (ε1r + ε2iB) =
=1
Σ[ε1ε2 (br −mB) + i (bB + mr)]
56
and therefore:
Im(S 1
0
)=
1
Σ(bB + m r) .
———————–
• Calculus of S ∗0 component:
S ∗0 = − 1
Tp0m
∗ = − 1
T
(− 1
Γl0l∗
),
therefore
|S ∗0 |2 =
1
|T |21
|Γ|2 |l0|2|l∗|2 ;
and since |T |2|Γ|2 =Σ
C(0)
and |l0|2 =1
4(ε1σ + ε2a) then:
|S ∗0 |2 =
√C(0)
1
4Σ(ε1σ + ε2a) |l∗|2 .
———————–
Now, using the first one of the (3.23) equations, we have
g00 = −1 +1
Σ
[2 (bB + mr)− 64 1
64√
C(0) (ε1σ + ε2a) |l∗|2]
,
which, compared with the corresponding component (3.15), leads to the relation:
|l∗|2 =q2
√C(0) (ε1σ + ε2a)
. (3.25)
This last expression, together with (3.20) and (3.21), gives
|l0|2 =1
4√
C(0)
(ε1σ + ε2a)
|l1|2 =1
4√
C(0)
(a2 + q2 − σ2
ε1σ + ε2a
)
|l∗|2 =q2
√C(0) (ε1σ + ε2a)
.
The first two of these last formulas, from (3.19) through the relation[Re
(l0l1
)]2+[
Im(l0l1
)]2= |l0|2|l1|2 , give the same expression for the square of σ as in (3.14). Hence
57
we have:
|l0|2 =1
4√
C(0)
(ε1σ + ε2a)
|l1|2 =1
4√
C(0)
(m2 + b2
ε1σ + ε2a
)
|l∗|2 =q2
√C(0) (ε1σ + ε2a)
.
Therefore, it is possible to write the lA vectors as:
l0 =1
2 4√
C(0)
Aeiψ
l1 =1
2 4√
C(0)A(ε1m + ε2ib) eiψ
l∗ =1
4√
C(0)Aqeiγ
,
where A.=√
ε1 σ + ε2 a is constant. Using the rescaling freedom mentioned at the be-
ginning of this section, and choosing l0 = 0 , then ψ = 0 , and C(0) = A4/16 . About γ
phase, recalling that q2 = e2 + g2 , it is naturally defined by e = ε3q cγ and g = ε4q sγ .
It results that all sign indicators ε# are inessential regarding the final result, hence they
are set in the following way: ε1 = ε3 = +1 , ε2 = ε4 = −1 , from which it follows:
C(0) =(σ − a)2
16.
In this way, we have obtained the expressions of the lA vectors in terms of physical
parameters:
l0 = 1
l1 =m− i b
σ − a
l∗ = 2e− i g
σ − a
(3.26)
Comparison of ga1 components.
The ensuing comparisons of ga1 with the corresponding components of McGuire-Ruffini
solution are just a way to verify completely the identification hypothesis between the
58
two solutions, since all the freedom available have been fixed. It is worth noting that
this identification could be valid at least of a linear coordinate transformations, involving
just the t and ϕ coordinates, and preserving the values of g00 component that we have
used to set the (3.26) relations. Calculi for these components are much more laborious
respect those already performed and will not be reported here. The calculus for g01
component gives a results which is a linear combination with constant coefficients of g00
and g01 components of McGuire-Ruffini solutions. This suggests the following coordinate
transformations (which leaves g00 component unchanged)6:
dt = dt′ + E dφ′
dφ = dφ′, where E =
q2
σ − a− 2a . (3.27)
It is easy to check that values of the E factor coincides, for the one-soliton case, with
the general formula, given at page 239 in [3], valid for a generic n-soliton solution. We
rewrite it here, according to our notation and bearing in mind that it depends on the
choice l0 = 1 ,
En = −i
n∑
k,l
γ−1kl
(1 + l(k)1 l(l)1
), where γkl =
4− 4 l(l)1 l(k)1 + l(l)∗ l(k)∗
4 (wl − wk). (3.28)
The transformation (3.27) changes the components gab and Aa in:
g′00 = g00
g′01 = E g00 + g01
g′11 = E 2g00 + 2 E g01 + g11
(3.29)
A′0 = A0
A′1 = E A0 + A1 .
(3.30)
As a final verification, the calculus of g11 inserted together with g00 and g01 into the
third of (3.23) yields g′11 . It results to be just equal to the corresponding component
given by (3.17).
3.2.4 Components of the electromagnetic potential.
To complete the construction of the one-soliton solution, it remains to calculate, from the
equations (3.24), the Aa components and then, through the (3.30), the electromagnetic
6This coordinate transformation can be obtained, independently from the comparison with theMcGuire-Ruffini solution, also by an asymptotical analysis of the behaviour of g01 .
59
potential A′a . Again, we skip the description of calculus giving directly the final result:
A′0 =
e r + g (b + a c θ)
Σ
A′1 =
me + b g
σ − a− 1
Σ g r2 c θ + e r (a s2θ − 2 b c θ) + g [ (a2 − b2) c θ + a b s2θ ]
(3.31)
where we recall that Σ = r2 + (b + a c θ)2 .
Chapter 4
A perturbative approach forstationary axially symmetric solitonsolutions
4.1 Some preliminary remarks.
It is important to note that in the one-soliton solution, given by (3.15)-(3.17) and (3.31),
σ appears only inside a constant addendum of A′1 which is physically irrelevant for the
determination of the electromagnetic fields; being always present in its second power
elsewhere, using the relation (3.14) it disappears. Because of this, we can ignore the
constrains to be imposed on the physical parameters to assure the reality of σ , looking
just at the final result. In any case this is possible just for one-soliton solutions since the
multi-soliton solutions will contain mixed products σkσl with k 6= l . The difficulties
in performing an analytical continuation to remove the constrains among the physical
parameters is a limitation of the soliton technique.
To illustrate the physical meaning of this limitation, we can consider, as simple example,
the solution of Reissner-Nordstrom. It is obtainable, as a particular case, from the
general one-soliton solution, leaving different from zero only the mass and electric charge
parameters. We thus have:
g00 = −1 +2m
r− e2
r2σ =
√−m2 + e2 .
Since, to assure σ to be real and different from zero then −m2 + e2 > 0 , it follows that
there are no values of coordinate r for which g00 changes its sign. This implies the
absence of an horizon and that the solution does not describe a blackhole but a naked
singularity or, in other words, as called conventionally in literature, a superextremal part
60
61
of Reissner-Nordstrom solution1.
Therefore a multi-soliton dressing will be able to produce only a superposition of super-
extremal sources which represents a weakness of this generating technique.
Apart from the reduction of the generality of the solutions that can be obtained, a
deeper meaning of this limitation appears clearer under the following additional physical
considerations.
Let us consider a two-soliton solution, representing the superposition of two Reissner-
Nordstrom like sources, that is, of two charged masses. The resulting metric will depend
on five independent parameters: the two masses m1 , m2 , the two charges e1 , e2 and
the distance d between the sources. It will obviously belong to the class of kind (3.1).
Moreover, because of the absence of any angular momentum, it will also be diagonal,
that is, static. In accordance with the thinking of classical physics, to have two sources in
static equilibrium, it is necessary to have a compensation between the attractive gravita-
tional force and the repulsive electrostatic force. In natural units, this condition reads as
the equality between the product of the two masses and the product of the two charges:
m1 m2 = e1 e2 . (4.1)
The relativistic generalization, given in [4], of the (4.1) is more complicate. Because of
the nonlinearity of the equations, it depends also on the distance parameter. At the
classical limit of weak fields, for great values of the parameter of distance2, the (4.1)
is recovered. This implies that the system has to be composed either by two extremal
sources or by a subextremal and a superextremal source, that is by a black hole and a
naked singularity.
The fact that the two-soliton dressing could yield fields generated by two superextremal
objects has as a consequence that, on the points of the segment of the axis between the
two sources, the elementary flatness of the solution is violated and not that the final
result is not a solution of Einstein-Maxwell equations. In other words the points of that
segment are conical singularities. That is, contracting a small circle, linked together with
the axis of symmetry ρ = 0 , to one of the points of the axis placed between the sources,
the limit of the ratio of the length of the circle to its radius times 2π does not go to the
unity. Different words can be found in literature to denote this kind of singular line as
strings, props, struts, rods, since to them is possible [54], [81] to associate a particular3
1The cases for −m2 + e2 = 0 and −m2 + e2 < 0 are called respectively extremal and subextremal,corresponding this last one to a black hole solution.
2It is worth noting that the equilibrium constrains give the possibility to have some particularconfiguration peculiar of the relativistic regime for which this limit does not exist.
3In General Relativity a stress generates gravitational fields.
62
kind of stress of topological nature, not realizable in nature4, which, even if does not
generate gravitational fields, balances the reciprocal actions between the sources which
are not able, by themselves, to remain in static equilibrium. This stress can be described
in terms of a force which shows, depending on the sign of the deficit of angle of the con-
ical singularity, a repulsive or attractive character. We will come back again on this to
illustrate in detail this connection in the chapter dedicated to the analysis of the stability
of the double Reissner-Nordstrom solution of Alekseev and Belinski.
It is now clear the weakness of the soliton dressing procedure. It would obviously be
preferable to dispose of a generating technique capable to yield regular electrovacuum
solutions and therefore the possibility to remove such kind of spurious singularity, giving,
in such a way, the balancing condition among the parameters of the solution.
It is important to take into account another important aspect. We have found the
reparameterization formulas (3.26) and the characterization of the imaginary part of the
poles (3.14), comparing directly the one-soliton generated solution with the known so-
lution of McGuire-Ruffini. To construct a two-soliton solution we could use the same
formulas for each soliton5. This will leads to an exact solution of the Einstein-Maxwell
equations in terms of the eleven parameters mk , bk , ak , ek , gk , plus the distance
between the two sources d . But they will loose their individual physical meaning due
to the non linearity of the interaction. Therefore, to obtain an interpretable final so-
lution either additionally considerations will be necessary or it needs to start from an
alternative reparameterization.
4.2 A proposal for a perturbative approach.
On the line of the considerations mentioned just above, hereafter we will describe a
proposal for a perturbative approach to look for hints to overcome some, or possibly
each one, of the difficulties presented by the solitonic technique. The motivation for this
attempt lies on considerations [10] concerning, for example, the experience coming from
the deduction of the Alekseev-Belinski solution [5]. The discrepancy between the rather
cumbersome way6 to obtain that solution and the rather surprising simplicity of the final
result suggests that the same result could be reached, in a simpler and direct way, by
means of the dressing procedure. Procedure that, then, could be applied to deal with
more general problems as that concerning the general balance conditions for two charged
4It does not correspond to any kind of gravitational mass.5Hence we have to restore in the notation the indexes to each parameter to indicate it refers to
which source or soliton.6We recall that Alekseev and Belinski did not use the dressing technique to find their solution.
63
and rotating sources.
We are going to construct a perturbative generating procedure based on the expan-
sion respect the Newton’s constant. We will denote it with γ . We recall that until
now we adopted geometrized units where the gravitational constant γ and the speed of
light c are set equal to one. Now, keeping again c = 1 , we have to reintroduce in its
right place γ . Anyway, for formal reasons we will explain soon after, we will work with
the square root of γ for which we will use the symbol κ . Relatively to the physical
(and unphysical) parameters with which we are dealing with, this can be done simply
performing the following substitution:
m → κ2 m, b → κ2 b , a → a , e → κ e , g → κ g , (4.2)
where
κ =√
γ .
We could perform the expansion in the κ parameter directly on the final product of
the generating technique7. Anyway we would find some hint to overcome the problems
carried by the solitonic technique. Resuming them here, they are: the reparameteriza-
tion between mathematical and physical constants, the characterization of the complex
poles and their analytical continuation. To solve these problems, it could be necessary
to modify the dressing procedure even at the level of the Lax Pair. We recall that the
proposal of Alekseev is not the only one. For this reason we will deal with the expansion
starting from a deeper level or the dressing procedure.
In this chapter, we will give an outline of the dressing procedure expanded in κ . We
can schematically separate it in two moments. One concerning with the expansion of the
generating matrix function ϕ , the other one regarding the expansion of the poles and of
the constants quantities. In the next chapter, we will apply this perturbative scheme to
dress a flat background. In this case, we can see that all the terms of the approximate
terms of the generating matrix will be formally equals.
7These expansions will present obviously only terms corresponding to even powers in κ ; that is,they will result to be expansions in γ .
64
4.3 Outline of the perturbative solitonic generating
technique.
Hereafter, we will adopt the following notation to denote the terms of each expansion:
f =0
f + κ1
f + κ22
f + ... .
4.3.1 Expansion of Einstein-Maxwell Equations.
Introducing the Newton’s constant γ in the Einstein-Maxwell equations, we have:
ηµν 1
α
(α gbcgac,µ
),ν
= −2 γ gbcηµνΦa,µΦc,ν
ηµν (α gacΦc,µ),ν = 0
. (4.3)
Remembering the expression (2.40) for the matrix X , and noticing that its out of di-
agonal block elements depend linearly by the electromagnetic potential, it is convenient
to introduce the control parameter κ . Hence we can read the (4.3) as the result of the
formal substitution Φ → κ Φ . This implies that, even if it should be natural to consider
just the expansions of the potentials in γ , from now on, with the introduction of the
control term κ , we will deal with even power expansions for the gravitational potentials
gab and odd power expansion for the electromagnetic potentials Φ . Hence, instead to
write
Φ → κ Φ = κ (0
Φ + γ2
Φ + ...)
changing the notation, we will directly put:
Φ → κ1
Φ + κ33
Φ + ... .
Noticing that α = ρ and hence α ≡ 0α , the equations (4.3) therefore splits, up to the
fifth order in κ , into the following systems:
ηµν 1
α
[α
0g bc 0
gac,µ
],ν
= 0
ηµν
(α
0g ac
1
Φc,µ
)
,ν
= 0 ,
(4.4)
65
ηµν 1
α
[α
(0g bc 2
gac,µ +2g bc 0
gac,µ
)],ν
= −2 ηµν 0g bc
1
Φa,µ
1
Φc,ν
ηµν
[α
(0g ac
3
Φc,µ +2g ac
1
Φc,µ
)]
,ν
= 0 ,
(4.5)
ηµν 1
α
[α
(0g bc 4
gac,µ +2g bc 2
gac,µ +4g bc 0
gac,µ
)],ν
=
= −2 ηµν
(0g bc
1
Φa,µ
3
Φc,ν +0g bc
3
Φa,µ
1
Φc,ν +2g bc
1
Φa,µ
1
Φc,ν
)
ηµν
[α
(0g ac
5
Φc,µ +2g ac
3
Φc,µ +4g ac
1
Φc,µ
)]
,ν
= 0 .
(4.6)
It is worth noting that, starting from the assumption of a decoupled problem between
the gravitational fields and electromagnetic fields, the0g potentials can be taken as a
generic solution of Einstein equations in vacuum given by the first of (4.4). Therefore the
equations of successive orders will give the first corrections for week interactions between
the two fields over this fixed generic background solution.
About the terms likekg ab , we recall that given a generical matrix M and its perturbative
representation M =0
M + κ1
M + κ22
M + ... , thenk
M −1k
M 6= I . That isk
M −1 6= (k
M)−1 .
The termsk
M −1 are instead defined by the condition:
M−1M = (0
M −1 + κ1
M −1 + κ22
M −1 + ...)(0
M + κ1
M + κ22
M + ...) = I . (4.7)
4.3.2 Lax Pair expansion.
Rewriting the Lax Pair (2.24) in the simplified manner
Πµψ = Aµψ , 8 (4.8)
and looking at the equations (2.27) and (2.31), it is clear that we have to expand ψ and
Aµ . Because of this expansion, the system (4.8) will split, at least of O(κ2) , into the
8Hence Aµ =e
λ2 − e α2
(λ ε σ
µ Kσ − α Kµ
).
66
following one:
Πµ
0
ψ =0
Aµ
0
ψ
Πµ
1
ψ =0
Aµ
1
ψ +1
Aµ
0
ψ .
(4.9)
It can be rewritten in the more compact notation as:
Πµψ = Aµ ψ , (4.10)
if
ψ.=
0
ψ 0
1
ψ0
ψ
and Aµ
.=
0
Aµ 0
1
Aµ
0
Aµ
. (4.11)
Notice that these are 6 × 6 matrixes. Hereafter, with the hat put on a symbol, we
will mean its expanded version without any specification about the order of expansion.
Hence, for example, with ψ we mean the 3(N + 1)× 3(N + 1) lower triangular matrix
ψ =
0
ψ1
ψ0
ψ 02
ψ1
ψ0
ψ...
......
. . .N
ψN−1
ψ . . . . . .0
ψ
,
if N is the order of expansion. When the structure of the covered9 symbols will have
some different structure from the above one, it will be specified explicitly. Therefore the
Kµ matrices present inside the Aµ coefficients of the (4.8) will have to be obviously
taken in their “hat” version inside Aµ .
It is now easy to represent the expanded compatibility conditions for the system (4.10)
as
ηµν(α X,µX
−1)
,ν= 0 . (4.12)
At the fifth order, they will reproduce the systems (4.4)-(4.6) if X has the structure
given by (2.40). The block components of X−1 are determined as described in the
example (4.7).
We recall that the dressing procedure work in general not only for 3 × 3 matrix but it
can be naturally extended to matrices of any dimension. This is the reason for which the
“hat” notation will straightforwardly lead to the expanded version of the exact generating
9I.e. dressing the hat.
67
procedure.
Hence, to pass from the expansion of the Lax Pair (2.24) to that of the (2.47), it is
sufficient to use “hat” version of the definition (2.41):
ψ =(X − 4 i Ω
)ϕ ,
where Ω = diag(Ω, Ω, ...) . Being also
Uµ = i e α ηµρερσX−1X,σ + 4 e (α2 X−1),µΩ ,
then we have:
∂ ϕ
∂xµ=
1
2 i
[w + β
(w + β)2 − e α2Uµ +
e α
(w + β)2 − e α2ηµρ ερσUσ
]ϕ . (4.13)
4.3.3 Expanded dressing procedure.
Now, taking a background solution ϕ(0) of the system (4.13), we can obtain the perturbed
dressed generating matrix through:
ϕ = χ ϕ(0) .
Here it is useful to specify the structure of the χ and χ−1 matrices. Their are given by:
χ = I +n∑
k=l
(1
w − wk
)Rk ,
χ−1 = I +n∑
k=l
(1
w − wk
)Sk .
The scalar factors are given by
(1
w − wk
)=
1
w − 0wk
+ κ
1wk
(w − 0wk)2
+ ... ,
(1
w − wk
)=
1
w − 0wk
+ κ
1wk
(w − 0wk)2
+ ... ,
since we have to expand in κ the poles too.
If we have two proportional matrices A and B through a scalar k such as A = kB ,
then for the expansion we have A = kB , where just up to the first order as an example:
k =
0
k 01
k0
k
and A =
0
k0
B 00
k1
B +1
k0
B0
k0
B
.
68
For what concerns the R and S matrices, we have that:
(Rk)BA = n
(k)A m(k)B , (Sk)
BA = p
(k)A q(k)B .
To have the right structure of these matrices, n(k)A , p
(k)A and and m(k)A , q(k)A are respec-
tively 3N ×N and N × 3N matrices, since the corresponding exact ones n(k)A and p
(k)A
are three dimensional column vectors and m(k)A and q(k)A are three dimensional row
vectors.
Therefore, taking as an example n(k)A and m(k)A we have for them, up to the first order,
the following structure:
n(k)A =
0n
(k)A 0
1n
(k)A
0n
(k)A
, n
(k)A =
0n
(k)A + κ
1n
(k)A + ... ,
m(k)A =
0m(k)A 0
1m(k)A 0
m(k)A
, m(k)A =
0m(k)A + κ
1m(0)A + ... .
Now, we dispose of the scheme which will enable us to construct the approximate terms
of soliton solutions.
Chapter 5
Generation of approximate solitonsolutions over a flat background.
In this chapter, we apply the perturbative methods, described in the previous chapter,
to generate the lower order solitonic corrections on a Minkowsky background space-time
together with a null background electromagnetic field. Note that the perturbative scheme
permits us to separate the dressing of the two fields. Hence the choice of null back-
ground electromagnetic field is not a consequence of the flatness of the background
space-time. That is, we could also take a non null electromagnetic fields decoupled with
the Minkowsky background as starting point. In the same manner, this freedom makes
the perturbative approach interesting for more general applications, as the generation of
pure electromagnetic soliton perturbed solutions on a curved seed space-time.
