completeness of general pp-wave spacetimes and their
TRANSCRIPT
This content has been downloaded from IOPscience. Please scroll down to see the full text.
Download details:
IP Address: 195.113.23.24
This content was downloaded on 23/11/2016 at 14:21
Please note that terms and conditions apply.
You may also be interested in:
Geodesics in nonexpanding impulsive gravitational waves with , Part I
Clemens Sämann, Roland Steinbauer, Alexander Lecke et al.
On the completeness of impulsive gravitational wave spacetimes
Clemens Sämann and Roland Steinbauer
The global uniqueness and C1-regularity of geodesics in expanding impulsive gravitational waves
J Podolský, C Sämann, R Steinbauer et al.
The global existence, uniqueness and 1-regularity of geodesics in nonexpanding impulsive
gravitational waves
J Podolský, C Sämann, R Steinbauer et al.
An iterative method to reconstruct the refractive index of a medium from time-of-flight
measurements
Udo Schröder and Thomas Schuster
General Kundt spacetimes in higher dimensions
Jií Podolský and Martin Žofka
Completeness of general pp-wave spacetimes and their impulsive limit
View the table of contents for this issue, or go to the journal homepage for more
2016 Class. Quantum Grav. 33 215006
(http://iopscience.iop.org/0264-9381/33/21/215006)
Home Search Collections Journals About Contact us My IOPscience
Completeness of general pp-wavespacetimes and their impulsive limit
Clemens Sämann1,3, Roland Steinbauer1 and Robert Švarc1,2
1 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090Vienna, Austria2 Institute of Theoretical Physics, Faculty of Mathematics and Physics, CharlesUniversity in Prague, VHolešovičkách2, 18000 Praha 8, Czech Republic
E-mail: [email protected], [email protected] and [email protected]
Received 7 July 2016, revised 12 August 2016Accepted for publication 12 September 2016Published 13 October 2016
AbstractWe investigate geodesic completeness in the full family of pp-wave (orBrinkmann) spacetimes in their extended and impulsive forms. This class ofgeometries contains the recently studied gyratonic pp-waves, modelling theexterior field of a spinning beam of null particles, as well as N-fronted waveswith parallel rays, which generalize classical pp-waves by allowing for ageneral wave surface. The problem of geodesic completeness reduces to thequestion of completeness of trajectories on a Riemannian manifold under anexternal force field. Building upon respective recent results, we derive com-pleteness criteria in terms of the spatial asymptotics of the profile function inthe extended case. In the impulsive case, we use a fixed-point argument toshow that, irrespective of the behaviour of the profile function, all geometriesin the class are complete.
Keywords: pp-waves, gyratons, geodesic completeness, low regularity,impulsive limit
(Some figures may appear in colour only in the online journal)
1. Introduction
Since the seminal work of Penrose [Pen65a], singularities in general relativity are usuallyunderstood as the presence of incomplete causal geodesics, i.e. geodesics which cannot beextended to all values of their parameter. In this work, we study geodesic completeness for a
Classical and Quantum Gravity
Class. Quantum Grav. 33 (2016) 215006 (27pp) doi:10.1088/0264-9381/33/21/215006
3 Author to whom any correspondence should be addressed.
0264-9381/16/215006+27$33.00 © 2016 IOP Publishing Ltd Printed in the UK 1
large class of spacetimes admitting a covariantly constant null vector field, forming the well-known pp-wave subclass of the Kundt family of non-twisting shear-free and non-expandinggeometries [Kun61, Kun62]. This subfamily includes, e.g. gyratonic pp-waves [FF05, FZ06],representing the exterior vacuum field of spinning particles moving with the speed of light,which may serve as an interesting toy model in high-energy physics [YZF07], but alsoN-fronted waves with parallel rays (NPWs): a generalization of classical pp-waves allowingfor an n-dimensional Riemannian manifold N as the wave surface [CFS03]. Remarkably, thisfamily of exact spacetimes has already been described by the original Brinkmann form[Bri25] of the pp-wave metric, given in equation (1), below.
We will consider geodesic completeness both in the extended case, i.e. where the profilefunctions are smooth, as well as in the impulsive case, i.e. when the metric functions arestrongly concentrated and of a distributional nature. The problem of completeness in this classof spacetimes can be reduced to a purely Riemannian problem, namely the question ofcompleteness of the motion on a Riemannian manifold under the influence of a time- andvelocity-dependent force. In the extended case we generalize recent results([CRS12, CRS13]) to the case at hand to provide completeness statements subject to con-ditions on the spatial fall-off of the profile function. We also consider the case of impulsivewaves in our class, which are models of short but violent bursts of gravitational radiationemitted by a spinning (beam of) ultrarelativistic particle(s). These models are also interestingfrom a purely mathematical point of view, since they are examples of geometries of lowregularity, which has attracted growing attention recently, see e.g.[CG12, KSSV15, Min15, Säm16, Sbi15]. Our main result here is that all impulsive geo-metries in the full class of pp-waves are geodesically complete irrespective of the spatialasymptotics of the profile function. This result confirms the effect (previously noted in similarsituations: see, for example, [PV99, SS12]) that the influence the spatial characteristics of theprofile function exert on the behaviour of the geodesics is wiped out in the impulsive limit. Inthis way, we prove a large class of interesting non-smooth geometries to be non-singular inview of the Penrose definition.