After an introductive section, where we will present the perturbed version of the for-
mulas relative to the final steps of the dressing procedure, we will apply the perturbed
procedure to one-soliton case. What we expect is to generate exactly the same perturbed
terms we could find by taking the exact one-soliton solution, presented in the third chap-
ter, and performing directly on it the expansion in the γ parameter; therefore it will
only play the role of an useful verification. Then, in the last section, we will start to
generate the approximate terms of the two-soliton solution. About them, we will only
give the results relative to the lowest order expansion, since the work about the first (in
γ ) or second and third (in κ ) order corrections is still in progress. The results of these
preliminary calculi, even if obviously yield no new physically relevant results, consist in
the reparameterization to obtain the usual complex representation of week electromag-
netic fields of two rotating charged sources on Minkowsky background.
For what we specified in section 4.2 at the end of page 62, the perturbative procedure
can be separated in two phases. The first one regarding the expansion of the generating
matrix, i.e. the determination of the components of the ϕ(0) matrix, the second one the
69
70
expansions of the poles and, as a consequences, the expansions of all that intermediate
quantities depending on them, by which the final fields are constructed. The choice of
a flat background, joined with the fact that the two-soliton solution will be constructed
by a simultaneous double dressing and not by an iterated dressing, will reduce our work
only to the second phase regarding the expansions of the poles. Therefore, using, as a
frame, the same enumeration of the perturbative procedure as that used for the exact
one, we can start directly from the step number 8 . All the results relatives to the previ-
ous ones (together with the choices of all the arbitrary constants), are exactly equal to
those found in the second chapter for the one-soliton exact solution.
5.1 Perturbative building block quantities
Here, we give the basic formulas to generate the approximate terms of the soliton solu-
tions at least up to the third order in κ . Since our main task is to construct then the
approximate terms of two-soliton solution, we leave the soliton indexes, intending them
running from 1 to 2. Hence, recalling the index notation we have:
i, j, h, k, ... = 1, 2 ; a, b, c, ... = 0, 1 ; A, B, C, ... = 0, 1, ∗ .
In the following formulas the parameterizations (3.14) and (3.26), extended to each
soliton, will not be assumed as in the n-soliton solution presented in [3]. The only
generally valid assumption that we can keep is to take l0k = 1 . Otherwise, we use them
just for dimensional considerations. In fact, their expansions suggests that both those of
the imaginary parts of the poles σk and those of the vectors l1 present only even order
terms, while l∗ only odd ones.
The set of expressions for all the approximate terms, listed hereafter, will be useful as a
formulary for algebraic computer computations.
Bipolar coordinates expansion
In general, for multi-soliton solution is still convenient to adopt multipolar coordinates
(rk, θk)1, of the kind (3.5). Thus, we have for each soliton:
ρ =√
R2k + σ2
k
√1− y2
k
z = Rk yk + ζk
, where
Rk = rk −mk
yk = cos θk
. (5.1)
1These systems of coordinates are obviously redundant.
71
In particular, for two solitons we can choose:
ζ1 = −d
2, ζ2 = +
d
2. (5.2)
Expansions of the poles, of the λk and Γk functions and of the lA vectors
Recalling the definitions (3.8) and (3.9), that is
λ(w) = z − w , Γ(w) =√
ρ2 + λ(w)2 ,
we denote their values on the poles wk = ζk + i σk , with
λk.= λ(w)|w=wk
, Γk.= Γ(w)|w=wk
,
therefore:
λk = Rk yk − i σk , Γk = Rk − i σk yk .
For the imaginary parts of the poles expansions we have, as metioned above, only even
order terms, i.e.
σk =0σk +
2σkκ
2 + O(κ4) ,
Therefore the expansions of the poles gives:
0wk = ζk + i
0σk ,
1wk = 0 ,
2wk = i
2σk ,
3w = 0 . (5.3)
The perturbed terms of the λk and Γk functions are respectively:
0
λk = rk yk − i0σk ,
1
λk = 0 ,2
λk = −mk yk − i2σk ,
3
λk = 0 ; (5.4)
0
Γk = rk − i0σk yk ,
1
Γk = 0 ,2
Γk = −mk − i2σk yk ,
3
Γk = 0 . (5.5)
For the lAk complex vectors we have:
l0k = 1 , l1k =2
l1k κ2 + O(κ4) , l∗k =1
l∗k κ +3
l∗k κ3 + O(κ4) ,
and therefore:
1
l 0 = 0 ,2
l 0 = 0 ,3
l 0 = 0 ;0
l 1 = 0 ,1
l 1 = 0 ,3
l 1 = 0 ;0
l ∗ = 0 ,2
l ∗ = 0 .
72
Expansions of mA(k) and p(k)A vectors
As prescribed by the step number 8, now we have to expand the mA(k) and p(k)A com-
plex vector functions as given by their respective expressions (2.64) and (2.65). In the
following formulas we will omit the soliton index k .
0mA :
0m0 = 4
0
Γ ,0m1 = 0 ,
0m∗ = 0 ; (5.6)
0pA :
0p0 =
10
Γ
,0p1 = i
0
λ0
Γ
,0p∗ = 0 . (5.7)
1mA :
1m0 = 0 ,
1m1 = 0 ,
1m∗ =
1
l∗ ; (5.8)
1pA :
1p0 = 0 ,
1p1 = 0 ,
1p∗ =
1
l∗ . (5.9)
2mA :
2m0 = 4
2
Γ + 4 i0
λ2
l1 ,2m1 = −4
2
l1 ,2m∗ = 0 ; (5.10)
2pA :
2p0 = −
2
Γ0
Γ 2
,2p1 = i
2
λ0
Γ
−0
λ2
Γ0
Γ 2
+
2
l1 ,2p∗ = 0 . (5.11)
3mA :
3m0 = 0 ,
3m1 = 0 ,
3m∗ =
3
l∗ ; (5.12)
3pA :
3p0 = 0 ,
3p1 = 0 ,
3p∗ =
3
l∗ . (5.13)
Expansions of Tkl and (T−1)kl matrices
Recalling the formula (2.57), we had that:
Tkl =p
(k)A m(l)A
wl − wk
Putting Nkl = p(k)A m(l)A and Wlk = wl− wk then: Tkl =
Nkl
Wlk
, or, omitting the indexes,
simply T =N
W. Being the expansions of N and W given by:
0
Nkl =0pA
(k) 0m(l)A ,
0
W lk =0wl −
0wk ,
1
Nkl =0pA
(k) 1m(l)A +
1pA
(k) 0m(l)A ,
1
W lk =1wl −
1wk ,
2
Nkl =0pA
(k) 2m(l)A +
1pA
(k) 1m(l)A +
2pA
(k) 0m(l)A ,
2
W lk =2wl −
2wk ,
3
Nkl =0pA
(k) 3m(l)A +
1pA
(k) 2m(l)A +
2pA
(k) 1m(l)A +
3pA
(k) 0m(l)A ,
3
W lk =2wl −
3wk ,
,
73
it is easy to see that both1
N ≡3
N ≡ 0 and1
W ≡3
W ≡ 0 , thus we have only even order
terms of the T matrix expansion:
0
T =
0
N0
W
,2
T =
2
N0
W
−2
W0
W
0
T , while1
T = 0 ,3
T = 0 .
Therefore, the terms of the expansion of the inverse matrix T−1 are, up to the fourth
order in κ , only0
T−1 and2
T−1 . This last one reduces to:
2
T−1 = −0
T−12
T0
T−1 .
Expansion of S BA matrix
Again, to lighten the notation, we rewrite the formula (2.88)
S BA = −
n∑
k,l=1
(T−1)kl p(l)A m(k)B ,
omitting all indexes, but leaving unchanged the order of the factors to remember then,
easily, where to reinsert them in the right place; hence we write simply:
S = −T−1p m .
The eliminations of all null terms leads to the following reduced expressions:
0
S = −0
T−1 0p
0m,
1
S = −0
T−1(
0p
1m +
1p
0m
),
2
S =
(0
T−12
T0
T−1
)0p
0m−
0
T−1(
0p
2m +
1p
1m +
2p
0m
),
3
S =
(0
T−12
T0
T−1
) (0p
1m +
1p
0m
)−
0
T−1(
0p
3m +
1p
2m +
2p
1m +
3p
0m
).
Expansions of metric tensor and electromagnetic potential
Inserting the terms of the expansions of mA and pA vectors into the formulas for the S BA
matrix, it results that some of its components are null. The resulting general expressions
for the corrections of the metric tensor components, as given by the dressing and hence
74
before the final coordinates transformation (3.27), are
0g00 = −1 + 2 Im(
0
S01)
0g01 = i
[0
S00 +
0
S11
]
0g10 = −i
[0
S11 +
0
S00
]
0g11 = ρ2 − 2 Im(
0
S10)
,
2g00 = 2 Im(
2
S01)− 4
1
S0∗
1
S∗0
2g01 = i
[2
S00 +
2
S11
]− 4
1
S0∗
1
S∗1
2g10 = −i
[2
S11 +
2
S00
]− 4
1
S1∗
1
S∗0
2g11 = −2 Im(
2
S10)− 4
1
S1∗
1
S∗1
.
It is worth noting that only the even order terms are present in the above final set of
formulas. It is, in fact, easy to verify that the righthand sides of the relations for the
odd terms, are all identically zero. Hereafter we write only those about the first order
corrections:1g00 = Im(
1
S01) ≡ 0
1g01 = i
[1
S00 +
1
S11
]− 4
1
S0∗
0
S∗1 ≡ 0
1g10 = −i
[1
S11 +
1
S00
]− 4
1
S1∗
0
S∗0 ≡ 0
1g11 = −2 Im(
1
S10) ≡ 0
.
Recalling that the dressing technique yields directly a real and symmetric metric tensor,
we can use the above formulas for the corrections of the g01 and g10 components as a
good test for the calculi. In particular, in our simple 0-order case, it is very easy to see
that such conditions are verified. For the second order, the relation equivalent to both
such conditions (equating the righthand side of2g01 and
2g10 ) is:
Re (2
S00 +
2
S11)− 4 Im (
1
S0∗ 1
S1∗) = 0 .
Anyway, it is to say that the usage of this equation to perform this test in terms of thek
lA vectors is not convenient since, already to the second order, it reduces to a very not
trivial expression. Instead, it is much more convenient to equate directly the final results
of the corrections of the out of diagonal components of the metric tensor and perform
some numerical test, after having chosen some random numerical values for the constants.
75
The formulas for the corrections of the electromagnetic potentials components are:
1
A0 = −2 Im (1
S0∗) ,
1
A1 = −2 Im (1
S1∗) ;
3
A0 = −2 Im (3
S0∗) ,
3
A1 = −2 Im (3
S1∗) .
From these components, as a final step, we will have to perform the coordinate transfor-
mation defined by (3.27). We have obviously to expand such transformations too. Hence,
from the defining relations (3.28), for the constant En appearing in such transformation,
we will obtain the correctionsk
En . Thus, from the expansions of the righthand side of
the (3.23) and (3.30), constructed by means of the combinations of these constant terms,
we will obtain the corrections for the fields components for each desired order.
The corrections for the bidimensional conformal factor f are:
0
f =0
C(0)
0
f (0)0
T0
T ,2
f =0
C(0)
[2
f (0)0
T0
T + 20
f (0) Re
(0
T2
T
)]+
2
C(0)
0
f (0)0
T0
T ,
Where now, we recall, T is the determinant of the Tkl matrix. For flat background,
being f (0) = 1 , we simply have:
0
f =0
C(0)
0
T0
T ,2
f = 20
C(0) Re
(0
T2
T
)+
2
C(0)
0
T0
T .
5.2 One-soliton approximate solution.
For this first deduction2 (which, we recall, is here performed only as a verification), we
keep the assumptions given by the (3.14) and (3.26) formulas. Hence, to determine the
perturbative terms of the m(k)A and p(k)A vectors, it is before necessary to expand σ
and the lA vectors, according to their determinations (3.14) and (3.26). Moreover, we
equate to zero the unphysical constants b and g . For this case of one-soliton solution,
we will preform the expansion up to O(κ4) . From (3.14) we have:
σ = ±(|a|+ 1
2
e2
|a| κ2
)+ O(κ4) .
Since, from the expansion of the lA vectors, we can see that their perturbative terms are
proportional to (±|a| − a)−1 , then we have to replace ±|a| with −a . The first terms
2Dealing with only one soliton, we will omit, in this section, to write the soliton index.
76
of the expansion of σ are then:
0σ = −a ,
2σ = −1
2
e2
a.
Therefore, since for one-soliton solution we put again ζ ≡ Re (w1) = 0 , the perturbative
terms of the unique pole are:
0w = −i a ,
2w = −i e2
2 a.
For the lA vectors we have:
0
l 0 = 1 ,1
l 0 = 0 ,2
l 0 = 0 ,3
l 0 = 0 ,
0
l 1 = 0 ,1
l 1 = 0 ,2
l 1 = −1
2
m
a,
3
l 1 = 0 ,
0
l ∗ = 0 ,1
l ∗ = −e
a,
2
l ∗ = 0 ,3
l ∗ =e3
4 a3.
While, for the expansions of λ and Γ functions, we obtain:
0
λ = r c θ + i a ,2
λ = −mc θ +i e2
2 a,
0
Γ = r + i a c θ ,2
Γ = −m +i e2
2 ac θ .
Hereafter we will list in a schematic manner the principal expressions and the results for
each order. We will denote withkg′ab and
k
A′a , the corrections of the dressed components,
withkgab and
k
Aa , the final components obtained by the transformation of coordinates
(3.27), defined by the constant (3.28) denoted with E1 =0
E1 +2
E1 κ2 + O(κ4) .
———————–
• 0-order0
T and0
S matrices
0
T =2
0
Γ
i0σ
0
Γ
0
SAB =
−2 i0σ 0 0
20σ
0
λ 0 0
0 0 0
77
Metric tensor components0gab
0g′00 = −1
0g′01 = 2
0σ
0g′11 = ρ2 − 4
0σ2
and, with0
E1 = 20σ , then
0g00 = −1
0g01 = 0
0g11 = ρ2
0
f =0
C(0)
0
f (0)0
T0
T =0
C(0)
0
f (0) 40σ2
Choosing0
C(0) =0σ2/4 = a2/4 , then
0
f = 1 .
Electromagnetic potential components0
Aa
Since0
Sa∗ = 0 then
0
Aa = 0 .
———————–
• 1-order1
T and1
S matrices
1
T = 01
SAB =
0 0 − i e
20
Γ
0 0e
0
λ
20
Γ
−2 i e0
Γ 0 0
Metric tensor components1gab
Since1
Sab = 0 and
1
T 11 = 0 , then1gab = 0 and
1
f = 0 .
78
Electromagnetic potential components1
Aa
1
A′0 =
e r
r2 + a2 c2θ
1
A′1 =
e a r (1 + c2θ)
r2 + a2 c2θ
1
A0 =1
A′0
1
A1 =0
E1
1
A′0 +
1
A′1
1
A0 =e r
r2 + a2 c2θ
1
A1 = − e a r s2θ
r2 + a2 c2θ
———————–
• 2-order0
T and0
S matrices
2
T =
0
Γ
2 i0σ
0
Γ2
(4 m− e2 r
a2+
3 i e2 c θ
a
)
In the following expressions for the components of2
SAB we put y = c θ .
2
S00 =
1
2 (r2 + a2y2)2 [m (r2 − a2) y − e2ry] + i [e2r2 + 3e2a2y2 − 2 m a2r(1 + y2)]
2
S01 =
m (a y + i r)
r2 + a2y2
2
S10 =
1
2 (r2 + a2y2) a
[−e2r3 + 2 ma2r2(3 + y2)− e2a2r (2 + 3y2)− 2 ma4(1− y2)]y+
+i a[−2 mr3(1− y2) + e2r2(3− 2y2)− 3 ma2r (2 + 3y2) + 5 e2a2y2
]
2
S11 =
1
r2 + a2y2[−m (r2 − a2) y + im a (1 + y2)]
2
Sa∗ = 0
79
Metric tensor components2gab
2g′00 =
2 mr − e2
r2 + a2y2
2g′01 =
−e2r2 + 4 ma2r (1 + y2)− e2a2(2 + 3y2)
2 (r2 + a2y2) a
2g′11 =
−2 e2r2 + 2 ma2r (1 + y2)2 − e2a2(1 + 4y2 + y4)
r2 + a2y2
Thus, since the terms of the expansion of the constant E1 for the transformation of
coordinate are0
E1 = −2 a ,1
E1 = 0 ,2
E1 =−e2
2 a,
then:2g00 =
2 mr − e2
r2 + a2c2θ
2g01 = −(2 mr − e2) a (1− c2θ)
r2 + a2c2θ
2g11 =
(2 mr − e2) a2(1− c2θ)2
r2 + a2c2θ
Choosing2
C(0) = e2/8 , then
2
f =2 mr − e2c2θ
r2 + a2c2θ
Electromagnetic potential components2
Aa
Since2
Sa∗ = 0 then
2
Aa = 0 .
———————–
80
• 3-order3
T and3
S matrices
1
T = 0 ,
3
Sab = 0
3
S0∗ = 0
3
Sa∗ =
e
4 (r2 + a2y2)a−e2y + i [m (r2 + a2y2)− e2r]
Metric tensor components1gab
Since3
Sab = 0 and
3
T 11 = 0 , then3gab = 0 and
3
f = 0 .
Electromagnetic potential components3
Aa
3
A′0 = 0
3
A′1 = − e
2 (r2 + a2y2) a[m (r2 + a2y2)− e2r]
Since3
E1 = 0 and thus
3
A1 =3
A′1 +
0
E1
3
A′0 +
2
E1
1
A′0 then
3
A0 = 0
3
A1 = −em
2 a
———————–
It results that all the boxed formulas found above are just equal to the corresponding
perturbative terms obtainable through the expansion of the exact one-soliton solution
described in chapter 3.
It is worth noting that the lowest order electromagnetic potential can be expressed as
the real part of the Lynden-Bell [60] potential3
Ψ = e/0
Γ =e
r + i a c θ. (5.14)
3It is the same of the Ernst electormagnetic complex potential for the Kerr-Newman solution [83].
81
It is easy to see that Ψ =0
Φ0 and hence that, since
0
A0 = Re Ψ and0
B0 = Im Ψ ,
the component0
A1 is deducible from Ψ by means of the integration of the system (2.17).
5.3 Two-soliton lowest order approximate solution
At the two-soliton case regard, we wish to underline the following fundamental aspect
of our strategy. Since the problems of it concerns the reparameterization and the char-
acterization of the poles, we will avoid to assume the same choices as in one-solitonic
case. Therefore, we reject the assumptions (made in [3]) to take the indexed two-soliton
versions of the (3.26) and (3.14). That is, namely, we will not use the definitions:
σ2k = −m2
k − b2k + a2
k + q2k , l1k =
mk − i bk
σk − ak
, l∗k = 2ek − i gk
σk − ak
, k = 1, 2 .
We will only preserve l0k = 1 .
The analysis of the two-soliton perturbed solution, limited to the lowest order, needs
only of starting formulas (5.3) for the poles, (5.4) (5.5) for the functions0
Γk ,1
Γk ,0
λk ,1
λk
and (5.6)-(5.9) for the vectors0m(k)A ,
0p
(k)A and
1m(k)A ,
1p
(k)A . The definitions (5.1) and
(5.2), concerning the coordinates and the positions of the poles on the axis, are obviously
assumed. Hereafter we will use the notation ξk = Re1
l∗k , ηk = Im1
l∗k . Therefore, the
only task of this lowest order analysis is to find the expressions to represent0σk and
ξk , ηk in terms of ak , ek4. These expressions will be deduced imposing the linear
superposition of the electromagnetic potentials as expressed by the (5.14), namely:
Ψ2−sol = Ψ1 + Ψ2 ,
that is:
Ψ2−sol =e1
r1 + i a1 c θ1
+e2
r2 + i a2 c θ2
. (5.15)
Since1σk = 0 and
1
Γk = 0 , hereafter we will simply denote with σk and Γk respectively0σk and
0
Γk , thus
Γk = rk − i σk yk , where yk = c θk .
Besides, we introduce the symbols:
σ+ = σ1 + σ2
σ− = σ1 − σ2
, and
Σ+ = d + i σ+
Σ− = d + i σ−
.
4We recall that the mk mass parameters can enter only at least with the second order of expansion.
82
These notations simplify the building block expressions in such a way that we have:
0
T =
−2 i Γ1
σ1 Γ1
4 Γ2
Σ+ Γ1
− 4 Γ1
Σ+ Γ2
−2 i Γ2
σ2 Γ2
, thus det0
T = − 4
σ1 σ2
|Σ− |2|Σ+ |2
Γ1 Γ2
Γ1 Γ2
and hence
0
T−1 =i
2 |Σ− |2
σ1 |Σ+ |2 Γ1
Γ1
−2 i σ1σ2 Σ+Γ2
Γ1
2 i σ1σ2 Σ+Γ1
Γ2
σ2 |Σ+ |2 Γ2
Γ2
.
For the0
SAB matrix, which, we recall, is defined by
0
SAB = −
2∑
k, l =1
(0
T −1)k l0pA
(l) 0m(k)B ,
since it results that
0p
(l)
A
0m
(k)B
=
4Γk
Γl
0 0
4 i0
λlΓk
Γl
0 0
0 0 0
,
it follows, after some simplification, that:
0
S00 = −2 i σ+ ,
0
S10 = 2 z σ+ + d σ− + 2 i σ2
+ .
These values yields the following components of the metric tensor, which, obviously,
results to be flat:
0g′00 = −1
0g′01 = 2 σ+
0g′11 = ρ2 − 4
0σ+
2
, and with0
E2 = 20σ+ , then
0g00 = −1
0g01 = 0
0g11 = ρ2
.
The calculi for the1
Sa∗ components, since
1
S0∗ = −(
0
T −1)k l0p0
(l) 1m(k)∗ ,
1
S1∗ = −(
0
T −1)k l0p1
(l) 1m(k)∗ ,
83
gives
1
S0∗ =− i
2 |Σ− |2[ ( |Σ+ |2 − 2 i σ2 Σ+
) σ1 l∗1Γ1
+( |Σ+ |2 + 2 i σ1 Σ+
) σ2 l∗2Γ2
], (5.16)
1
S1∗ =
1
2 |Σ− |2[(|Σ+ |2
0
λ1 − 2 i σ2 Σ+
0
λ2
)σ1 l∗1Γ1
+
(|Σ+ |2
0
λ2 + 2 i σ1 Σ+
0
λ1
)σ2 l∗2Γ2
]. (5.17)
To impose the condition (5.15), we can use only the component1
S0∗ . In fact
Re(Ψ2−sol) =1
A0 = −2 Im(1
S0∗) , Im(Ψ2−sol) =
1
B0 = +2 Re(1
S0∗) .
Thus, from the conditions
Re
(e1
Γ1
+e2
Γ2
)= −2 Im(
1
S0∗) = Re (2 i
1
S0∗) ,
Im
(e1
Γ1
+e2
Γ2
)= +2 Re(
1
S0∗) = Im (2 i
1
S0∗) ,
that is
Ψ2−sol =e1
Γ1
+e2
Γ2
= 2 i1
S0∗ ,
it follows that:
0σk = −ak ,
ξ1 = − d2 + a+a−d2 + a2
+
e1
a1
η1 = − d (a+ − a−)
d2 + a2+
e1
a1
,
ξ2 = − d2 − a+a−d2 + a2
+
e2
a2
η2 = +d (a+ + a−)
d2 + a2+
e2
a2
. (5.18)
It is possible to check that these results are coherent with the1
S1∗ component. In fact
substituting (5.18) in (5.17) it follows that
1
S1∗ =
1
2
[(0
λ1 − 2 i a2
)e1
Γ1
+
(0
λ2 − 2 i a1
)e2
Γ2
].
Moreover, we have1
A1 =0
E2
1
A0′ +
1
A1′ = −2 Im(
0
E2
1
S0∗ +
1
S1∗)
and finally1
A1 = −Im
(0
λ1e1
Γ1
+0
λ2e2
Γ2
).
84
It is now easy to verify that this expression is the same as that deducible by means of
the integration of the system (2.18) which, for this component, becomes
1
A1,ρ = −ρ1
B0,z
1
A1,z = ρ1
B0,ρ
.
Equilibrium conditions
The equilibrium condition can now be found looking for the distance for which the
interaction potential assumes extremal values. It can be expressed as the opposite of the
total Lagrangian of the system
Vint = −L =1
16 π
∫Fij F ij dV . (5.19)
The scalar invariant in the above volume integral is constructed with the non null co-
variant and controvariant components of electromagnetic tensor field Fµa = Aa,µ and
F µa =1
fηµνgac Ac,ν . The integral (5.19) gives5
Vint =e1e2 d
d 2 + a 2+
.
Therefore, the distances between the two sources for which this potential assumes ex-
tremal values is
d = ±a+ . (5.20)
These distances will correspond to stable or unstable configurations depending by the
signs of the charges and the magnitudes of the angular momentums.
Substituting the values given by (5.20) into the expressions (5.18) it results:
1
l∗1 = −e1
a1
(a1 ± i a2
a+
),
1
l∗2 = −e2
a2
(a2 ∓ i a1
a+
).
5This result is present in [79]; as pointed by its author, it is worth to mention, about this calculation,L. Samuelsson.
Chapter 6
Electric force lines of the doubleReissner-Nordstrom exact solution
6.1 Introduction
The new solution which has been recently found by Alekseev and Belinski [4] (in the
following denoted with AB) has solved the long standing problem of the static equilibrium
of two charged masses in the context of General Relativity (GR).
While in the Newtonian theory the equilibrium condition is simply m1m2 = e1e2 , in the
relativistic regime the problem is much more complicate because one has to solve the
full system of the Einstein-Maxwell equations:
Rij − 12R gij = 2
(FikFj
k + 12FlmF lmgij
)
(√−gF ik),k =
√−gji
(6.1)
and find a static solution with two sources. Furthermore, in general this solution will
present conic singularities at the symmetry axis1; to find the equilibrium condition is
equivalent to require the absence of any conic singularity, i.e. the axis has to be el-
ementary flat. This means that there must be neither “struts” nor “strings”(see [81]
for the rigorous relation between the value of the angle deficit and the effective energy-
momentum tensor of these struts and strings) which prevent the two bodies to fall or
run away each other.
The key point to understand the main differences between classic and relativistic regime
is the repulsive nature of gravity in GR near a naked singularity. This can be seen just
by looking at the Reissner-Nordstrom (RN) metric
gtt = 1− 2M
r+
Q2
r2, (6.2)
1It is called “conic singularity” because the ratio between a small circumference around the axis andits radius is not 2π (as for a circle painted on a cone around its vertex).
85
86
where gravity is repulsive for r < Q2
M: it is for that reason that the equilibrium is allowed
only at certain distances. Indeed, e.g. if one considers the geodesic of a neutral particle
on that background, it is easy to find a (stable) equilibrium precisely at
rc =Q2
M. (6.3)
For charged particles an equilibrium is also possible at a fixed distance [21]; in these
cases it can be both stable or unstable, according to the choice of the parameters. In the
AB-solution, the Newtonian equilibrium condition is restored taking the limit of large
distance between the two singularities.
Although in principle such exact solution could be found already many years ago - by
using the Inverse Scattering Method (ISM) or the Integral Equation Method (IEM) -
practically nobody was able to eliminate the conic singularity in a reasonable explicit
way. Indeed, the important achievement of the AB-solution is the extreme compactness
of all the formulas, despite of complexity of calculations by which it was found [5]. They
get the wanted task using the IEM which presents some advantages with respect to the
ISM2.
As they showed, the equilibrium is possible, apart from the well-known Majumdar-
Papapetrou case where the charge of each source is equal to its mass, only for a naked
singularity near a black hole (b.h.). We excluded from our analysis the b.h.-b.h. and
naked-naked configurations since they do not exist at all in the equilibrium state.
This chapter is organized as follows: we give a brief historical review of the works in
literature (Sec. 6.2) (this section can be skipped by the ones interested only to the
physical contents); to make easier the reading, we also add the reproduction of the
Alekseev-Belinski solution in Sec. 6.3; we give some details clarifying the use of the
coordinates systems involved (Sec. 6.4); then we recall the definition of the electric field
in GR (Sec. 6.5) and finally we graph the plots of the electric force lines in the various
qualitatively different cases (Sec. 6.6). More precisely, in this last section, we consider
at the beginning the general case with two charges, firstly with e1e2 > 0 and then
e1e2 < 0 ; and finally that in which only one object (the naked singularity) is charged.
This last case was presented in different form in [6]. Of each configuration, we present
also the limit in which one source has a much smaller mass and charge than the other.
In particular we consider the limit case of a small charged particle near a Schwarzschild
black hole, finding electric force lines plots congruent with the Hanni-Ruffini [43] ones.
2In the ISM there are also some unphysical parameters (NUT parameter, magnetic charge) and therotation which are not easy to be eliminated.
87
6.2 Some Historical Remarks
The problem of the equilibrium of two charged masses and their resulting gravitational
and electric fields has a long history in GR literature (see table 6.1). It is possible to
distinguish two different kind of results: approximate results, and exact solutions.
Table 6.1: Some historical remarks.
Perturbation Methods Exact Solutions
(1927) Copson: Electric field of a test chargenear a Schwarzschild b.h.
(1947) Majumdar-Papapetrou: mi = ei
(1973) Hanni-Ruffini: Electric force lines of atest charge near a Schwarzschild b.h.
(1976) Linet: A correction of Copson solu-tion
(1978) Belinski-Zakharov: Vacuum solitons(1979) Hauser-Ernst: Integral equation
method for rational axis data(1980) Alekseev: Electrovacuum solitons(1984) Sibgatulling: Integral equation
method for rational axis data(1985) Alekseev: Integral equation method
for rational monodromy data(1993) Bonnor: Equilibrium of a test particle
on RN background(1997) Perry-Cooperstock: Equilibrium is
possible (three numerical example)(2007) Bini-Geralico-Ruffini: Equilibrium of
a test charge on RN with back-reaction until first order
Alekseev-Belinski: Exact solution forequilibrium (without strut) of two RNsources
In the contest of the approximate results, the first to be mentioned is the one of Copson
[28], who gave in 1927 the electric potential of a test charge on the Schwarzschild back-
ground (therefore it was neglected the backreaction of the particle on the metric tensor).
That work was important because it gave the potential in a closed analytic form, however
that result was not completely correct because it implicated that the black hole would
have an induced charge: the correct potential was given by Linet [59] only in 1976 —the
electric potential of the AB-solution indeed reduces to that form in the limit in which
the naked singularity source can be considered as a test particle.
88
In 1973 Hanni and Ruffini [43] gave for the first time the plots of the electric force lines3,
again for a test particle near a Schwarzschild black hole (but they used a multipole ex-
pansion of the electric potential).
Later a certain number of papers have been published in which different authors (us-
ing both exact generating techniques and approximate one, like Post-Newtonian (PN)
and Parameterized-Post-Newtonian (PPN) approaches) arrived to different conclusions
about the possibility/impossibility of an equilibrium configuration. However no final
statements were achieved because of the use of supplementary hypothesis or for the in-
completeness of the analysis.
In 1993 the already mentioned article of Bonnor [21] gave an important hint to clarify
the problem: studying the equilibrium configurations in the test particle limit, namely a
test charge on the RN background, he pointed out that equilibrium configurations were
possible when the ratio e/m was less than unity for the background and more than unity
for the particle, or viceversa; he showed also that equilibrium was possible for charges
of opposite signs too. It is worth noting that the Alekseev-Belinski solution confirms
practically word-by-word (from a qualitative point of view) that picture.
Then in 1997 Perry and Cooperstock [73] found three numerical example showing that
the equilibrium was possible for naked-b.h. configurations using an exact solution.
Finally it is to mention the Bini-Geralico-Ruffini articles [18, 17], in which the authors
found, using the Zerilli perturbative approach, the correction to the test particle ap-
proximation, considering the back-reaction of the particle to the background until the
first order. Surprisingly they found that the Bonnor condition remain unchanged also
considering these corrections.
For what concerns the exact solutions history, the first two important articles were the
ones of Majumdar and Papapetrou [63, 72], which exhibited the fields of an arbitrary
number of sources in reciprocal equilibrium, each one with mi = ei .
For many years that was the only exact result known, the next step was made by Belinski
and Zakharov [14, 15] in 1978 with the foundation of the Inverse Scattering Method in
General Relativity (purely gravitational), which was then extended also to the Einstein-
Maxwell equations by Alekseev [1] (see [9] for a self-consistent review). This method al-
lows to find stationary, axially symmetric solutions with an arbitrary number of sources.
From this time in principle the solution of our problem was available. However, practi-
cally, the constraints necessary to eliminate the rotation, the conic singularity and the
unphysical parameters (NUT parameter, magnetic charge) were too complicate to be
handled analytically.
The next step was made by Ernst and Hauser [46, 47], Sibgatullin [80] and Alekseev
3We follow this work for the construction of the plots of the present solution.
89
[2], who developed different integral equation methods for constructing of solutions of
Einstein-Maxwell equations. (The first method of such kind for pure gravity was already
formulated in [14, 15]). The method of [2] was used by Alekseev and Belinski to find
the present solution [4] (see also [5]), the important achievement of which is the extreme
simplicity of the formulas and of the equilibrium condition.
6.3 Summary of the Alekseev-Belinski formulas
The following (6.4)-(6.13) formulas are the reproduction of formulas (1)-(10) of [4].
The solution, which can be interpreted as the non-linear superposition of two RN source
at a fixed distance on the z-axis, is of the form
ds2 = Hdt2 − ρ2
Hdϕ2 − f(dρ2 + dz2) (6.4)
(6.5)
At = Φ, Aϕ = Aρ = Az = 0 (6.6)
where H , f and Φ are real function of ρ and z only. In what follows m1, m2 and
e1, e2 are the physical masses and charges of each source respectively4; the masses
include also the interaction energy therefore Mtot = m1 + m2 , and Qtot = e1 + e2 . It
is convenient to use the spheroidal coordinates (r1, θ1) and (r2, θ2) which are linked to
the Weyl coordinates (ρ, z) by:
ρ =√
(r1 −m1)2 − σ21 sin θ1
z = z1 + (r1 −m1) cos θ1
ρ =
√(r2 −m2)2 − σ2
2 sin θ2
z = z2 + (r2 −m2) cos θ2
(6.7)
By definition l ≡ z2 − z1 is the distance, expressed in the Weyl coordinate z , between
the two objects. Then, the explicit solution is:
H =[(r1 −m1)
2 − σ21 + γ2 sin2 θ2][(r2 −m2)
2 − σ22 + γ2 sin2 θ1]
D2(6.8)
Φ =[(e1 − γ)(r2 −m2) + (e2 + γ)(r1 −m1) + γ(m1 cos θ1 + m2 cos θ2)]
D(6.9)
f =D2
[(r1 −m1)2 − σ21 cos2 θ1][(r2 −m2)2 − σ2
2 cos2 θ2](6.10)
4The expressions were found with the help of the Gauss theorem.
90
where
D = r1r2 − (e1 − γ − γ cos θ2)(e2 + γ − γ cos θ1) , (6.11)
while γ , σ1 and σ2 are defined by:
γ = (m2e1 −m1e2)(l + m1 + m2)−1 ,
σ21 = m2
1 − e21 + 2e1γ , σ2
2 = m22 − e2
2 − 2e2γ .
(6.12)
It is easy to see that (fH)ρ=0 = 1 on the whole axis, i.e. automatically there is no conic
singularity. The above formulas give the solution satisfying the Einstein-Maxwell system
only under the equilibrium condition
m1m2 = (e1 − γ)(e2 + γ). (6.13)
Each of the parameters σ1 and σ2 can be either real (in the case of a black hole) or
imaginary (for a naked singularity); however in the following it will be always
σ21 > 0 , σ2
2 < 0, and σ1 > 0 (6.14)
i.e. the first source is “dressed” and the second is “naked”. Since we want to deal only
with separable objects, we require also the non-overcrossing condition
l − σ1 > 0 (6.15)
(it means that the naked singularity must be outside the horizon). Using (6.13), the
distance l can be written as a function of the other parameters by the very simple
formula:
l = −m1 −m2 +m1e2 −m2e1
2(m1m2 − e1e2)
[(e2 − e1)±
√(e1 + e2)2 − 4 m1m2
]; (6.16)
we always choose the sign in front of the root in (6.16) in order to satisfy the non-
overcrossing condition (6.15). From (6.16) it is clear that the parameters must satisfy
the restriction
(e2 + e1)2 > 4 m1m2 . (6.17)
6.4 Some further details of the solution
The solution has a very simple form, the only price to pay is just the simultaneous use
of two pairs of coordinates. Obviously for practical purposes, as for the electric lines
plot, one needs the use of only one system. We choose (r1, θ1) , the one related to the
91
black hole (which is centered on the origin, since we took z1 = 0 for simplicity, and
consequently z2 = l ). The linking relations are:
r2 −m2 = 1√2
√b2 +
√b4 − 4σ2
2(z − z2)2
cos θ2 = (z − z2)(r2 −m2)−1
(6.18)
where b2 ≡ ρ2 +σ22 +(z−z2)
2 , while ρ and z have to be expressed using the first couple
of (6.7). We take the plus sign of the roots in the first of (6.18) since r1 and r2 must
coincide at infinity.
The peculiarity of the coordinates used needs a clarification in order to understand the
physical property of the solution, first of all where the “true” divergences are and what
happens on the horizons.
Using (r1, θ1) : These coordinates are centered on the black hole and can be considered
as the natural generalization of the Schwarzschild ones. For the peculiar choice of the
(r1, θ1)− coordinates, the horizon remains a perfect circle (it can be seen also analytically
that H vanishes at rh = m1±σ1 as for the single RN black hole). However the spherical
symmetry is only apparent, indeed the invariants have a θ1-dependence and vary on the
horizon. In this frame is not possible to reach the inside of the spheroid r2 < m2 (we
called the surface r2 = m2 the ‘critical spheroid’, as in [4]), therefore the second source
(the naked RN centered in z = z2 ), appears squeezed “inside” a horizontal segment that
cuts the vertical axis: this happens because the naked singularity lies inside the region
not covered by (r1, θ1) .
Table 6.2: The two peculiar regions in Weyl and in the spheroidal coordinates.
Physical description Location
Horizonρ = 0, z1 − σ1 ≤ z ≤ z1 + σ1
or equivalentlyr1 = m1 + σ1 , ∀θ1
Critical spheroidof the naked singularity
0 ≤ ρ ≤ Im(σ2) , z = z2
or equivalentlyr2 = m2 ,∀θ2
92
Using (r2, θ2) : Conversely, using (r2, θ2) , the ‘critical spheroid’ of the naked RN will
appear as a sphere of coordinates r2 = m2 , while the black hole horizon as a segment
squeezed on the axis: in this case it is the ‘critical spheroid’ of the first source, i.e.
r1 < m1 , that cannot be reached. Again, that has nothing to do with physics but
just with the choice of the coordinate system). In table 6.2 we localize the two peculiar
Table 6.3: Characteristic points of the first source.
Note the degeneracy of the Weyl coordinates.For the numerical evaluation we used m1 = 1, e1 = 0.7, m2 = 0.3, e2 = 0.44,l = 5 (the same used for fig.1).The central singularity of the b.h. is split in two points:
r(I)1 =
m1+m2+l−√
(m1+m2+l)2−4e1e2
2, and r
(II)1 =
m1−m2−l+√
(m1−m2−l)2−4e1e2
2.
Description (r1, θ1) (ρ, z) H Φ f F ijFij
Two branch points
r1 = m1
θ1 = 0, π(0, 0) finite fin. fin. fin.
Equatorial pointof the ext. hor.
r1 = m1+ σ1
θ1 = π/2(0, 0) 0 fin. fin. fin.
Equatorial pointof the int. hor.
r1 = m1− σ1
θ1 = π/2(0, 0) 0 fin. fin. fin.
B.h. singularity I
r(CI)1 = 0.0497
θ1 = 0(0, r1−m1) +∞ −∞ 0 −∞
B.h. singularity II
r(CII)1 = 0.0594
θ1 = π(0,−r1+ m1) +∞ −∞ 0 −∞
North poleof the ext. horizon
r1 = m1+ σ1
θ1 = 0(0, σ1) 0 fin. +∞ fin.
South poleof the int. horizon
r1 = m1− σ1
θ1 = π(0, σ1) 0 fin. +∞ 0
North poleof the int. horizon
r1 = m1− σ1
θ1 = 0(0,−σ1) 0 fin. +∞ 0
South poleof the ext. horizon
r1 = m1+ σ1
θ1 = π(0,−σ1) 0 fin. +∞ fin.
regions (the horizon and the critic spheroid), using Weyl coordinates, with the respective
translations in (r1, θ1) or (r2, θ2) ; while in tables 6.3-6.4 we give a detailed description
of the relevant physical quantities in the notable points of these two zones. It is also
93
to note the “degeneracy” of the Weyl coordinates: to the same point in (ρ, z) it can
corresponds different values of the spheroidal coordinates.