This article is structured as follows. In section 2 we introduce the full pp-wave (orBrinkmann) metric and summarize its basic geometric properties and its algebraic structure.Then, in section 3 we deal with geodesic completeness in the extended case. We reviewprevious works and apply recent results on the completeness of solutions to second-orderequations on Riemannian manifolds to the geodesic equations for the full Brinkmann metric.In particular, we establish completeness for a large class of physically reasonable (extended)gyratonic pp-waves. In section 4 we introduce impulsive geometries within our general classof pp-wave solutions, thereby leaving the realm of classical smooth Lorentzian geometry. Insection 5 we consider the regularized version of these geometries and establish their com-pleteness, combining the results of section 3 with a fixed-point argument. Finally, in section 6we explicitly calculate the limits of the complete regularized geodesics given in section 5 andrelate them to the geodesics of the background spacetime. Physically, this amounts to cal-culating the geodesics in the distributional model of the impulsive wave.
2. The spacetime metric
In this section we introduce the full pp-wave (or Brinkmann metric) ([Bri25]) and reviewsome of its basic geometric properties along with relevant subclasses and special cases whichhave been treated extensively in the literature.
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
2
To begin with, let (N, h) be a smooth connected n-dimensional Riemannian manifold. Weconsider the spacetime = ´M N g,2( ) where the line element is given by
= - + +ds h dx dx dudr x u du x u dudx2 , 2 , . 1iji j
ii2 2( ) ( ) ( )
Here, = = ¼x x x x, ,i n1( ) are coordinates on N and u r, are global coordinates on 2.Moreover ´ N: and ´ N:i are smooth functions. We fix a timeorientation on M by defining the null vector field ¶r to be future-directed.
Some immediate geometric properties of the spacetime (1) are the following: ¶r is thegenerator of the null hypersurfaces of constant u, = @ ´P u u u N0 0( ) ≔ { } , i.e.¶ = - = -u ugradr and it is covariantly constant. The latter property is the definingcondition for pp-waves (plane-fronted waves with parallel rays), and this is why we will referto (1) as the full pp-wave (see, for example, [GP09], p 324 and section 18.5).
The null geodesic generators of P u0( ) form a non-expanding, shear-free and twist-freecongruence and the family of n-dimensional space-like submanifolds ´N u0{ } orthogonal to¶r has the interpretation of wave surfaces. Consequently, we will also refer to N as the wavesurface, and to h and x i as the spatial metric and coordinates, respectively.
Using the coordinates x u r, ,i( ), the inverse metric takes the form
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟= -
-
mngh g
g g
00 0 1
1, 2
ij ri
ri rr
( )
where h ij denotes the components of the inverse spatial metric -h 1 on N, and we have = - +g hrr ij
i j and =g hri ijj. The non-vanishing Christoffel symbols then are
G = - -
G = - - G = -
G = - G = - G = G
g
g
h h
,
, ,
, , , 3
uur ri
i u i u
ujr ri
i j j i j jkr
j k
uui ik
k u k uji ik
k j j k jki N
jki
,12 ,
12 ,
12 , ,
12 ,
,12 ,
12 , ,
( )
( )( )
( ) ( )
( ∣∣ )
( )
where G N( ) denotes the Christoffel symbols of the Riemannian metric h on N, stands for thecovariant derivative of h, and i, and u, denote derivatives with respect to the i-th spatialdirection and with respect to u, respectively.
From the vanishing of all Christoffel symbols of the form Gmnu , we immediately find that
for any geodesic g =s x s u s r s, ,i( ) ( ( ) ( ) ( )) in (1) we have =u 0. Hence, there are geodesicswith =u s u0( ) that are entirely contained in the null hypersurface P u0( ) and thus are eitherspace-like or the null generators of P u0( ). Observe that in case = 0i , the r-component ofthese geodesics also becomes affine, = +r s r r s0 0( ) ˙ . All other geodesics may be rescaled totake the form
g =s x s s r s, , . 4i( ) ( ( ) ( )) ( )
The coordinate function u is increasing along any future-directed causal curve=c t x t u t r t, ,i( ) ( ( ) ( ) ( )) since = = u c g u c u, 0( ) ( ) and it is even strictly increasing
along any future-directed time-like curve. Hence, (M,g) is chronological. Moreover, u isstrictly increasing along all future-directed causal geodesics of the form (4). So, in case = 0i there is no closed null geodesic segment and the spacetime is even causal.