Table 6.4: Characteristic points of the naked source:
The first three points correspond all to (ρ = 0, z = z2 = l) .The same numerical values of table 6.3 are used.
Description (r2, θ2) (ρ, z) H Φ f F ijFij
Naked sing.
r2 =
l+Mtot−√
(l+Mtot)2−4e1e2
2
θ2 = π(0, l) +∞ −∞ 0 −∞
Crossing of thecut with the axis(up border)
r2 = m2
θ2 = 0(0, l) fin. fin. fin. fin.
Crossing of thecut with the axis(down border)
r2 = m2
θ2 = π(0, l) fin. fin. fin. fin.
Extremesof the cut
r2 = m2
θ2 = π/2(Im σ2, l) fin. fin. +∞ fin
The electromagnetic invariant
In order to understand where the charges are located it is useful to consider the electro-
magnetic invariant F = F ijFij/2 . For the solution (6.4) it has the form:
F = − [(r1 −m1)2 − σ2
1](∂r1Φ)2 + (∂θ1Φ)2
f H [(r1 −m1)2 − σ21 cos2 θ1]
. (6.19)
It can be seen numerically (see tables 6.3,6.4) that it diverges inside the horizon and
inside the critical spheroid of the naked RN 5.
It is also worth noting that on the critical spheroid, although in the (r1, θ1) representa-
tion it is a line, the up- and down-limit of F do not coincide, since they correspond to
different points of the physical space-time.
Looking at F it is possible to see that no real discontinuity exists on the horizon, indeed
it diverges only on the central singularities.
The other invariant, εijklFijFkl = E ·B , is identically zero.
5The spheroid, i.e. the line 0 < ρ < Im(σ1) , z = z2, seems apparently regular in (r1, θ1) coordinatesjust because its interior can be reached only using (r2, θ2)
94
6.5 Electric force lines definition
Just to understand better the meaning of the plots, we want to recall the definition of
the electrical vector (which is not a trivial choice in GR). Following [43], we define the
electric field as the three non-diagonal time-like components of the controvariant tensor
F ij :
Eα = Fα0 , α = 1, 2, 3 . (6.20)
That identification is geometrically justified by the Gauss theorem generalized to the
curved manifolds [88]:
4πQ =
∫
C
*F =
∫
C
∗Fij dxi ∧ dxj , (6.21)
where ∗Fij = 1/2εijklFkl√−g is the dual tensor of F ij . Then it is natural to define the
force lines in the usual way as the trajectories of the dynamical system:
d
dλr1 = Er1
d
dλθ1 = Eθ1
(6.22)
or equivalently by
dr1
dθ1
=Er1
Eθ1,
Er1
Eθ1=
[(r1 −m1)
2 − σ21
] ∂r1Φ
∂θ1Φ. (6.23)
Then, from the equation of motion for this problem, restricting to our case, we have:
F r1t u
t dθ1 − F θ1t u
t dr1 = 0 , (6.24)
having used the coordinates xi = (t, ϕ, r1, θ1) .
The physical interpretation (Christodoulou-Ruffini, quoted in [43]) is the following: a
force line is a line tangent to the direction of the electric force measured by a free-falling
test charge momentarily at rest, with initial 4-velocity
ut = (√
gtt, 0, 0, 0). (6.25)
Note that such interpretation is valid only for gtt > 0 , for this reason we have not plotted
the lines inside the horizon.
In the (t, ϕ, r1, θ1) coordinates the metric (6.4) becomes
ds2 = H dt2 − ρ2
Hdϕ2 − f [(r1 −m1)
2 − σ21 cos2 θ1]
[dr1
2
(r1 −m1)2 − σ21
+ dθ12
](6.26)
95
while the electric potential remains unchanged. Then for the electric field we have:
Eϕ = 0
Er1 = gttgr1r1 ∂At
∂r1
Eθ1 = gttgθ1θ1 ∂At
∂θ1
. (6.27)
Therefore the force lines are given by the solution of
dr1
dθ1
=[(r1 −m1)
2 − σ21
] ∂r1At
∂θ1At
. (6.28)
It is worth noting that the force lines depend only on the two ratios ∂r1At/∂θ1At and
gr1r1/gθ1θ1 (indeed the conformal factor f and neither gtt nor gϕϕ do not appear in
(6.28).
6.6 Plots of the force lines
In the plots, what we called “second source” (i.e. the naked RN) is always up, while the
“first source” (i.e. the black hole) is always down and centered on the origin. The lines
are plotted in (x, y) Cartesian coordinates defined as
x = r1 sin θ1
y = r1 cos θ1
(6.29)
(they coincide with (ρ, z) defined in (6.7) when r1 →∞ ).
In the plots we have used geometrical units ( G = c = 1 ), in which the unitary length is
given by the Schwarzschild mass m1 = 1 .
The graphical Faraday criterium is used, namely we plotted the electric force lines such
thatNumber of lines from the first source
Number of lines from the second source∼= e1
e2
.
The separatrix
In general, when there are two charges, the electric force diagram will present a separa-
trix, which is a force line which reach asymptotically a saddle point of the potential and
separates the lines of the two charges in the case they have the same sign. In the case of
opposite sign charge, it delimits the region in which the lines flow from one to the other
source. We marked these separatrix lines in bold; may be it is worth to mention that on
the saddle point they have an invariant definition since on that point F = 0 .
96
Inside the horizon
In the following plots the force lines are graphed only outside the horizon since there
it is no more possible to consider a static observer; the physical interpretation given in
Sec. 6.5 does not hold because (6.25) becomes imaginary. However, when the separatrix
starts from the inside of the horizon, the study of that region is important to understand
the difference between cases with the same or opposite charges. Therefore, in the case of
fig. 6.3, in which the saddle point is inside the horizon, we calculated the point where the
separatrix touches the horizon, and we plotted the diagram just from there. (This was
possible because mathematically the eqn. (6.28) is well defined also inside the horizon).
In the following three sub-sections we analyze the three qualitatively different sub-cases:
e1e2 > 0 (6.6.1), e1e2 < 0 (6.6.2), and finally e1 = 0 (6.6.3).
6.6.1 Two charges of equal sign ( e1e2 > 0 )
General case: two comparable RN sources
Let us consider the case in which the two RN sources have charges and masses of com-
parable dimensions
m1 ≈ m2 e1 ≈ e2
m21 > e2
1 m22 < e2
2
e1e2 > 0 .
(6.30)
This is the closest case to the classical picture, indeed here the equilibrium is mainly due
to the classical balance of the electrostatic force and gravitational field. The resulting
plot is given in fig.(6.1).
The qualitative behavior of the force lines does not change with the changing of the
distance l .
Small charge (naked) near a RN black hole
Here and in the following we say ”small” charge and not ‘test’ charge because the exact
nature of the solution automatically takes in account all the back-reaction terms even if
they can be very small (while the ‘test’ limit is in general referred as the one in which all
those terms are completely neglected). The equilibrium configurations of this case (see
97
10
8
6
6
4
4
2
0
2
-2
-4
0-2-4-6
Figure 6.1: Force lines in the general case (6.30), when the two RN have charges of thesame sign. Note that the critical spheroid in that coordinate representation (6.29) is anhorizontal segment. The bold line is the separatrix. The circle on the bottom is theexternal horizon of the first source. Parameters used: m1 = 1 , e1 = 0.7 , m2 = 0.3 ,e2 = 0.44 , l = 5 .
fig. 6.2), with
m1 >> m2 , |e1| >> |e2|
m21 > e2
1 m22 < e2
2
e1e2 > 0 ,
(6.31)
have been studied in the test particle approximation first in [21], and recently in [18, 17],
where they took in account also the back-reaction of the test particle.
98
6
4
4
2
2
0
-2
0-2-4
Figure 6.2: Force lines of a small charge near a RN with horizon, case (6.31). Parametersused: m1 = 1 , e1 = 0.1 , m2 = 10−3 , e2 = 1.3 · 10−2 , l = 2.5 ). The bold line is theseparatrix.
Small charge (with horizon) near a naked RN
This case does not exist for e1e2 > 0 .
6.6.2 Two charges of opposite sign ( e1e2 < 0 )
Although it is easy to show that in the previous cases with e1e2 > 0 the implications
m21 > e2
1 ⇒ σ21 > 0
m22 < e2
2 ⇒ σ22 < 0 ,
(6.32)
are always true, it is not so if e1e2 < 0 . However in the following we considered two
cases in which (6.32) holds.
99
Two comparable RN sources
This case, with
m1 ≈ m2 e1 ≈ −e2
m21 > e2
1 m22 < e2
2
e1e2 < 0 ,
(6.33)
is the case in which the relativistic effects are much evident since here also the electric
force is attractive (see fig. (6.3)): in this case the equilibrium is due to the repulsive
nature of the naked singularity.
10
8
6
6
4
4
2
0
2
-2
-4
0-2-4-6
Figure 6.3: Force lines in the general case (6.33), with charges of the opposite sign.Parameters used: m1 = 1 , e1 = 0.05 , m2 = 0.3 , e2 = −1.66 , l = 5 . The bold line isthe separatrix, which now encircles also the central singularity of the b.h.: inside thatregion the lines go from one charge to the other. Outside that region the lines go frome2 to infinity (some of them pass also through the horizon).
100
Small charge near a RN
It is also possible to find values that corresponds to a small charge with horizon near a
naked RN:m1 << m2 , |e1| << |e2| ,
m21 > e2
1 , m22 < e2
2 ,
e1e2 < 0 .
(6.34)
However in this case it would be useless to plot the force lines because the electric field
is trivially Coulombian (the first source is weakly interacting both gravitationally and
electrically).
The inverse case, namely a small charge naked near a RN with horizon, does not exist
for particles lying outside the horizon (i.e. requiring l > σ1 ), as noted by Bonnor [21].
6.6.3 Cases with only one charge
In the following we will consider the cases with a naked singularity near a neutral black
hole; they are qualitatively different from the previous ones since now there is no separa-
trix and the electric flux over the horizon surface is zero. In the particular case in which
the first source is neutral (i.e. e1 = 0 ), the equilibrium distance is even simpler,
l = −m1 −m2 +e22
2m2
1±
√1− 2 m1
(e22
2m2
)−1 , (6.35)
which can be always satisfied for sufficiently large values of the charge parameter e2 .
101
RN near a Schwarzschild black hole (comparable masses)
10
8
6
6
4
4
2
0
2
-2
-4
0-2-4-6
Figure 6.4: Force lines for the values (6.36). The blank circle of radius 2m1 is theSchwarzschild horizon. Parameters used: m1 = 1 , m2 = 0.3 , e2 = 1.5 , l = 5 .
Thanks to the exact nature of the solution, it is very interesting also the case in which
the RN source has comparable mass with the Schwarzschild black hole, say
m1 ≈ m2 , e1 = 0 ,
σ1 = m1 , m22 < e2
2 ,
(6.36)
indeed this case cannot be achieved by a perturbative approach, see fig. 4. It is possible
to see that the electric lines are just slightly deformed by the gravitational field.
Small charge near a Schwarzschild black hole
We can also consider the small-charge limit,
m1 >> m2 , e2 e1 = 0 ,
σ1 = m1 , m22 < e2
2 ,
(6.37)
i.e. the second source is a small RN naked singularity. That is the only case in which we
have a good comparing in literature, since it is the only case already studied (as we know)
102
by using the force lines plots [43], although by a perturbative approach. Strictly speaking
the Hanni-Ruffini case refers to a slightly different situation, since they considered a
particle momentarily at rest in the Schwarzschild metric, while the AB solution is exactly
static6. However the present solution confirms very nearly their multipole expansion,
since we find that the plots are in practice coincident. In order to have the best possible
comparing we considered the same distances between the charge and the horizon (figg. 5-
7). Since now l is not an independent parameter we fixed the masses values m1 = 1 and
m2 = 10−4 , then varying the distance we found (using (6.16)) the respetive parameter
e2 . The test particle is at z = l , or equivalently at r1 = l + m1 . (Just to clarify the
link with [43]’s notations: their r is our r1 , and their M is our m1 ).
10
10
5
5
0
-5
0-10
-5-10
Figure 6.5: Force lines for the values (6.37), with l = 3 m1 , i.e. in the spheroidalcoordinates the particle is in r1 = 4 m1 . The circle of radius 2m1 is the Schwarzschildhorizon. The plots are practically identical to the ones found by Hanni and Ruffini.
6From another point of view, Hanni-Ruffini do not use (6.35) to determine the fourth parameter(because in their approximation the fourth parameter, say m2 , is considered arbitrarily small, thereforeit is not present at all)
103
10
10
5
5
0
-5
0-10
-5-10
Figure 6.6: Now the distance is l = 2 m1 , or equivalently the charge is in r1 = 3 m1 .
10
10
5
5
0
-5
0-10
-5-10
Figure 6.7: Now the distance is l = 1.2 m1 , or equivalently the charge is in r1 = 2.2 m1 .
From eq. (6.23), considering that now σ1 = m1 , it is easy to see that the corrections
104
to the Hanni-Ruffini approximation are limited only on the exact form of the At poten-
tial, since to use the Schwarzschild metric or the functions H and f given in (7.115)
and (6.10) does not change the force lines.
6.7 Final remarks
The main result of our analysis is that the exact solution seems to confirm quite strictly
the test-charge approximation on the RN background (see. e.g. [21]), which seems to
give a good test of the exact picture.
Size of the naked singularity Sometimes in literature has been guessed ([36], [68])
that e2/2m should be considered as a ‘critical radius’ of the naked singularity inside of
which the RN solution has no physical meaning since it should be matched with a more
realistic matter field tensor, in order to avoid the well known problems of a point-like
source, as the divergence of the electric energy.
If the quantity e22/2m2 can be roughly considered as the physical size of the RN charge,
then from formula (6.35) it is easy to see that the equilibrium configurations exist only
for e22/2m2 larger than the Schwarzschild radius ( 2m1 ). This seems to suggest that a
real ‘small’ charge limit cannot be achieved, in the sense that the particle can be ‘small’
only gravitationally (and electrically), but not geometrically because it would have a size
larger than the Schwarzschild horizon.
However this is just a speculation since further investigations should be done to model
the interior of a realistic RN source and find its radius.
Coordinate dependence of the plots Any plot of the force lines change drastically
for different choices of the coordinates. However, what is interesting is to compare
different situation by using the same coordinate representation, e.g. as we did for the
Hanni-Ruffini case.
Stability If the solution would be unstable that would mean that it is a completely
academic problem, since the equilibrium will be physically not allowed. However in the
geodesic/test particle approximation, which gives the essential features of the problem,
the equilibrium is stable, therefore at least in some range of values it should be the same
also in the exact case (indeed the exact solution smoothly converge to the test particle
approximation in the limit ei , mi → 0 , with ei/mi finite, i = 1 or 2 ).
Anyway a systematic, even if not complete, analysis of stable configurations it is reported
in the next chapter. There we will see that, with respect to the different cases examined
above, a number of exact configurations are indeed stable.
Chapter 7
A stability analysis of the doubleReissner-NordstromAlekseev-Belinski exact solution
Introduction
In the static double Reissner-Nordstrom (RN) solution1of Alekseev-Belinski [4] (AB),
the equilibrium condition which ensure the absence of any conic singularity, implies that
the distance between the two sources has to be a function of the other four parameters
of mass and charge of the two sources: l0 = l0(m1,m2, e1, e2) (see below equation (7.7)).
In the following we will study the stability of this solution in a very restrictive sense, i.e.
with respect to spatial displacements of the two sources. Indeed there can be a lot of
different perturbations, as rotational ones, anyway we think that these are some of the
most physically significant. Our analysis make no use of the usual perturbative methods
(i.e. to put a perturbation in the Einstein-Maxwell equations and see how they evolve in
time), we use instead the dynamical properties of the conical singularity, following the
Sokolov-Starobinski definition [81].
If the two sources are placed in a distance different as regards the equilibrium one, say
l = l0 + x , then one can have still a static solution but in that case it appears a conic
singularity between the two bodies [5], namely on ρ = 0, z ∈ [z1, z2] (let us suppose
that source-1 is in z1 = 0 and source-2 in z2 = l ). This is interpreted as a strut or a
string to which it can be associated a force. It is called “strut” if it exerts a compensative
pressure, “string” if it exerts a tension; since the difference between these two situations
depends only by a sign, hereafter, we will use simply the word ”strut”.
Now, we assume that in the reality there are no struts if the two sources are displaced
1We recall that in this solution, the sources have to be a black hole (b.h.) and a naked singularity(n.s.).
105
106
from the equilibrium position, but that the two sources oscillate near the distance l0
(stable configuration), or go far away or collapse (unstable configurations). Then we
assume that the force exercised by the two bodies one to the other will be precisely the
opposite of FStrut , say
FBodies = −FStrut ; (7.1)
indeed the eventual presence of a strut with such a force would balance the re-
pulsion/attraction of the bodies, keeping the system exactly in “equilibrium”, with
Ftot = FBodies + FStrut = 0 . Therefore by means of the knowledge of FBodies , the
analysis of the equilibrium follows the usual procedure of classical mechanics.
What reported in this chapter is an extended version of part of [74].
7.1 Force of the strut.
To calculate the force of the strut we need to calculate the energy-momentum tensor
T ji on this segment2. Since we know the metric (from the AB solution in his general
5-parameters formulation [5]), we can define T ji , by means of the Einstein equations
(G = 1 , c = 1 )
8π T ji = Rj
i − 1/2 δji R , (7.2)
in terms of a Dirac δ-function, as described in [81].
If the parameter l 6= l0 , then the AB metric presents a conic singularity in the segmenta
of the axis between the two sources. This means that, if we expand the solution near
the axis, the line element corresponding to the two-dimensional space-like surface t, z =
const. , has the form:
ds2 = dρ2 + a2ρ2dϕ2 , ρ ' 0 , 0 < z < l , (7.3)
which is, for a < 1 , the line element of a conic surface. The coefficient a is a constant.
In terms of the δ parameter of the AB solution it is given by
a =1
(√
fH)|ρ=0
=1 + 2δ
1− 2δ. (7.4)
In general, when a 6= 1 between the two sources, it represents a deficit of angle specific
of the conical singularity.
2In this chapter we adopt the following notations: signat. = (−+ ++) ; i, j, ... = 0, 1, 2, 3 ,α, β, ... = 1, 2 ;
(x0, x1, x2, x3) = (t, ρ, ϕ, z
).
107
We recall, from [5], that:
δ =m1m2 − (e1 − γ)(e2 + γ)
(l0 + x)2 −m21 −m2
2 + (e1 − γ)2 + (e2 + γ)2, (7.5)
γ =m2e1 −m1e2
l0 + x + m1 + m2
, (7.6)
l0 = −m1 −m2 +m1e2 −m2e1
2(m1m2 − e1e2)
[(e2 − e1)±
√(e1 + e2)2 − 4 m1m2
]. (7.7)
The calculation of Rji for the line element
ds2 = −dt2 + dρ2 + a2ρ2dϕ2 + dz2 (7.8)
which approximates the AB solution near the axis, gives Rji = 0 , R = 0 , and thus
T ji = 0 . However , we can introduce a distribution-like source using the Gauss-Bonnet
theorem as in [81]:∫
S
Kdσ = 2π −∫
∂S
kg ds . (7.9)
In this formula K is the Gaussian curvature which results to be the half of the Ricci
scalar, K = R/2 ; kg is the geodesic curvature:
kg = εαβ
(d2xα
ds
dxβ
ds+ Γα
µν
dxµ
ds
dxν
ds
dxβ
ds
) (gαβ
dxα
ds
dxβ
ds
)−1/2
; (7.10)
S is a small disk centered on the vertex of the conic surface and ∂S is its border.
Since the right-hand side of (7.9) yields:
2π −∫
∂S
kg ds = 2 π (1− a) , (7.11)
introducing the Dirac delta-function δD(ρ) , normalized by∫ ∞
0
∫ 2π
0
δD(ρ) ρ dρ dϕ = 1 , (7.12)
we can define, by means of (7.9), the Gaussian curvature as:
K = 2π1− a
aδD(ρ) . (7.13)
It results that K = R/2 = R11 = R2
2 , hence Gji = −1/2 R diag (1, 0, 0, 1) . Thus, the
Einstein equations yiels
T 33 = −1− a
4 aδD(ρ) , (7.14)
108
using which, we find the expression of the force3:
FStrut =
∫ ε
0
∫ 2π
0
T 33 a ρ dρ dϕ =
1
4(a− 1). (7.15)
From (7.4) and (7.5), we obtain:
FStrut =δ
1− 2δ=
m1m2 − (e1 − γ)(e2 + γ)
l2 − (m1 + m2)2 + (e1 + e2)2, (7.16)
where we recall that l = l0+x and l0 is the equilibrium distance. For x = 0 than l = l0 ,
the value of which is obtained by the equilibrium condition: m1m2 = (e1 − γ)(e2 + γ) .
Therefore, in this case, it naturally follows that FStrut = 0 .
It is worth to mention the work of Manko [64], where this results is given, at least of the
notation used therein and of the sign4.
It is also useful to recall the work of Letelier and Oliveira [58], where a confusion in the
literature about different definitions of the force of the string is reported and discussed.
Anyway, these differences, since reduce to a positive rescaling of the expression of the
forces, do not affect our following analysis.
Classical limit of the force
Choosing an equilibrium configurations with all the parameters of mass and charge dif-
ferent from zero, and expanding the formulas (7.16) for l going to infinity, we find
−FStrut = FBodies = −m1 m2
l2+
e1 e2
l2. (7.17)
That is, we obtain the resultant between the Newtonian gravitational force law and
the Coulomb electrostatic one. The limit given by (7.17), together with the relations
Mtot = m1 + m2 , shows that the mk parameters, deduced by Alekseev and Belinski by
means of Komar integrals, have to be ever positive and that they coincide with Newtonian
masses.
3In the following formula ε is a small positive number for which the local approximation (7.3) holds.4In that work the force given by the formula (30) is declared to be ”the interaction force between
the costituents” and, then, for the uncharged case, the formula (31) gives ”the known expression for theinteraction force between two Schwarzschild black holes”. By means of this last formula it is easy to seethat, at large separation distance, we do not obtain Newtonian expression of the gravitational force butits opposite; hence, the formula (30), which coincide with our (7.16), is just the expression describingthe compensative forces exerted by the strut.
109
7.2 Analysis of equilibrium in the AB solution
The stability can now be deduced from the sign of the derivative of the force w.r.t. x ;
obviously we have to evaluate this quantity on the equilibrium point x = 0 :
(∂xFStrut)| x=0 =m2e1 −m1e2
(l0 + m1 + m2)2
[2γ0 − e1 + e2
l20 −m21 −m2
2 + (e1 − γ0)2 + (e2 + γ0)2
]; (7.18)
where γ0 is γ evaluated on x = 0 . The stability condition is:
(∂xFStrut)| x=0 < 0 . (7.19)
The previous formula can be simplified without loss of generality using the following
considerations5:
1. Using the arbitrariness of the electric charge’s sign definition:
e2 > 0 ; (7.20)
2. Since we are considering a black hole and a naked singularity:
|e1|m1
< 1 <e2
m2
; (7.21)
3. Separability requirement:
l0 > σ1 ; (7.22)
4. Finally, the existence of a real l0 needs [71]:
(e2 + e1)2 > 4 m1m2 . (7.23)
We recall that, as defined in [5] ,
σ12 = m1
2 − (e1 − γ)2 + γ2 , σ22 = m2
2 − (e2 + γ)2 + γ2 . (7.24)
From (7.20) and (7.21), we find that for all the configurations according with them, it
results that
γ0 < 0 . (7.25)
Then, from Eq.(7.22) it is easy to show that the denominator in Eq.(7.113) is always
positive, i.e.
l20 −m21 −m2
2 + (e1 − γ0)2 + (e2 + γ0)
2 = l20 − σ21 − σ2
2 + 2γ20 > 0 (7.26)
5Obviously a necessary condition for the stability is also that the equilibrium exists.
110
(indeed σ22 < 0 because source-two is naked). Thus, stability condition (7.113) can be
reduced to the following one:
2γ0 − e1 + e2 > 0 . (7.27)
If the previous inequality is not fulfilled, it means that the configuration is unstable.
Inequality (7.118) can be rewritten as:
(m2 −m1)(e1 + e2) + (e2 − e1) l0 > 0 ; (7.28)
for commodity in the following we define the quantity
X = (m2 −m1)(e1 + e2) + (e2 − e1) l0 ,
which is anyway an irrational 4-parameters quantity.
Resuming, the analysis of the stable configurations implies the discussion of the following
system of inequalities:
e2 > 0
|e1|m1
< 1 <e2
m2
l0 > σ1
(e2 + e1)2 > 4 m1m2
X > 0
. (7.29)
Being two of the inequalities present in the above system irrational, a complete classifica-
tion of the stable configurations represents a non trivial task; it will be object of a future
work. Hereafter, we limit ourselves in presenting the discussion of a list of qualitatively
different situations.
A. Equal signed charges: e1 > 0 , e2 > 0 .
This is the only case in which we found also unstable equilibria.
A.1 m1 < m2 : b.h. smaller than n.s..
If m1 < m2 then, necessarily from(7.21), e1 < e2 : consequently X > 0 is always
satisfied and the equilibrium is always stable.
A.2 m1 > m2 : b.h. larger than n.s..
111
A.2.1 m1 > m2 and e1 < e2 .
Numerically we found only stable equilibrium (when it exist).
A.2.2 m1 > m2 and e1 > e2 .
In this case X is always negative and thus the equilibrium unstable.
A particular situation of this sub-case is the small-particle limit (that is
m2 = α e2 , e2 → 0 , with 0 < α < 1 constant); it agrees with the in-
stability found by Bonnor [21] .
B. Opposite signed charges: e1 < 0, e2 > 0 .
In this case we suspect that the equilibrium is always stable (however there is one subcase
in which we was not able to demonstrate it analytically).
Since X now is
X = (m2 −m1)(e2 − |e1|) + (e2 + |e1|) l0 , (7.30)
then we can consider the two different sub-cases: m1 < m2 and m1 > m2 .
B.1 m1 < m2 : b.h. smaller than n.s.
If m1 < m2 , then from condition (7.21) we have necessarily e1 < e2 , which implies
that X is always positive.
B.2 m1 > m2 : b.h. larger than n.s.
In this case we need to consider the two different configurations: |e1| ≷ e2 .
B.2.1 m1 > m2 , |e1| > e2 .
If |e1| > e2 , then one can see at first sight from (7.119) that X is always
positive. Anyway, we was not able in finding any numerical configurations
satisfying the separability condition (7.22).
B.2.2 m1 > m2 , |e1| < e2 .
If |e1| < e2 we are not able to demonstrate that X is always positive, but
we can say that at least for enough large values of e2 this is true, because
lime2→∞
l0 ' m1
|e1| e2 , (7.31)
and thus X → m1
|e1| e22 → +∞ .
C. One charge only e1 = 0
This case is always stable. Indeed, considering e1 = 0 , the stability condition (7.28)
112
becomes:
l0 + m2 > m1 . (7.32)
Then, considering the separability condition, which is now:
l0 > σ1 = m1 , (7.33)
it is immediate to see that (7.125) is always true.
7.3 Final remarks
The above analysis, despite it does not give a complete classification of all possible
equilibrium configurations, yields the following results.
1. Most of the cases are stable.
2. The only unstable case we found is that describing two sources with equal signed
charges and with the parameters of mass and charge of the black hole being both
greater of the corresponding ones of the naked singularity. This result agrees with
Bonnor limit.
3. The one-charge case is always stable.
4. In our previous chapter, we drawn some plots of the electric force lines. It results
that all that cases (taking the same numerical values of the parameters) are all
stable, except the case plotted in figures (6.1) and (6.2). They corresponds to the
case pointed out above at item number two.
5. This criterion of stability, although very peculiar, can be considered at least a
necessary condition for the stability in general sense. Anyway, it becomes also a
sufficient criterion in the limit in which one source is much smaller than the other.
Conclusions and prospects
In the first part of this thesis, in the framework of the solitonic Alekseev technique,
generating exact solutions of the electrovacuum symmetry reduced Einstein-Maxwell
equations, we have presented a procedure to construct approximate terms of such kind
of exact solutions. These terms are relative to the expansions respect to a control pa-
rameter interpretable as the Newtonian gravitational constant or, equivalently, as the
amplitude of the electromagnetic fields. Therefore this perturbative tool gives a way to
compute approximate solutions, belonging to such class of symmetries, both for weakly
coupled gravitational and electromagnetic fields and for weak electromagnetic fields over
curved gravitational backgrounds.
We have tested this procedure computing these approximate terms for the well known
one-soliton stationary exact solution representing a Kerr-Newman black hole. We have
hence deduced the lowest order terms relative to the fields generated by two Kerr-
Newman sources placed at fixed distance on the axis of symmetry. The achieved results
are the weak fields generated by two spinning charged sources in Minskowsky space-time
and the reparameterization between the mathematical parameters and the physical one.
This first deduction represents the first step of a work, still in progress, consisting in the
derivations of the next order terms. Its hoped task is to find a way to find a repara-
meterization for such weak fields solutions, the consequent equilibrium conditions and
thus, some sufficient conditions about the physical configurations to obtain regular static
fields; that is, fields generated by sources in equilibrium without any conical singularity
presents on the axis. Investigations of the solitonic technique, from the prospective of
this approximate approach, could also give some further indications about some peculiar
aspects of its. For example, the problems of the analytical continuations of the poles,
the relation between the Alekseev technique and the Belinski-Zakharov one.
The second part of the thesis has concerned the analysis of some features of the
Alekseev-Belinski solution describing the electrovacuum fields generated by two Riessner-
Nordstrom like sources in static equilibrium. In particular we have presented the be-
haviors of the electric field force lines relative to qualitatively different configurations,
following a classification with respect to the signs and the magnitude of the physical
113
114
parameters of mass and charge of the two sources. We have singled out those cases
which can not exist, like a small charge black hole near a naked equally signed charged
singularity or a small charge naked source near a opposite signed charged black hole.
Besides, we have obtained, and hence confirmed exactly, some results found in the past
by Bonnor and Hanni-Ruffini by means of test particle approximation approaches.
We have then performed an analysis of the stability of these configurations respect small
displacements from the equilibrium distance between the two sources. We have showed
that most of the configurations discussed are stable and singled out cases which are al-
ways unstable. However, due to the complexity of the stability conditions, which consist
in a system of irrational inequalities, the analysis performed is not complete. Namely,
we have found cases for which, until now, we have not been able in finding the discrim-
inating conditions between stable and unstable configurations. A complete analysis of
stable (in this sense) configurations could be an interesting task for a future research.
Again, with respect to this stability analysis, we have found and confirmed previous
results found by Bonnor by means of the test particle approximation.
It is a well known thing that the the Reissner-Nordstrom solution posses a slight physical
significance because, from an astrophysical point of view, celestial bodies are practically
uncharged. Besides, it is rather senseless to use this solution as a framework to find
model for elementary particles. We know that even for an electron, the particle with the
largest charge per unit mass, the gravitational corrections are significant at scales much
smaller respect those at which quantum field theories dominate. However, this does not
exclude the possibility to look for some results, at least at a semiclassical level of approx-
imation; a certain number of works dealing with such kind of investigations has been
produced1. The interest for investigations on the Reissner-Nordstrom solution is given
principally by the fact that it is a simple example of a solution of the Einstein-Maxwell
equations. Besides, as a remarkable fact, it has given new significant insights for the
development of the black holes theory. We believe that, in the context of this theory, the
new solution of Alekseev-Belinski could yield further new interesting results. We recall
that the analysis relevant to this solution that we have presented is restricted to fields
generated by separate sources. At this regard, an interesting topic for future research
could be, for example, the investigation on the possibility to have non-separate sources
configurations; at present, it is still an open question.
1See, for example, the bibliography cited in the introduction of [13]. The reader can find it belowas the first work added in the ”Additional contributions” section.
Additional contributions
Hereafter two works produced in collaboration with V.A. Belinski and M. Pizzi during
the last two years are added in their preprint-version.
In the first one [12, 13], we demonstrate an alternative (with respect to the ones ex-
isting in literature) derivation of exact solution of the Einstein-Maxwell equations for
the motion of a charged spherical membrane with tangential tension. We stress that the
physically acceptable range of parameters for which the static and stable state of the
membrane producing the Reissner-Nordstrom repulsive gravity effect exists. The con-
crete realization of such state for the Nambu-Goto membrane is described. The point is
that membrane is able to cut out the central naked singularity region and at the same
time to join in appropriate way the Reissner-Nordstrom repulsive region.
We have obtained a model of an everywhere-regular material source exhibiting a repulsive
gravitational force in the vicinity of its surface: this construction gives a more sensible
physical status to the the naked singularity part of the Reissner-Nordstrom solution.
In the second one [75], we describe the equation of motion of two charged spherical
shells with tangential pressure in the field of a central Reissner-Nordstrom source. We
solve the problem of determining the motion of the two shells after the intersection by
solving the related Einstein-Maxwell equations and by requiring a physical continuity
condition on the shells velocities.
We consider also four applications: post-Newtonian and ultra-relativistic approxima-
tions, a test-shell case, and the ejection mechanism of one shell.
This work is a direct generalization of Barkov-Belinski-Bisnovati-Kogan [7] paper.
115
I. Charged membrane as a sourcefor repulsive gravity
Introduction
One of the interesting effects of relativistic gravity which has no analogue in the New-
tonian theory is the presence of gravitational repulsive forces. The classical example is
the Reissner-Nordstrom (RN) field in the region close enough to the central singularity.
Indeed, in the RN metric
−ds2 = −f c2dt2 + f−1 dr2 + r2(dθ2 + sin2 θdφ2) (7.34)
where
f = 1− 2kM
c2r+
kQ2
c4r2, (7.35)
the radial motion of a test neutral particle follows the equation:
d2r
ds2= −1
2
df
dr=
k
c4r2
(Q2
r−Mc2
)(7.36)
from where one can see the appearance of repulsive force in the region of small r. In this
zone the gradient of the gravitational potential f(r) is negative and the gravitational
force in Eq.(7.36) is directed toward the outside of the central source.
For the RN naked singularity case (Q2 > kM2), in which we are interested in the
present paper, the potential f(r) is everywhere positive and has a minimum at the point
r = Q2/Mc2. Therefore at this point a neutral particle can stay at rest in the state of
stable equilibrium (the detailed study can be found in [27, 21]).
It is an interesting and nontrivial fact that the same sort of stationary equilibrium state
due to the repulsive gravity exists also as an exact asymptotically flat two-body solution
of the Einstein Maxwell equations which describes a Schwarzschild black hole situated at
rest in the field of a RN naked singularity without any strut or string between these two
objects [4, 6]. However, solutions of this kind have the feature that the object creating
the repelling region has naked singularity and this last property has no clear physical
interpretation. Consequently the pertinent question is whether the repelling phenomenon
116
117
around a charged source arises only due to the presence of the naked singularity or it
can be also a feature of physically reasonable structure of the space-time and matter.
By other words the question is whether or not it is possible to construct a regular material
source which can block the central singularity and join the external repulsive region in
a proper way. Then we are interested to construct a body with the following properties:
1. inside the body there are no singularities;
2. outside the body there is the RN field (7.34)-(7.35), corresponding to the case
Q2 > kM2;
3. the radius of the body is less than Q2/Mc2, so between the surface of the body
and the sphere r = Q2/Mc2 arises the repulsive region;
4. such stationary state of the body is stable with respect to collapse or expansion.
In this paper we propose a new model for such body in the form of spherically symmetric
thin membrane with positive tension. We assert that there exists a physically acceptable
range of parameters for which all the above four conditions (1)-(4) can be satisfied. We
illustrate this conclusion by the especially transparent case of a Nambu-Goto membrane
with equation of state ε = τ .
Then the existence of everywhere-regular material sources possessing RN “antigravity”
properties in the vicinity of their surfaces attribute to this phenomenon and to the RN
naked singularity solution more sensible physical status.
It is necessary to mention that at least two exact solutions of Einstein-Maxwell equations
representing a compact continuous spherically symmetric distribution of charged matter
under the tension producing the gravitationally repulsive forces inside the matter as
well as in some region outside of it already exist in the literature. These are solutions
constructed in [26] and [85]. A more detailed study of these two results can be found
in [42]. An interesting possibility to have a gravitationally repulsive core of electrically
neutral but viscous matter has been communicated in [77]. It is worth to remark that
the first (to our knowledge) mentioning of the gravitational repulsive force due to the
presence of electric field was made already in 1937 in [48] in connection to the nonlinear
model of electrodynamics of Born-Infield type. One of the first paper where a repulsive
phenomenon in the framework of the conventional Einstein-Maxwell theory has been
mentioned is [32]. The general investigation of the different aspects of this phenomenon
apart from the already mentioned references can be found also in the more detailed works
[51, 52, 38, 78]. Some part of these papers is dedicated to a possibility of construction a
classical model for electron. This is doubtful enterprise, however, because the intrinsic
structure of electron is a matter out of classical physics. Nonetheless the mathematical
118
results obtained are useful and can be applied to the physically sensible situations, e.g.
for construction the models of macroscopical objects.
Equation of motion of a membrane with empty space
inside
The equation of motion for the most general case of a thin charged spherically symmetric
fluid shell with tangential pressure moving in the RN field have been derived 38 years
ago by J.E. Chase [24]. The corresponding dynamics for a charged elastic membrane
with tension follows from his equation simply by the change of the sign of the pressure.
We derived, however, the membrane’s dynamics again using a different approach.
Chase used the geometrical method which have been applied to the description of sin-
gular surfaces in relativistic gravity in [53] and have been elaborated in [32, 56] for some
special cases of charged shells. An essential development of the Israel approach in appli-
cation to the cosmological domain walls can be found in the series of works of V.Berezin,
V.Kuzmin and I. Tkachev, see [16] and references therein. Our treatment follows the
method more habitual for physicists which have been used in [7], where the motion of
a neutral fluid shell in a Schwarzschild field was derived by the direct integration of the
Einstein equations with appropriate δ-shaped source. Now we generalized this approach
for the charged membrane and charged central source.
Of course, the membrane’s equation of motion that we obtained coincides with that of
Chase. Nonetheless the different approach to the same problem often has a methodolog-
ical value and gives new details. We hope that our case makes no exception, then for
an interested reader we put the main steps of our derivation in Appendix A (where we
considered a general case with central source).
In this section we study only the particular solution in which there is no central body,
that is inside the membrane we have flat space-time.
Although the basic formulas of this section follow from the Appendix A under restriction
Min = Qin = 0 the exposition we give here is more or less self-consistent. Only the defi-
nitions of 4-dimensional membrane’s energy density and tension need some clarification
which can be found in Appendix A.
For the thin spherically symmetric membrane with empty space inside and with radius
which depends on time the metrics inside, outside and on membrane are:
− (ds2)in = −Γ2(t)c2dt2 + dr2 + r2(dθ2 + sin2 θdφ2) (7.37)
− (ds2)out = −f(r)c2dt2 + f−1(r)dr2 + r2(dθ2 + sin2 θdφ2) (7.38)
− (ds2)on = −c2dη2 + r20(η)(dθ2 + sin2 θdφ2) (7.39)
119
In the interval (7.113) η is the proper time of the membrane. The factor Γ2 in (7.110) is
necessary to ensure the continuity of the global time coordinate t through the membrane.
The metric coefficient f(r) in the region outside the membrane is given by Eq.(7.35).
Matching conditions for the intervals (7.110)-(7.113) through the membrane’s surface
are:
[(ds2)in]r=r0(η) = [(ds2)out]r=r0(η) = (ds2)on (7.40)
If the equation of motion of the membrane r = r0(η) is known, then from these conditions
the connection t(η) between global and proper times and factor Γ(t) follow easily:
Γ(t) =f(r0)
√1 + c−2(r0,η)2
√f(r0) + c−2(r0,η)2
(7.41)
d t
dη=
√f(r0) + c−2(r0,η)2
f(r0)(7.42)
The differential equation for the function r0(η) follows from Einstein-Maxwell equations
with energy-momentum tensor and charge current concentrated on the surface of the
membrane. It is:
Mc2 = µ(r0)c2
√1 +
(d r0
c dη
)2
+Q2
2r0
− kµ2(r0)
2r0
(7.43)
Here µ(r0) > 0 is the effective rest mass of the membrane in the radially comoving
frame. This quantity includes the membrane’s rest mass as well as all kinds of interaction
mass-energies between membrane’s constituents, that is those intrinsic energies which are
responsible for the tension. The constants Q and M are the total charge of the membrane
and total relativistic mass of the system. These are the same constants which appeared
earlier in Eq.(7.35). The membrane’s energy density ε and tension τ are (see Appendix
A for a further clarification):
ε = ε0(r0)δ[r − r0(η)] τ = τ0(r0)δ[r − r0(η)] (7.44)
where
ε0 =µ(r0)c
2
8πr20
[1√
1 + c−2(r0,η)2+
f(r0)√f(r0) + c−2(r0,η)2
](7.45)
τ0(r0) =dµ(r0)
dr0
r0ε0(r0)
2µ(r0)(7.46)
The electromagnetic potentials have the form Ar = Aθ = Aφ = 0, At = At(t, r) and for
the electric field strength ∂At/∂r the solution is
∂At
∂r=
Qr2 for r > r0(η)
0 for r < r0(η)
(7.47)
120
The formulas (7.110)-(7.126) give the complete solution of the problem for the case of
empty space inside the membrane.