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
3
The non-vanishing components of the Ricci tensor for (1) are
= = = - +R R R h R h h h, , ,
5
ij ijN
uimn
m i n uumn
m u n mnkl mn
k m l n, ,12 , ,( )
( )
( )[ ] ∣∣ ∣∣ ∣∣ [ ] [ ]
where the square brackets, as usual, denote antisymmetrization. The Ricci scalar of (1)corresponds to that of the transverse space, i.e. =R R N( ). Also, the metric (1) belongs to theclass of VSI (vanishing scalar invariant) spacetimes if N is flat.
Next, we employ the algebraic Petrov classification ([OPP13, PŠ13]) to the pp-wavemetric (1). We project the Weyl tensor onto the natural null frame
= ¶ = ¶ + ¶ = ¶ + ¶k l m m, , , 6r r u i ii
i r i12
( ) ( )( ) ( )
where d=h m mij ii
jj
ij( ) ( ) and find that the highest boost-weight irreducible components Y0ij,
Y T1 i and Y1ijk˜ vanish and the Brinkmann spacetimes are thus necessarily at least of algebraictype II with = ¶k r being a double-degenerated null direction, in fact Y = 02ij and (1) is oftype II(d). More precisely, without employing any field equations, the non-vanishing Weylscalars are
Y =+
R , 7S n nN
21
1( )
( )( )
Y = -m m R h R , 8T n ii
jj
ijN
n ijN
21 1
ij ( )˜ ( )( ) ( )( ) ( )( )
Y = m m m m C , 9ii
jj
kk
ll
ijklN
2ijkl˜ ( )( ) ( ) ( ) ( )( )
Y = m h , 10T n ii kl
i k l31
,i ( )( ) [ ] ∣∣
Y = - --
m m m h h h , 11ii
jj
kk
k j i nmn
ij k m n ik j m n3 ,1
1 , ,ijk ( )˜ ( ) ( )( ) ( ) ( ) [ ] ∣∣ [ ] ∣∣ [ ] ∣∣
Y = - + +
- - + +
m m h
h h h . 12
ii
jj
ij i u jmn
m i n j
n ijkl
kl k u lmn
m k n l
412 , , ,
1 12 , , ,
ij
)(( ) ( )
( ) ( ) ∣∣ ( ∣∣ ) [ ] [ ]
∣∣ ∣∣ [ ] [ ]
The boost-weight zero components Y Y Y, ,S T2 2 2ij ijkl˜ ˜( ) are entirely governed by the properties ofthe transverse space N; the components of boost-weight −1, Y T3 i, and Y3ijk˜ , are governed bythe off-diagonal terms i; while the boost-weight −2 component Y4ij depends on all themetric functions and i. The conditions under which the geometry (1) becomesalgebraically more special are summarized in table 1.
Moreover, if the vacuum Einstein field equations (withΛ = 0) are employed some of theconditions are satisfied identically (cf. (5) and table 2).
The spacetimes (1) were originally considered by Brinkmann in the context of conformalmappings of Einstein spaces. Over the decades, starting with Peres [Per59], several specialcases of (1) have been used as exact models of spacetimes with gravitational waves.
The most prominent case arises when the wave surface N is flat 2 and the off-diagonalterms i vanish. In fact, writing =x x y,i ( ), these ‘classical’ pp-waves,
= + - +ds dx dy dudr x y u du2 , , , 132 2 2 2( ) ( )have become textbook examples of exact gravitational wave spacetimes (see, for example,[GP09, Chapter17]). The Petrov type is N (see table 1) without the application of the field
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
4
equations, hence in any theory of gravity. Moreover, the Ricci tensor simplifies tod= -R 1 2uu
ijij,( ) (see (5)) and its source can be interpreted as any type of null matter or
radiation. The vacuum Einstein equations reduce to the two-dimensional (flat) Laplacian sothat (13) with harmonic represents pure gravitational waves. Such solutions are mostconveniently written using the complex coordinate z = +x iy with z z z= +u F u F u, , ,( ) ( ) (¯ ) and F a combination of terms of the form
Table 1. The conditions refining possible algebraic (sub)types of the Brinkmann geo-metry (1). No field equations are employed. The relations II(b), III(b) are identities forn=2 and II(c) for n=2, 3, respectively. When II(abc) are satisfied simultaneously,the transverse space has to be flat.