Finally we would like to stress the following important point. As follows from discussion
in Appendix A, the signs of the square roots√
1 + c−2(r0,η)2 and√
f(r0) + c−2(r0,η)2
coincide with the signs of the time component u0 of the 4-velocity of the membrane
evaluated from inside and outside of the membrane respectively. The component u0 is
a continuous quantity by definition and can not change the sign when passing through
the membrane’s surface. Besides, for macroscopical objects we are interested in in this
paper u0 should be positive. Consequently the both aforementioned square roots should
be positive. From another side it is easy to show that equation (7.119) can be written
also in the following equivalent form
Mc2 = µc2
√f(r0) +
(d r0
c dη
)2
+Q2
2r0
+kµ2
2r0
(7.48)
Then from this expression and from (7.119) follows that both square roots will be positive
if and only if
Mc2 − Q2
2r0
− kµ2
2r0
> 0 (7.49)
This is unavoidable constraint which must be adopted as additional condition for any
physically realizable solution of the equation of motion (7.119) in classical macroscopical
realm.
Nambu-Goto membrane with “antigravity” effect
To proceed further we must specify the function µ(r0), which is equivalent to specifying
an equation of state, as can be seen from (7.125).
Let us analyze the membrane with equation of state ε = τ . This model can be inter-
preted as “bare” Nambu-Goto charged membrane [55, 49], or as Zeldovich-Kobzarev-
Okun charged domain wall [90]. It follows from (7.125) that for such type of membrane
we have:
µ = γr20 (7.50)
where γ is an arbitrary constant. In this and next section we consider only the case of
positive constants γ and M :
γ > 0 , M > 0 . (7.51)
121
The sign of Q is of no matter since the charge appear everywhere in square. Now we
write the equation of motion (7.119) in the following form:
4
(d r0
c dη
)2
−(
kγr0
c2+
2M
γr20
− Q2
c2γr30
)2
= −4 . (7.52)
Formally this can be considered as the equation of motion of a non-relativistic particle
with the “mass” equal to 8 moving in the potential U(r0),
U(r0) = −(
kγr0
c2+
2M
γr20
− Q2
c2γr30
)2
(7.53)
and under that condition that particle is forced to live on the “total energy” level equal
to minus four.
For the existence of the stable stationary state we are interested in, the following condi-
tions should hold:
1. The gravitational field in the exterior region should correspond to the super-
extreme RN metric:
Q2 > kM2. (7.54)
2. The potential U(r0) should have a local minimum at some value r0 = Rmin. The
form (7.53) of U(r0) permit this if and only if
kγ2Q6 < (Mc2)4. (7.55)
Under this restriction the potential U(r0) has three extrema, two maxima at points
r0 = R(1)max and r0 = R
(2)max and a minimum which is located between them: R
(1)max <
Rmin < R(2)max. We show the shape of the potential U(r0) for this case in Fig.1.
The equation U(r0) = 0 has only one real root and this is also the first local
maximum R(1)max. The minimum and the second maximum are coming as two other
roots of the equation dUdr0
= 0.
The equation for Rmin is:
kγ2R4min − 4Mc2Rmin + 3Q2 = 0 . (7.56)
This fourth order equation has only two real solutions and Rmin is the smaller one.
3. For the stationary position of the membrane at the minimum of the potential we
must ensure the relation U(Rmin) = −4 which is:
kγ
c2Rmin +
2M
γR−2
min −Q2
c2γR−3
min = 2 (7.57)
(the minus two in the r.h.s. of (7.131) would be incompatible with Eq.(7.56) under
condition (7.127)).
122
Figure 7.1: The membrane’s motion can be described as the motion of a non-relativisticpoint particle in the potential U(r0).
4. To have repulsive region it is necessary for the membrane’s radius Rmin to be
less than the minimum of the gravitational potential f(r), that is less than the
quantity Q2/Mc2. In this case outside of the membrane surface in the region
Rmin < r < Q2/Mc2 we have the repulsive effect. Then we demand:
Rmin <Q2
Mc2. (7.58)
5. Also the additional constraint (7.49) should be satisfied. This means that for our
stationary solution we have to satisfy the inequality:
Mc2 − Q2
2Rmin
− kγ2
2R3
min > 0 . (7.59)
6. We have also another condition: that the electric field nearby the membrane
should be not too large, otherwise the stability of the model would be destroyed
123
by the strong macroscopical consequences of quantum effects, e.g. by the inten-
sive electron-positron pair creation. This condition (which was suggested by J.A.
Wheeler long time ago [8]) is:
Q
R2min
<< Ecr , Ecr =m2
ec3
ee~, (7.60)
where me and ee are the electron’s mass and charge). Ecr is the well known critical
electric field above which the intensive process of pair creation starts.
To satisfy these six conditions we have to find a physically acceptable domain in the
space of the four parameters M , Q, γ and Rmin. The point is that such domain indeed
exists and it is wide enough. If we introduce the dimensionless radius of the stationary
membrane x as
kγ
c2Rmin = x , (7.61)
then one can check directly that the first five of the above formulated conditions will be
satisfied under the following three constraints:
x < 1 (7.62)
M =c4
k2γ(3x2 − 2x3) (7.63)
Q2 =c8
k3γ2(4x3 − 3x4) (7.64)
The last two of these relations are just the equations (7.56) and (7.131) but written in
the form resolved with respect to M and Q2.
The formulas (7.138)-(7.141) shows that for the first five conditions it is convenient to
take x < 1 and γ as independent parameters, and then to calculate the mass and charge
necessary to obtain the model we need.
As for the last constraint (7.136) it gives some restriction also for parameter γ:
kγ2 <<x
4− 3xE2
cr . (7.65)
The energy density ε for the stationary state at r0 = Rmin, expressed in terms of para-
meters x and γ, is:
ε =γc2
8π(1 +
√x2 − 2x + 1)δ(r −Rmin). (7.66)
124
Appendix
For the spherically symmetric case the metric6 is:
−(ds)2in = g00c
2dt2 + g11dr2 + r2(dθ2 + sin2 θdφ2) (7.67)
where g00 and g11 depend only on t, r and the standard notation for the coordinates is:
(x0, x1, x2, x3) = (ct, r, θ, φ) . (7.68)
The Electromagnetic tensor Fik has the form:
Fik = Ak,i − Ai,k (7.69)
and the Einstein-Maxwell equations are:
Rki −
1
2Rδk
i =8πk
c4T k
i (7.70)
(F ik);k =4π
cρui (7.71)
The energy-momentum tensor for a spherical charged membrane with energy density ε
and tangential tension τ is:
T ki = ε uiu
k − (δ2i δ
k2 + δ3
i δk3)τ +
1
4π(FilF
kl − 1
4δki FlmF lm) (7.72)
and for the membrane’s 4-velocity ui we have:
u0 = u0(t, r), u1 = u1(t, r), u2 = u3 = 0 , uiui = −1 . (7.73)
The main step is to define the 4-invariant charge and energy densities ρ and ε. After
that, the tension τ follows automatically from the Einstein-Maxwell equations and from
the equation of state. To construct ρ and ε we apply the Landau-Lifschitz procedure
[57].
The charge dq in the 3-volume element dV =√
g11g22g33 dx1dx2dx3 is a 4-invariant quan-
tity by definition (although dV is not a 4-scalar). The three-dimensional charge density
ρ(3) can be introduced by the relation dq = ρ(3)dV . Consequently, for the spherically
symmetric membrane case it is:
ρ(3) =Qδ(r − r0)
4πr2√
g11
, (7.74)
6We use the notations in which the interval is written as −ds2 = gikdxidxk and metric signature is(−, +, +,+), i.e. the time-time component g00 is negative. The norm of a time-like vector is negative.The Roman indices take values 0, 1, 2, 3. The Newtonian constant is denoted by k. The simple partialderivatives we designated by a comma, while covariant derivatives by semicolon.
125
where Q is the electric charge of the membrane and r0 is the membrane’s radius. Indeed
it is easy to check that Q =∫
ρ(3)dV as it should be7.
Since ρ(3)dV is a 4-scalar the quantities ρ(3)dV dxi represent a 4-vector. With the use of
the previous formula we obtain:
cρ(3)dV dxi =cQδ(r − r0)
4πr2u0√−g00g11
ui√−g d4x , (7.75)
where g is the 4-metric’s determinant. The last formula shows that the factor in front
of ui√−gd4x is a 4-scalar. This scalar is nothing else but the 4-invariant charge density
ρ which appeared in the Maxwell equation (7.71):
ρ =cQδ[r − r0(t)]
4πr2u0√−g00g11
. (7.76)
For the electric current jk we have jk = ρuk.
The 4-scalar energy density ε which figure in the energy-momentum 4-tensor (7.72) can
be constructed exactly in the same way if we observe that the rest energy of the matter
in a 3-volume element dV (i.e. the sum of the all kinds of the internal energies of this
element in the reference system in which this element is at rest) is a 4-invariant quantity
by definition. Then we can introduce the 3-dimensional rest energy density (the direct
analogue of the previous charge density ρ(3)) which under integration over 3-volume gives
the total rest energy µc2 of the membrane. Then µc2 is the sum of the all kinds of internal
energies of the membrane in the radially comoving system in which membrane is at rest.
In this way we obtain:
ε =µc2δ[r − r0(t)]
4πr2u0√−g00g11
. (7.77)
Clearly the effective rest mass µ of the membrane in the presence of a tension depends
on the membrane radius r0(t).
In the case of spherical symmetry the electromagnetic potentials Ai can be taken in the
form:
A0 = A0(t, r), A1 = A2 = A3 = 0, (7.78)
which gives only one nonvanishing component for the electromagnetic tensor Fik, namely
F10 (and its antisymmetric partner F01):
F10 = A0,1 . (7.79)
Now, we enter with definitions (7.67)-(7.69) and (7.72)-(7.79) into the Einstein-Maxwell
equations (7.70)-(7.71) to calculate the solution. These calculations need special care
7The δ-function in curved metric (7.67) is defined by the usual relation∫
δ(r − r0)dr = 1. Suchδ-function has dimension cm−1.
126
since we are dealing with distributions in application to the non-linear theory. In general
this is not a trivial task (see e.g. [84, 39, 82]), however, for particular case of spherical
symmetry everything is tractable and can be done easily thanks to the specially simple
structure of the field equations. The resulting solution contains four arbitrary constants
of integration Min, Qin and Mout, Qout which have an obvious interpretation as mass and
charge of a central RN source and the total mass and charge of the whole system (the
central body together with the membrane) respectively. The membrane’s charge Q is
simply the difference of Qout and Qin:
Q = Qout −Qin . (7.80)
To represent the solution in compact form we use the proper time η of the membrane,
denoting the membrane’s equation of motion as r = r0(η), and introducing the following
notations:
φin(r) = 1− 2k Min
c2r+
kQ2in
c4r2
φout(r) = 1− 2k Mout
c2r+
kQ2out
c4r2
(7.81)
Sin(η) =√
φin(r0) + c−2(r0,η)2
Sout(η) =√
φout(r0) + c−2(r0,η)2
(7.82)
We consider the global time t in (7.67) as continuous quantity when passing through the
membrane. Then the intervals inside, outside and on the membrane are:
− (ds2)in = −Γ2(t)φin(r)c2dt2 +dr2
φin(r)+ r2(dθ2 + sin2 θdφ2) (7.83)
− (ds2)out = −φout(r)c2dt2 +
dr2
φout(r)+ r2(dθ2 + sin2 θdφ2) (7.84)
− (ds2)on = −c2dη2 + r20(η)(dθ2 + sin2 θdφ2) (7.85)
The matching conditions for these intervals through the membrane are:
[(ds2)in]r=r0(η) = [(ds2)out]r=r0(η) = (ds2)on (7.86)
Using the relations (7.86), the factor Γ(t) in (7.83) and the connection t(η) between
global and proper times can be expressed through the membrane’s radius r0(η):
dt
dη=
Sout
φout(r0), (7.87)
Γ(t) =φout(r0)Sin
φin(r0)Sout
. (7.88)
127
Namely the continuity conditions (7.86) and continuous character of the time variable t
are responsible for the appearance of the term Γ2(t) in g00 in Eq.(7.87). Since this term
depends only on time, it can be easily removed by passing to the internal time variable
tin by the transformation
Γdt = dtin , (7.89)
which can be found with the help of (7.87) after the function r0(η) became known. In
terms of the variables (tin, r) also the internal metric (7.83) takes the standard RN form.
As it was already mentioned, the membrane’s effective rest mass µ which appeared in the
energy density (7.77) depends on the membrane radius. The concrete form of the function
µ(r0) is not known in advance and its specification is equivalent to the specification of the
equation of state. For an arbitrary µ(r0) the Einstein-Maxwell equations (7.70)-(7.71)
give the following equation of motion for the membrane:
Moutc2 −Minc
2 =1
2(Sin + Sout)µc2 +
QQin
r0
+Q2
2r0
, (7.90)
together with the condition that both square roots Sin and Sout defined by (7.82), should
have the same sign. The provenance of this condition is due to the fact that the signs of
Sin and Sout are nothing else but the signs of the time-component of u0 of the membrane’s
4-velocity when it is seen from the inside (r → r0 − 0) and outside (r → r0 + 0) of the
membrane surface respectively. In our approach (with continuous coordinates t, r) we
can consider the 4-velocity ui as a field continuous through the surface of the membrane.
We can define ui everywhere in space-time simply by smooth parallel transport from the
membrane’s surface, no matter that the membrane is concentrated only at the points
r = r0. This concentration is ensured not by ui but due to the δ-functions in the densities
ρ and ε. Since u0 can not change sign passing through the membrane, Sin and Sout should
have the same sign.
Of course, we need to know the fields u0 and u1 only on the membrane, and there they
are:
u0 = t,η , u1 = c−1r0,η . (7.91)
It is easy to check that the matching conditions (7.87) and (7.88) are nothing else but
the demand that the normalization constraint uiui = −1 should hold independently from
which side we approach the surface of the membrane.
It is worth to be remarked that the Einstein-Maxwell equations also demand for the
trajectory r0(η) the second order (in time) differential equation of motion. However,
this last one represents simply the result of the differentiation in time of the first order
128
equation (7.90). Then this second-order equation we can forget safely.
The resulting expressions for the energy density and tension are:
ε =µc2
8πr20
[φin(r0)
Sin
+φout(r0)
Sout
]δ[r − r0(η)] (7.92)
τ =r0
2µ
dµ
dr0
ε . (7.93)
The electric field F10 outside the membrane is:
F10 =Qout
r2, r > r0 . (7.94)
Inside the membrane we have:
F10 =Qin
r2
dtindt
, r < r0 , (7.95)
where the factor dtindt
depends only on time and can be calculated from the relations
(7.88) and (7.89). The origin of this factor is due to the fact that we use the time t as
continuous global time including the region inside the membrane. If we describe the in-
ternal metric in terms of internal time tin the field strength F10 would be simply Qin/r2.
The formulas (7.81)-(7.85), (7.87), (7.88) and (7.90)-(7.95) provide the complete solution
of the problem. It is worth explaining briefly the main steps of our integration procedure
that we applied to the Einstein-Maxwell equations.
As in any spherically symmetric problem it is convenient to use, instead of the full original
Einstein equations (7.70), only its (00), (1
1) and (10) components, and the hydrodynamical
equations T ki;k = 0. All the remaining components of equations (7.70) after that will be
satisfied identically either due to the Bianchi identities or due to the symmetry of the
problem. Then the solution for g11 together with the basic eq.(7.90) follows from (00) and
(10) components of Einstein equations (7.70), and after that the solution for g00 follows
from the difference of the (00) and (1
1) components of (7.70). The solution for the electric
field F10 is the result of the Maxwell equations (7.71). The hydrodynamical equations
T ki;k = 0 give only two relations. The first one simply express the tension τ in terms of
other quantities and this is the formula (7.93). The second one results in the already
mentioned second order differential equation for r0(η) which represents the differentia-
tion in time of the first order equation (7.90). Then this second order equation is of no
importance.
We remark also that the procedure described above need a caution because the sym-
bolic function are involved. Nevertheless everything going well under the following three
standard operation rules with such functions:
1. ddx
θ(x) = δ(x) ,
129
2. F (x)δ(x) = 12[F (−0) + F (+0)]δ(x) ,
3. ddx
θ2(x) = 2θ(x)δ(x) = δ(x) .
(To call the third rule as the standard one is a little exaggeration; however it works
well and final results indeed coincide with those obtained in literature by different ap-
proaches). Originally we obtained the solution in global form using the step function
θ(x) and only after that we represented the results separately in the regions r > r0 and
r < r0. However, since θ(x) is defined also at the point x = 0 [θ(0) = 1/2], we found by
the way the values for the metric and electric field also at the points of the membrane’s
surface. Such global form is:
1
g11
= 1− 2kMin
c2r− 2k(Mout −Min)
c2rθ[r − r0(η)] +
k
c4r2Qin + Qθ[r − r0(η)]2 (7.96)
1√−g00g11
=1
Γ+
(1− 1
Γ
)θ[r − r0(η)] (7.97)
F10 =
√−g00g11
r2Qin + Qθ[r − r0(η)] , (7.98)
to which should be added the equation (7.90). This equation arise as self-consistency
condition for the (00) and (1
0) components of Einstein equations, which can be verified by
the direct substitution into these components of the above global expressions together
with eqs. (7.91)-(7.93).
Finally it should be mentioned that the membrane’s equation of motion (7.90) can be
written in the following two equivalent forms:
µc2Sin = Moutc2 −Minc
2 − QinQ
r0
− Q2
2r0
+kµ2
2r0
(7.99)
µc2Sout = Moutc2 −Minc
2 − QinQ
r0
− Q2
2r0
− kµ2
2r0
. (7.100)
Each of these two equations is equivalent to (7.90) which can be checked easily by simple
algebraic manipulations. For practical calculations we can use only one of these equa-
tions, however, in addition it is necessary to ensure the same sign for both quantities
Sin and Sout. (For a membrane with empty space inside they both should be positive).
More convenient is relation (7.99) which we write as
Moutc2 =Minc
2 + µc2√
φin(r0) + c−2(r0,η)2
+QinQ
r0
+Q2
2r0
− kµ2
2r0
(7.101)
130
This is the equation obtained by Chase [24] with the aid of a different derivation proce-
dure which makes use of Gauss-Codazzi equations (see Israel [53]).
Eqn.(7.101) is interesting because in spite of the fact that µ depends on time (or on r0)
this equation looks like an usual integral of motion, that is as if µ was a constant. Rela-
tion (7.101) expresses the conservation of the total energy Moutc2 of the system which is
the sum of the five familiar constituents: 1) the rest energy of the central body, 2) the
kinetic energy of the membrane together with its gravitational potential energy in the
gravitational field of the central body, 3)the electric interaction energy between mem-
brane and central source, 4) the positive electric self-interaction energy of the membrane,
and 5) the negative gravitational self-interaction energy of the membrane.
Conclusions
1. We showed that exists a possibility to have a spherically charged membrane in sta-
ble stationary state producing RN repulsive gravitational force outside its surface and
having flat space inside. To construct such model one should take a pair of constants
0 < x < 1 and γ > 0 satisfying the inequality (7.142) and calculate from (7.137) and
(7.140)-(7.141) the membrane’s radius Rmin, total mass M and charge Q.
2. The equation of motion (7.119) can be used also for the description of the oscil-
lation of the membrane in the potential well ABC (see fig.1) above the equilibrium point
C. If we slightly increase the total membrane’s energy Mc2 then the potential U(r0)
around its minimum (i.e. the point C and its vicinity) will be shifted slightly down but
he level ”minus four” in Eq.(7.53) on which the system lives will remain at the same
position. Then the membrane will oscillate between the new shifted walls AC and CB.
3. It is easy to see that in the general dynamical state the membrane can live only
inside the potential well ABC. All regions outside ABC are forbidden. In the region to
the right from the point R(2)max and above the potential U(r0) any location of the mem-
brane is impossible due to the fact that inequality (7.49) is violated there.
This means that a membrane of considered type in principle can not have the radius (no
matter in which state) greater than R(2)max. In turn for R
(2)max it is easy to obtain from the
potential (20) the upper limit R(2)max < c2
kγ
(4k2γM
c4
)1/3
.
The same violation of the inequality (7.49) take place in the domain between R(1)max and
R(2)max and above the segment AB. The motion in the region to the left from the point
R(1)max and above the curve U(r0) is forbidden again due to the same violation of the con-
dition (7.49). This means that a membrane of considered type in principle can not have
131
the radius less than R(1)max. In particular there is no way for a membrane with positive
effective rest mass µ to collapse to the point r0 = 0 leaving outside the field correspond-
ing to the RN naked singularity solution. This conclusion is in agreement with the main
result of the paper [7].
4. Although we claimed that the stationary state of a membrane constructed is sta-
ble this stability should be understood in a very restrict sense, that is as stability in the
framework of the dynamics described by the equation (7.119). We do not know what
will happen to our membrane after the whole set of arbitrary perturbations will be given.
5. In general the arbitrary perturbations will change also the equation of state. We
investigated a membrane with equation of state ε = τ . However this case can be con-
sidered only as “bare” Nambu-Goto membrane, by other words as a toy model. In the
papers [55, 49, 76, 23, 87, 50] it was shown that arbitrary perturbations essentially renor-
malize the form of the equation of state of the strings and membranes. Moreover for
the membranes [49] (differently from the strings) the fixed points of the renormalization
group for the transverse and longitudinal perturbations does not coincide, which means
that for the general “wiggly” membrane there is no equation of state of the type ε = ε(τ)
at all.
6. We also would like to stress that for appearance of repulsive force the presence of
electric field is of no principal necessity. For example the repulsive gravitational forces
arise also in neutral viscous fluid [77] and in the course of interaction between electrically
neutral topological gravitational solitons [11].
7. From the conditions (21)-(26) also follows that in addition to the inequality (25)
the radius Rmin of the shell in the stable stationary state cannot be less than Q2
2Mc2. A
simple analysis shows that there is no way for Rmin to be arbitrarily small keeping some
finite non-zero value for M and Q.
II. Intersection of self-gravitatingcharged shells in aReissner-Nordstrom field
Introduction
The mathematical model that we analyze in this paper describes the dynamic evolu-
tion of two spherical shells of charged matter which freely move outside the field of a
central Reissner-Nordstrom (RN) source. Microscopically these shells are assumed to
be composed by charged particles which move on elliptical orbits with a collective vari-
able radius. The angular motion, distributed uniformly and isotropically on the shell
surfaces, is mathematically described by a tangential-pressure term in the energy mo-
mentum tensor of the Einstein equations. The definition of the shell implies that all the
particles have the same following three ratios: energy/mass, angular momentum/mass,
and charge/mass. Indeed, since the equations of motion for any singled-out particle ”a”
are
dtads
=1
−mac2gtt(ra)(Ea + eaA0(ra)) (7.102)
(dra
ds
)2
=1
m2ac
4(Ea + eaA0(ra))
2
(1
−gtt(ra)grr(ra)
)−
(l2a
m2ac
2
1
r2+ 1
)1
grr(ra)(7.103)
(dθa
ds
)2
=l2a
m2ac
2
1
r4− k2
a
m2ac
2
1
r4 sin2 θa
(7.104)
dϕa
ds=
ka
mac
1
r2 sin2 θa
(7.105)
(gtt and grr are the components of a spherical symmetric metric and A0 is the electric
potential; ka and la are arbitrary constants), it is easy to see that the radial motion for
all particles is the same if
Ea
ma
= const,ea
ma
= const,|la|ma
= const, ∀a, (7.106)
132
133
where each const. does not depend on the index a. Therefore, if at the beginning the
particles are on the same radius ra = R0, then the shell will evolve “coherently”, i.e. all
particles will evolve with the same radius.
Now the problem we are interested in is to find the exchange of energy between the
two shells after the intersection. Indeed the motion of the shells before and after the
intersection can be easily deduced from the equation of motion for just one shell, which
equation has been found many years ago by Chase [24] with a geometrical method first
used by Israel [53].
What we achieve in the present paper is the determination of the constant parameters
after the intersection knowing just the parameters before the intersection. Actually the
unknown parameter is only one, m21, which is the Schwarzschild mass parameter mea-
sured by an observer between the shells after the intersection. This parameter is strictly
related to the energy transfer which takes place in the crossing, and it is found imposing
a proper continuity condition on the shells velocities.
In the model we assume that the emission of electromagnetic waves is negligible, and
that there are no other interactions between the two shells apart the gravitational and
electrostatic ones. In particular the shells, during the intersection, are assumed to be
”transparent” each other (i.e. no scattering processes).
The paper is divided as follows: in Sec.2 we preliminarily discuss the one-shell case;
in Sec.3, which is the central part of this article, we find the unknown parameter m21;
then, Secs.4-7 are devoted to some applications: post-Newtonian approximation, zero
effective masses case (i.e. ultra-relativistic case), test-shell case, and finally the ejection
mechanism.
In this paper we deal only with the mathematical aspects of the problem; some astro-
physical applications of charged shells in the field of a RN black hole have been considered
in [25].
A gravitating charged shell with tangential pressure
The motion of a thin charged dust-shell with a central RN singularity was firstly studied
by De La Cruz and Israel [32], while the case with tangential pressure was achieved by
Chase [24] in 1970. All these authors used the extrinsic curvature tensor and the Gauss-
Codazzi equations. However we followed a different way, indeed the same solution can be
found also by using δ and θ distributions and then by direct integration of the Einstein-
Maxwell equations (see [7] and the appendix in [13]). This method has the advantage
of a clearer physical interpretation, and it is also straightforward in the calculations;
134
however in the following we will give only the main passages.
Let there be a central body of mass min and charge ein and let a spherical massive
charged shell with charge e move outside this body. It is clear in advance that the field
internal to the shell will be RN, while externally we will have again a RN metric but
with different mass and charge parameters mout and eout = ein +e. Using the coordinates
x0 = ct and r, which are continuous when passing through the shell, we can write the
intervals inside, outside, and on the shell as
− (ds)2in = −eT (t)fin(r)c2dt2 + f−1
in (r)dr2 + r2dΩ2 (7.107)
− (ds)2out = −fout(r)c
2dt2 + f−1out(r)dr2 + r2dΩ2 (7.108)
− (ds)2on = −c2dτ 2 + r0(τ)2dΩ2 (7.109)
where we denoted
dΩ2 = dθ2 + sin2 θdφ2
and
fin = 1− 2Gmin
c2r+
Ge2in
c4r2, fout = 1− 2
Gmout
c2r+
G(ein + e)2
c4r2. (7.110)
In the interval (7.109), τ is the proper time of the shell. The “dilaton” factor eT (t) in
(7.107) is required to ensure the continuity of the time coordinate t through the shell. If
the equation of motion for the shell is
r = R0(t), (7.111)
then joining the angular part of the three intervals (7.107)-(7.109), one has
r0(τ) = R0[t(τ)], (7.112)
where the function t(τ) describes the relationship between the global time and the proper
time of the shell. Joining the radial-time parts of the intervals (7.107)-(7.108) on the
shell requires that the following relations hold:
fin(r0)
(d t
dτ
)2
eT (t) − f−1in (r0)
(d r0
cdτ
)2
= 1 , (7.113)
fout(r0)
(d t
dτ
)2
− f−1out(r0)
(d r0
cdτ
)2
= 1 . (7.114)
If the equation of motion for the shell, i.e. the function r0(τ), is known, then the function
t(τ) follows from (7.114) and consequently T (t) can be deduced by (7.113). Thus the
problem consist only in determining r0(τ), which can be done by direct integration of
the Einstein-Maxwell equations
Rki − 1
2Rgk
i = 8πGc4
T ki
(√−gF ik),k =
√−g 4πcρui
(7.115)
135
with the energy-momentum tensor given by:
T ki = ε uiu
k + (δ2i δ
k2 + δ3
i δk3)p + T (el) k
i (7.116)
T (el) ki =
1
4π(FilF
kl − 1
4δki FlmF lm) . (7.117)
Here on we employ the following notations:
−ds2 = gikdxidxk, gik has signature (−, +, +, +)
xk = (ct, r, θ, ϕ) i, j, k... = 0, 1, 2, 3
p ≡ p(R0) = pθ = pϕ =tangential pressure (pr = 0)
Fik = Ak,i − Ai,k
The above equations are to be solved for the metric
−ds2 = g00(t, r)c2dt2 + g11(t, r)dr2 + r2dΩ2, (7.118)
and for the potential
A0 = A0(t, r), A1 = A2 = A3 = 0. (7.119)
As follows from the Landau-Lifshitz approach [57] (see [7]) the energy distribution of the
shell is
ε =M(t)c2δ[r −R0(t)]
4πr2u0√−g00g11
, (7.120)
while its charge density is
ρ =c eδ[r −R0(t)]
4πr2u0√−g00g11
, (7.121)
where δ is the standard δ-function. In the absence of tangential pressure p, the quantity
M in Eqn.(7.120) would be a constant, but in presence of pressure, Mc2 includes the rest
energy along with the energy (in the radially comoving frame) of the tangential motions
of the particles that produce this pressure.
It can be checked that the Einstein part of (7.115) actually lead to the solution (7.107)-
(7.109) with, in addition, the “joint condition”
√fin(r0) +
(d r0
cdτ
)2
+
√fout(r0) +
(d r0
cdτ
)2
= 2(mout −min)
µ(τ)− e2 + 2eein
µ(τ)c2r0
, (7.122)
where we denoted
µ(τ) = M [t(τ)], (7.123)
136
while
mout −min = E/c2 (7.124)
is a constant which can be interpreted as the total amount of energy of the shell. Then,
from the Maxwell side of (7.115) the only non-vanishing component of the electric field
is
F01 = −√−g00g11
r2ein + eθ[r −R0(t)] (7.125)
(θ(x) is the standard step function). Finally, the equations T ki;k = 0 can be reduced to
the only one relation:
p = −dM
dt
c2δ[r −R0(t)]
8πru1√−g00g11
(7.126)
We will not treat here the steady case (i.e. r0 = const) which should be treated sepa-
rately; thus in the following we will assume always r0 6= const. .
The joint condition (7.122) can be written in several different forms: two of them, which
will be useful in the following, are
√fin(r0) +
(d r0
cdτ
)2
=(mout −min)
µ(τ)+
Gµ2(τ)− e2 − 2eein
2µ(τ)c2r0
(7.127)
and √fout(r0) +
(d r0
cdτ
)2
=(mout −min)
µ(τ)− Gµ2(τ) + e2 + 2eein
2µ(τ)c2r0
. (7.128)
As in [7], all the radicals encountered here are taken positive, since for astrophysical
considerations only these cases are meaningful. To proceed further, we must specify the
equation of state, i.e. the function µ(τ). Here we consider a particle-made shell, therefore
the sum of kinetic and rest energy of all the particles is
Mc2 =∑
a
(mac
2
√1 +
p2a
m2ac
2
), (7.129)
where pa is the tangential momentum of each particle (the electric interaction between
the particles is already taken into account by the self-energy term of, e.g., (7.127), thus
one has not to include it in M too). From the definition of the shell (see Introduction)
it follows:p2
a
m2a
=l2a
m2aR
20
=const
R20
, (7.130)
the square root in (7.129) does not depend on the index a; then defining
∑a
mac2 = mc2,
∑a
|la| = L,
137
formula (7.129) can be re-written (remembering definition (7.123) too) as
µ(τ) =
√m2 +
L2
c2r20(τ)
. (7.131)
Thus, now, one can determine the function r0(τ) from equation (7.122) (or from one of the
equivalent forms (7.127)-(7.128)) if the initial radius of the shell and the six arbitrary
constants min, mout, m, ein, e and L are specified. Accordingly with (7.120), (7.126),
(7.123) and (7.131), the equation of state that relates the shell energy density ε to the
tangential pressure p is
p =ε
2
L2
m2c2R20
(1 +
L2
m2c2R20
)−1
(7.132)
as in the uncharged case, i.e. the presence of the charges do not modify the relation
between energy density and pressure (indeed the presence of the charge is hidden in the
equation of motion). Note that when the shell expands to infinity (R0 →∞) the angular
momentum becomes irrelevant and the equation of state tends to the dust case p << ε.
The shells intersection
Let us now consider the case of two shells which move in the field of a central charged
mass. The generalization from the previous (single-shell) case is straightforward if the
shells do not intersect: indeed the outer shell do not affect the motion of the inner one,
while the inner one appears from outside just as a RN metric. Therefore the principal
aim of this section is to consider the intersection eventuality and to predict the motion
of the two shells after the crossing, having specified the initial conditions before the
crossing. After the intersection one has a new unknown constant that has to be found
by imposing opportune joining conditions as now we are going to explain (the analysis
follows step by step the [7]’s one). Let us previously analyze the space-time in the (t, r)
coordinates (which are continuous through the shells). We define the point O ≡ (t∗, r∗)
as the intersection point; then the space-time is divided in four regions (see Fig.1):
COB (r > R1, r > R2),
COA (R1 < r < R2),
AOD (r < R1, r < R2),
BOD (R2 < r < R1).
(7.133)
138
Figure 7.2: The four region in which it is divided the spacetime; the two lines representthe trajectories of shell-1 and shell-2.
Correspondingly to these regions we have the metric in form (7.114) but with different
coefficients g00 and g11:
g(COB)00 = −fout(r) , g
(COB)11 = f−1
out(r) (7.134)
g(COA)00 = −eT1(t)f12(r) , g
(COA)11 = f−1
12 (r) (7.135)
g(AOD)00 = −eT0(t)fin(r) , g
(AOD)11 = f−1
in (r) (7.136)
g(BOD)00 = −eT2(t)f21(r) , g
(BOD)11 = f−1
21 (r) (7.137)
The dilaton factor T (t) allows to cover all the space-time with only one t-coordinate;
here, fin and fout are the same as those in (7.110) while f12 and f21 are given by similar
expressions:
f12 = 1− 2Gm12
c2r+
G(ein + e1)2
c4r2(7.138)
f21 = 1− 2Gm21
c2r+
G(ein + e2)2
c4r2(7.139)
As we said, the parameters min, m12, mout, ein, e1, e2 are assumed to be specified at the
beginning, while m21 is the actual unknown constant which has yet to be determined
from the joining conditions on (t∗, r∗).
139
Before the intersection
Let us write the equation of motion for the two shells before the intersection (shell-1
inner and shell-2 outer). This can be made easily adapting the (7.128) and (7.127) to
the present case:
√f12(r1) +
(d r1
cdτ1
)2
=(m12 −min)
M1
− GM21 + e2
1 + 2eine1
2M1c2r1
(7.140)
for shell 1, while for shell 2
√f12(r2) +
(d r2
cdτ2
)2
=(m12 −min)
M2
+GM2
2 − e22 − 2(ein + e1)e2
2M2c2r2
(7.141)
with
M1 =
√m2
1 +L2
1
c2r21
, M2 =
√m2
2 +L2
2
c2r22
. (7.142)
Here, τ1 and τ2 are the proper times of the first and second shells respectively, while
r1(τ1) = R1[t(τ1)] and r2(τ2) = R2[t(τ2)]. Now we have to impose the joining conditions
for the intervals on both the shells. For the first shell (on curve AO) one has:
eT1(t)f12(r1)
(d t
dτ1
)2
− f−112 (r1)
(d r1
cdτ1
)2
= 1 (7.143)
eT0(t)fin(r1)
(d t
dτ1
)2
− f−1in (r1)
(d r1
cdτ1
)2
= 1; (7.144)
while for the second shell:
fout(r2)
(d t
dτ2
)2
− f−1out(r2)
(d r2
cdτ2
)2
= 1 (7.145)
eT1(t)f12(r2)
(d t
dτ2
)2
− f−112 (r2)
(d r2
cdτ2
)2
= 1. (7.146)
If all free parameters and initial data to Eqs.(7.140)-(7.142) were specified and if the
functions r1(τ1) and r2(τ2) were derived, then their substitution in (7.143)-(7.146) gives
the functions τ1(t), τ2(t) and T1(t), T0(t), which is enough for determining the motion of
the shells before the intersection. Therefore the intersection point (t∗, r∗) can be found
by solving the system r∗ = r1(τ1(t∗))
r∗ = r2(τ2(t∗)) ,(7.147)
which we assume that has a solution.
140
After the intersection
The equation of motion for the shells after the intersection time t∗ can be constructed
in the same way again by turning to Eqns.(7.127) and (7.128), and introducing the new
parameter m21 which characterize the “Schwarschild mass” seen by an observer in the
region BOD. We use Eq.(7.127) for (now outer) shell 1 and Eq.(7.128) for (now inner)
shell 2:√
f21(r1) +
(d r1
cdτ1
)2
=(mout −m21)
M1
+GM2
1 − e21 − 2e1(ein + e2)
2M1c2r1
, (7.148)
√f21(r2) +
(d r2
cdτ2
)2
=(m21 −min)
M2
− GM22 + e2
2 + 2e2ein
2M2c2r2
. (7.149)
Naturally, M1(r1) and M2(r2) are given by the same expression of (7.142) but now they
have to be calculated on r1(τ1) and r2(τ2) after the intersection.
Joining the intervals on the first shell (on curve OB) yields
fout(r1)
(d t
dτ1
)2
− f−1out(r1)
(d r1
cdτ1
)2
= 1 (7.150)
eT2(t)f21(r1)
(d t
dτ1
)2
− f−121 (r1)
(d r1
cdτ1
)2
= 1. (7.151)
Then, joining the second shell (on curve OB) we obtain:
eT2(t)f21(r2)
(d t
dτ2
)2
− f−121 (r2)
(d r2
cdτ2
)2
= 1 (7.152)
eT0(t)fin(r2)
(d t
dτ2
)2
− f−1in (r2)
(d r2
cdτ2
)2
= 1. (7.153)
Since the initial data to Eqs.(7.148) and (7.149) have already been specified (from the
previous evolution), then the evolution of the shells after the intersection would be de-
termined from Eqs.(7.148)-(7.153) if parameter m21 were known. Thus we need an
additional physical condition from which we could determine m21.
This condition follows from the fact that the Christoffel symbols (i.e. the accelerations)
of the shells have only finite discontinuities (finite jumps), therefore the relative velocity
of the shells must remain continuous through the crossing point.
In the presence of two shells, we can construct one more invariant than in the single
shell case (where only uiui = −1 was possible): the scalar product between the two
4-velocities of the shells. We can also avoid to apply the parallel transport if we evaluate
141
the 4-velocities on the intersection point (t∗, r∗). The continuity condition can be found
imposing that the scalar product has to have the same value when evaluated in both the
two limits t → t−∗ and t → t+∗ .
Determination of Q
Let us start determining the quantity
Q ≡ g(COA)00 u0
AOu0CO + g
(COA)11 u1
AOu1COt=t∗,r=r1=r2=r∗ , (7.154)
which is the scalar product of the two 4-velocities evaluated in the intersection point from
the region AOC (along the curves AO and CO). Written explicitly, the unit tangent
vector to trajectory AO is
uiAO = (u0
AO, u1AO, u2
AO, u3AO)
=
(d t
dτ1
,d r1
cdτ1
, 0, 0
)
t≤t∗
, (7.155)
while for the trajectory CO we have
uiCO = (u0
CO, u1CO, u2
CO, u3CO)
=
(d t
dτ2
,d r2
cdτ2
, 0, 0
)
t≤t∗
. (7.156)
The fact that these are actually unit vectors follows from the joining equations (7.143)
and (7.146).
The components of the vector (7.155) can be easily expressed from Eqs.(7.140) and
(7.143) as
(d t
dτ1
)
t≤t∗
=e−T1(t)/2
M1(r1)f12(r1)
(m12 −min − GM2
1 (r1) + e21 + 2e1ein
2c2r1
)(7.157)
(d r1
cdτ1
)
t≤t∗
=
=δ1
M1(r1)f12(r1)
√(m12 −min − GM2
1 (r1) + e21 + 2e1ein
2c2r1
)2
−M21 (r1)f12(r1)
(7.158)
where
δ1 = sgn
(d r1
cdτ1
)
t≤t∗
. (7.159)
142
Analogously, for the components of vector (7.156), we obtain the following expressions
from Eqs.(7.141) and (7.146):
(d t
dτ2
)
t≤t∗
=e−T1(t)/2
M2(r2)f12(r2)
(mout −m12 +
GM22 (r2)− e2
2 − 2e2(ein + e1)
2c2r2
)(7.160)
(d r2
cdτ2
)
t≤t∗
=δ2
M2(r2)f12(r2)·
·√(
mout −m12 +GM2
2 (r2)− e22 − 2e2(ein + e1)
2c2r2
)2
−M22 (r2)f12(r2) (7.161)
δ2 = sgn
(d r2
cdτ2
)
t≤t∗
. (7.162)
Thus, from the preceding results, we obtain:
Q = −1M1M2f12
·
·(
m12 −min − GM21 +e2
1+2e1ein
2c2r∗
)(mout −m12 +
GM22−e2
2−2e2(ein+e1)
2c2r∗
)+
−δ1δ2
√(m12 −min − GM2
1 +e21+2e1ein
2c2r∗
)2
−M21 f12
√(mout −m12 +
GM22−e2
2−2e2(ein+e1)
2c2r∗
)2
−M22 f12
;
(7.163)
here and in the following we omit the coordinate dependence of fa, Ma etc., implicitly
assuming that they have to be evaluated on (t∗, r∗) where not differently indicated.
Determination of Q′
It is possible to apply the same procedure to the region BOD (i.e. after the intersection
time), finding the quantity
Q′ ≡ g(BOD)00 u0
OBu0OD + g
(BOD)11 u1
OBu1ODt=t∗,r=r1=r2=r∗ . (7.164)
Now the unit tangent vectors to trajectories OB and OD are8:
uiOB = (u0
OB, u1OB, u2
OB, u3OB)
=
(d t
dτ1
,d r1
cdτ1
, 0, 0
)
t≥t∗
, (7.165)
8Obviously, when we say t ≥ t∗, we tacitly assume before a (possible) second intersection.
143
and
uiOD = (u0
OD, u1OD, u2
OD, u3OD)
=
(d t
dτ2
,d r2
cdτ2
, 0, 0
)
t≥t∗
; (7.166)
from the joining conditions (7.151) and (7.152) it is possible to see that these are actually
unit vectors. The components of uiOB can be deduced from Eqs.(7.148) and (7.151),
while the components of uiOD from Eqs.(7.149) and (7.152). Then, using the metric in
the region BOD, it is possible to calculate the scalar product
Q′ = −1M1M2f21
·
·(
mout −m21 +GM2
1−e21−2e1(ein+e2)
2c2r∗
)(m21 −min − GM2
2 +e22+2e2ein
2c2r∗
)+
−δ′1δ′2
√(mout −m21 +
GM21−e2
1−2e1(ein+e2)
2c2r∗
)2
−M21 f21
√(m21 −min − GM2
2 +e22+2e2ein
2c2r∗
)2
−M22 f21
,
(7.167)
where δ′1 and δ′2 have been defined as in (7.159) and (7.162), but for t ≥ t∗. We introduced
these symbols only for generality, but actually we are interested only in the case with9
δ′1 = δ1, δ′2 = δ2 . (7.168)
The necessary continuity requirement is thus
Q = Q′, (7.169)
then, since r∗ is assumed to be known, this equation allows to find m21.
Physical meaning of Q and Q′
Using standard definition for the shell velocities before the intersection one has
(v1
c
)2
=g
(COA)11 (r1)
−g(COA)00 (r1)
(d r1
cdt
)2
(7.170)
(v2
c
)2
=g
(COA)11 (r2)
−g(COA)00 (r2)
(d r2
cdt
)2
, (7.171)
9This is the only possible case if one excludes v1(t∗) = v2(t∗) = 0, because there are non disconti-nuities in the velocities.
144
and similarly for the velocities after the intersection,
(v′1c
)2
=g
(BOD)11 (r1)
−g(BOD)00 (r1)
(d r1
cdt
)2
(7.172)
(v′2c
)2
=g
(BOD)11 (r2)
−g(BOD)00 (r2)
(d r2
cdt
)2
. (7.173)
Then it is easy to obtain from the definitions (7.154) and (7.164), that10
Q =
v1v2/c
2 − 1√1− v2
1/c2√
1− v22/c
2
t=t∗,r1=r2=r∗
(7.174)
and
Q′ =
v′1v
′2/c
2 − 1√1− (v′1)2/c2
√1− (v′2)2/c2
t=t∗,r1=r2=r∗
. (7.175)
Determination of P and P′
First of all it is convenient to introduce new symbols to simplify the expressions of Q
and Q’. With
q1 ≡ −GM21 + e2
1 + 2e1ein
2c2r∗
q2 ≡ GM22 − e2
2 − 2e2(ein + e1)
2c2r∗,
and
q′1 ≡GM2
1 − e21 − 2e1(ein + e2)
2c2r∗
q′2 ≡ −GM22 + e2
2 + 2e2ein
2c2r∗,
then Q and Q′ can be re-written as
Q = −1M1M2f12
·
·
(m12 −min + q1) (mout −m12 + q2) +
−δ1δ2
√(m12 −min + q1)
2 −M21 f12
√(mout −m12 + q2)
2 −M22 f12
(7.176)
10It is also worth noting that√
Q2 − 1/Q = −|v1/c−v2/c|/(1−v1v2/c2), which is the relative velocitydefinition of two “particles” in relativistic mechanics.
145
andQ′ = −1
M1M2f21·
·
(mout −m21 + q′1) (m21 −min + q′2) +
−δ′1δ′2
√(mout −m21 + q′1)
2 −M21 f12
√(m21 −min + q′2)
2 −M22 f12
,
(7.177)
Now, in principle is possible to find m21 by squaring and solving Q = Q′ (which is a
quartic equation). However the procedure is cumbersome and moreover it is not possible
with Eq.(7.169) alone to determine the sign of the roots. Fortunately, as in the non-
charged case, it is possible to follow another easier way. Indeed, it is possible to introduce
two other invariants, say P and P ′, similar to Q and Q′, which are constructed using the
scalar products of the 4-velocities of the two shell, but now taking the limit to (t∗, r∗)
from the AOD and COB regions respectively. More explicitly, we define
P ≡ g(AOD)00 u0
AOu0OD + g
(AOD)11 u1
AOu1ODt=t∗,r=r1=r2=r∗ , (7.178)
and
P ′ ≡ g(COB)00 u0
COu0OB + g
(COB)11 u1
COu1OBt=t∗,r=r1=r2=r∗ . (7.179)
Then, the same continuity requirement of Eq.(7.169) implies that it must hold also that
Q = P , P = P ′ . (7.180)
Following the same method used to find Q and Q′, after some calculations, one arrives
toP = −1
M1M2fin·
·
(m12 −min + p1) (m21 −min + p2) +
−δ1δ′2
√(m12 −min + p1)
2 −M21 fin
√(m21 −min + p2)
2 −M22 fin
(7.181)
146
andP ′ = −1
M1M2fin·
·
(mout −m21 + p′1) (mout −m12 + p′2) +
−δ′1δ2
√(mout −m21 + p′1)
2 −M21 fout
√(mout −m12 + p′2)
2 −M22 fout
,
(7.182)
where we have denoted
p1 ≡ GM21 − e2
1 − 2e1ein
2c2r∗
p2 ≡ GM22 − e2
2 − 2e2ein
2c2r∗,
and
p′1 ≡ −GM21 + e2
1 + 2e1(ein + e2)
2c2r∗
p′2 ≡ −GM22 + e2
2 + 2e2(ein + e1)
2c2r∗.
Determination of m21; the energy transfer
Thus the complete set of continuity conditions at the point of intersection can be written
as
Q = Q′, Q = P, Q = P ′. (7.183)
It turns out that this three quartic equations for the unknown parameter m21 have only
one common root. It is possible to find the solution using hyperbolic functions. The
final result is remarkably simple:
m21 = min + mout −m12 − e1e2
c2r∗− GM1M2
c2r∗Q , (7.184)
or equivalently, in terms of f21:
f21 = fin + fout − f12 + 2G2M1M2
c4r2∗Q . (7.185)
It can be easily seen from Eqn.(7.184) that the charge ein of the central singularity does
not affect the result (but it affects the equation of the motion of the shells and thus
Q). Formula (7.184) solves the problem of determining the mass parameter m21 from
the quantities specified at the evolutionary stage before intersection. It is then possible
147
to determine the energy transfer between the shells. Indeed the energy of shell 1 and 2
before the intersection are, respectively
E1 = (m12 −min)c2 , E2 = (mout −m12)c2 , (7.186)
while, after the intersection
E ′1 = (mout −m21)c
2 , E ′2 = (m21 −min)c2 . (7.187)
The conservation of total energy is automatically ensured by the above formulas, indeed
E1 + E2 = E ′1 + E ′
2 . (7.188)
Then it is natural to define the exchange energy as
∆E = E ′2 − E2 = −(E ′
1 − E1) . (7.189)
Then, from Eqn.(7.184) and the above definitions, it follows that
∆E = −e1e2
r∗− GM1M2
r∗Q . (7.190)
It is also useful (especially for the Newtonian approximation) to use Eqn.(7.174) and
re-express ∆E as:
∆E = −e1e2
r∗− GM1M2
r∗
v1v2/c
2 − 1√1− v2
1/c2√
1− v22/c
2
r=r∗
. (7.191)
Post-Newtonian approximation
For slow velocities of the shells it is interesting to consider the Post-Newtonian limit of
Eqn.(7.191):
∆E =Gm1m2 − e1e2
r∗+
+1
2c2
Gm1m2
r∗[v1(r∗)− v2(r∗)]2 +
Gm2L21
m1r3∗+
Gm1L22
m2r3∗
+ o
(1
c4
). (7.192)
It is worth noting that only the zeroth order in 1/c2 changes with respect to the un-
charged case (because of the Coulomb term −e1e2/r∗), while all the other orders remain
unchanged, being of kinetic origin; m1 and m2 are the rest masses of the shells, indeed
we have used for the masses M1 and M2 the definitions (7.142).
148
It can be also useful to re-express all the quantities in a Newtonian language and consider
only the zeroth order in 1/c2, e.g. we can expand the energy as
E = mc2 + E + o
(1
c2
), (7.193)
where m and E do not depend on c. Therefore, similarly, we can define at the first order
in 1/c2
m12 −min = m1 +E1
c2, mout −m12 = m2 +
E2
c2, (7.194)
mout −m21 = m1 +E ′1c2
, m21 −min = m1 +E ′2c2
. (7.195)
Then it follows also that the energy conservation law takes the form
E1 + E2 = E ′1 + E ′2 , (7.196)
and Eqn.(7.189) becomes
E ′1 = E1 −∆E , E ′2 = E2 + ∆E , (7.197)
where ∆E = (∆E)c→∞. Thus from the above formulas and definitions it is clear that
∆E =Gm1m2 − e1e2
r∗. (7.198)
Pressureless shells with zero effective masses (L1 =
L2 = 0 and M1 = M2 = 0)
It is interesting also to consider the case in which the motion of the particles of the
shells is only radial (i.e. L1 = L2 = 0) and the rest masses are negligible with respect to
the kinetic energies and to the charges —indeed this is the case for two shells composed
by (ultra)rela-tivistic electrons and positrons. In this case the effective masses can be
replaced by
M1 = M2 = λ , (7.199)
where λ is a parameter arbitrary small.
From Eqn.(7.190), with Q expressed by formula (7.163), it is easy to find that the energy
transfer in this case is
∆E = −e1e2
r∗+
c4r∗2Gf12
(fin − f12)(f12 − fout) + o(λ2) , (7.200)
149
having assumed that the shells have opposite-directed velocities, i.e.
δ1δ2 = −1 . (7.201)
Otherwise, if the shells goes in the same direction, i.e.
δ1δ2 = 1 , (7.202)
then Eqn.(7.190) becomes simply
∆E = −e1e2
r∗+ o(λ2) ; (7.203)
obviously the previous formulas make sense only if r∗ exists. We want to underline the
presence of the term o(λ2), because, strictly speaking, a charge cannot have zero rest
mass, therefore we are in the case of just small effective masses. As expected, in the
case of vanishing charges (e1 = e2 = 0), Eqn.(7.203) gives zero at λ = 0 because this is
the case of two photon-shells which go in the same direction and therefore cannot never
intersect.
The intersection of a test shell with a gravitating one
One-shell case
Let us consider firstly the case of a test shell on the RN field. This limit has the only
aim to show that the shell’s equation of motion (7.127) actually reduce to the simple
test-particle case; the limit can be obtained by putting
m → λm , e → λm , L → λL , (mout −min)c2 → λE (7.204)
with λ → 0. Then, considering also (7.131), we find that Eqn.(7.127) becomes
E = µc2
√fin(r0) +
(d r0
cdτ
)2
+eein
r0
− λGµ2 − e2
2r0
, (7.205)
now, putting λ = 0 the self-energy term is killed; then re-writing Eqn.(7.205) using the
more familiar Schwarzschild time t and Eqn.(7.112),
E = c2
√m2 +
L2
c2R20(t)
√f 3
in(R0)
f 2in(R0)−
(dR0
cdt
)2 +eein
R0
+ o(λ) , (7.206)
it is easy to recognize that Eqn.(7.206) coincides with the first integral of motion of a
test-charge particle on the Reinssner-Nordstrom background, where E is the conserved
energy of the particle, m the rest mass, e the charge and L the angular momentum.
150
Two-shell case, with one test-shell
Now we can deal with the more interesting two-shell case, in which shell-2 is considered
”test”. To gain this limit we have to put
m2 → λm2 , e2 → λm2 , L2 → λL2 , (7.207)
(mout −m12)c2 → λE2 , (m21 −min)c2 → λE ′
2 .
Then, using Eqn.(7.184) with Q given by formula (7.163), one obtains
∆E = − e1e2
r∗+ 1
r∗f12·
·(
E1 − GM21 +e2
1+2e1ein
2c2r∗
)(E2 − e2(ein+e1)
c2r∗+ λ
GM22−e2
2
2c2r∗
)+
−δ1δ2
√(E1 − GM2
1 +e21+2e1ein
2c2r∗
)2
−M21 f12
√(E2 − e2(ein+e1)
c2r∗+ λ
GM22−e2
2
2c2r∗
)2
−M22 f12
.
(7.208)
Thus, only the self-energy terms of shell-2 are killed by λ = 0.
Now, it is worth noting the following fact: shell-1 does not have any discontinuity when
it intersect the shell-2 (this is natural because shell-2 is “test” and does not affect the
metric), on the other hand shell-2 undergoes a discontinuity in the metric when it cross
shell-1 and consequently it has an actual discontinuity in the velocity. It is easy to
calculate this gap; indeed using the definition (7.171) of velocity v2 [with the time d tdτ2
given by the joint condition (7.146)], with metric coefficient (7.135), and with the help
the first integral of motion (7.141), one finds
v22(r2) = 1− fout(r2)
(E2
M2(r2)− e2(e1 + ein)2
M2(r2)r2
)−2
+ o(λ) , t ≤ t∗ , (7.209)
where we have used f12 = fout + o(λ); in the same way, using (7.173), (7.152), (7.137),
and (7.149), the velocity v′2 (after the intersection) is
[v′2(r2)]2 = 1− fin(r2)
(E ′
2
M2(r2)− e2(e1 + ein)2
M2(r2)r2
)−2
+ o(λ) , t ≥ t∗ , (7.210)
where E ′2 can be expressed in function of E2 with the help of (7.208). From the previous
formulas it is clear that in general
v′2(r∗)− v2(r∗) 6= 0 . (7.211)
151
Shell ejection
The exchange in energy of the shells during the intersection makes possible that one
initially bounded shell can acquire enough energy to escape to infinity.
The shell ejection mechanism can take place also in the Newtonian regime. In this case,
from Eqs.(7.197)-(7.198) it results that
E ′1 = E1 − Gm1m2 − e1e2
r′∗, E ′2 = E2 +
Gm1m2 − e1e2
r′∗, (7.212)
and then, after the first intersection
E ′′1 = E ′1 + Gm1m2−e1e2
r′′∗= E1 − (Gm1m2 − e1e2)
(1r′∗− 1
r′′∗
)
E ′′2 = E ′2 − Gm1m2−e1e2
r′′∗= E2 + (Gm1m2 − e1e2)
(1r′∗− 1
r′′∗
),
(7.213)
where we have denoted the radius of the first and second intersection with r′∗ and r′′∗respectively. In the following we will consider only the case
Gm1m2 − e1e2 > 0 , (7.214)
this is e.g. the case in which the two shells have opposite charges. Thus, also in the case
E1, E2 < 0, if
r′′∗ > r′∗ , (7.215)
and if the initial condition were in such a way that r′∗ is enough small and r′′∗ not too
much close to r′∗, then it is possible to have E ′′2 > 0, i.e. the ejection of the second shell.
Let us now assume that r′′∗ > r′∗, and consider a “semi-relativistic” case in which at the
first intersection we use the full relativistic formulas11,
E ′1 = E1 − M1(r′∗)M2(r′∗)
r′∗(−Q) + e1e2
r′∗
E ′2 = E2 + M1(r′∗)M2(r′∗)
r′∗(−Q)− e1e2
r′∗,
(7.216)
while at the second intersection we use the Newtonian approximation,
E ′′1 = E ′
1 + Gm1m2−e1e2
r′′∗
= E1 −[
M1(r′∗)M2(r′∗)(−Q)−e1e2
r′∗− Gm1m2−e1e2
r′′∗
]
E ′′2 = E ′
2 − Gm1m2−e1e2
r′′∗
= E2 +[
M1(r′∗)M2(r′∗)(−Q)−e1e2
r′∗− Gm1m2−e1e2
r′′∗
].
(7.217)
11Remember that −Q = 1 + o(1/c2).
152
This approximation is always justified if the radius of the second intersection r′′∗ is enough
large. Now, it is remarkable that whatever the value of r′∗ is, the first term in the square
brackets in Eqn.(7.217) satisfies the inequality
M1(r′∗)M2(r
′∗)(−Q)− e1e2
r′∗>
Gm1m2 − e1e2
r′∗. (7.218)
Comparing the expressions (7.217), (7.218) and (7.213) it is possible to see that in the
relativistic regime the shell ejection possibility is even greater than in the Newtonian
case. Furthermore, it is worth noting that the presence of the charge do not change
qualitatively the pure gravitational analysis, but just magnifies the ejection effect.
Gm1m2 − e1e2 < 0 case
Let us consider also briefly the case in which the shells are equal-signed charged and
the repulsion overcome the gravity attraction, i.e. Gm1m2 − e1e2 < 0. In this case the
ejection can happen only after an odd number of intersections.
E.g. after three intersections, from the previous formulas we have, in the Newtonian
approximation:
E ′′′1 = E1 − (Gm1m2 − e1e2)
(1
r′∗− 1
r′′∗+
1
r′′′∗
). (7.219)
Obviously this formula has a meaning only if
1
r′∗<
1
r′∗− 1
r′′∗+
1
r′′′∗, (7.220)
otherwise the ejection happens at the first intersection (and then there would not be
other crossings, and no r′′∗ , r′′′∗ ), or never more; if Eqn.(7.220) is true, then it means that
the barycenter of the two shells is falling into the center singularity.
Conclusions
We have found the energy exchange between two charged crossing shells (formula
(7.191)). Then we have studied special cases of physical interest in which the formu-
las simplify: the non relativistic case, the massless shells, the test shell, and finally the
ejection mechanism in a semi-Newtonian regime: we found that the ejection mechanism
is more efficient in the charged case than in the neutral one if the charges have opposite
sign (because the energy transfer is larger due to the Coulomb interaction).
Personal works
Publications
1. M. Pizzi and A. Paolino, Equilibrium configurations in the double Reissner-Nordstromexact solution, International Journal of Modern Physics A (IJMPA) 23-8 (2008), 1222.
2. A. Paolino and M. Pizzi, Electric force lines of the double Reissner-Nordstrom exactsolution, International Journal of Modern Physics D (IJMPD) 17-8 (2008), 1159.
3. V.A. Belinski, M. Pizzi and A. Paolino, A membrane model of the Reissner-Nordstromsingularity with repulsive gravity, accepted for pub. by International Journal of ModernPhysics (IJMPD).
4. V.A. Belinski, M. Pizzi and A. Paolino, Charged membrane as a source for repulsivegravity, proceeding of III Stuckelberg Workshop (2008).
5. M. Pizzi and A. Paolino, Intersections of self-gravitating charged shells in a Reissner-Nordstrom field, submitted to International Journal of Modern Physics D (IJMPD).
Talks and posters
1. A. Paolino and M. Pizzi, Electric force lines of the double Reissner-Nordstrom solution,at the II Stueckelberg Workshop, (2007); delivered by the coauthor M. Pizzi.
2. M. Pizzi and A. Paolino, Electric force lines in the Alekseev-Belinski solution, Poster-Section at the APS april meeting, St. Louis, Missouri (USA), (2008); presented by thecoauthor M. Pizzi.
3. V.A. Belinski, M. Pizzi and A. Paolino, Charged membrane and repulsive gravity, at theIII Stuckelberg Workshop (2008); delivered by the coauthor V.A. Belinski.
4. M. Pizzi and A. Paolino, Elettric Force Lines and Stability in the Alekseev-Belinski so-lution, at the III Stuckelberg Workshop (2008); delivered by the coauthor M. Pizzi.
153
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