(Sub)type Condition
II a( ) =R 0N( )
II b( ) =R h RijN
n ijN1( ) ( )
II c( ) =C 0ijklN( )
II d( ) always
III II abcd( )
III a( ) =h 0kli k l,[ ]
III b( ) = --
h h hk j i nmn
ij k m n ik j m n,1
1 , ,( )[ ] [ ] [ ]
N III ab( )
O - + + hij i u jmn
m i n j12 , , ,( ) [ ] [ ]
= - + +h h hn ij
klkl k u l
mnm k n l
1 12 , , ,( )[ ] [ ]
Table 2. The algebraic structure of the spacetimes (1) restricted by Einstein’s vacuumfield equations. The algebraic type becomes, in general, II(abd). Moreover, the con-dition II(c) is identically satisfied in n = 2, 3 and III(b) in n=2, respectively. Con-dition II(c) necessarily leads to a flat transverse space.
(sub)type Condition
II a( ) alwaysII b( ) alwaysII c( ) =C 0ijkl
N( )
II d( ) always
III II abcd( )
III a( ) alwaysIII b( ) = 0k j i,[ ]
N III ab( )
O - + + =h 0ij i u jmn
m i n j12 , , ,( ) [ ] [ ]
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
5
å åz a z m z b z= - +=
¥-
=
¥
F u u u u, log 14m
mm
mm
m
1 2
( ) ( ) ( ) ( ) ( )
with arbitrary ‘profile functions’ am, bm, μ of u. Here, the inverse-power terms represent pp-waves generated by sources with multipole structure (see, for example, [PG98]) moving alongthe axis, which is clearly singular. Hence, it has to be removed from the spacetime which nowhas spatial part =N 02⧹{ }. The same is true for the extended Aichelburg–Sexl [AS71]solution represented by the logarithmic term. Finally, the polynomial terms are non-singularwith b2 representing plane waves (see below). The higher-order terms ( m 3) lead tounbounded curvature at infinity and display chaotic behaviour of geodesics (e.g. [PV98])consequently seem to be physically less relevant.
In the special case of being quadratic in x y, , one arrives at plane waves, i.e.
= + - +ds dx dy dudr u x x du2 , 15iji j2 2 2 2( ) ( )
whereij is a real symmetric ´2 2( )-matrix-valued function on. Here, the curvature tensorcomponents are constant along the wave surfaces and the spacetime is Ricci-flat provided thetrace of the profile function vanishes, = 0i
i , in which case we speak of (purely)gravitational plane waves. While being complete by virtue of the linearity of the geodesicequations, plane waves ‘remarkably’ fail to be globally hyperbolic ([Pen65b]) due to afocussing effect of the null geodesics. This phenomenon has been investigated thoroughly forgravitational plane waves in a series of papers by Ehrlich and Emch ([EE92b, EE92a, EE93];see also [BEE96, Chapter 13]), who were able to determine their precise position on thecausal ladder: They are causally continuous, but not causally simple. In this analysis,however, the high degree of symmetries of the flat wave surface was extensively used and the‘stability’ of the respective properties of plane waves within larger classes of solutionsremained obscure.
Partly to clarify these matters Flores and Sánchez, in part together with Candela, in aseries of papers ([CFS03, CFS04, FS03, FS06]) introduced more general models which theycalled (general) plane-fronted waves. Indeed, they generalize pp-waves (13) by replacing theflat two-dimensional wave surface by an arbitrary n-dimensional Riemannian manifold N, i.e.they are defined as (1) with vanishing off-diagonal terms i,
= - +ds h dx dx dudr x u du2 , . 16iji j2 2( ) ( )
Motivated by the geometric interpretation given above, and following [SS12], we call thesemodels N-fronted waves with parallel rays (NPWs). The Petrov type of these geometries isnow II(d) (see table 1) and the vacuum field equations become D = =h 0h
ijij , i.e. the
Laplace equation for on (N, h) which, in addition, has to be Ricci-flat. Hence, vacuumNPWs are necessarily of Petrov type II(abd) and are of type N if and only if (N, h), inaddition, is conformally flat, and hence flat. It turns out that the behaviour of at spatialinfinity is decisive for many of the global properties of NPWs with quadratic behaviourmarking the critical case: NPWs are causal, but not necessarily distinguishing: they arestrongly causal if - behaves, at most, quadratically at spatial infinity4 and they are globallyhyperbolic if - is subquadratic and N is complete. Similarly, the global behaviour ofgeodesics in NPWs is governed by the behaviour of at spatial infinity. The respectiveresults will be discussed in the next section together with the stronger and newer results of[CRS12, CRS13].
A complementary generalization of pp-waves (13) was considered by Bonnor ([Bon70])and independently by Frolov and his collaborators in [FF05, FIZ05, FZ06, YZF07]. Here, the
4 For precise definitions of these conditions, see section 3, below.
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
6
n-dimensional transverse space N is considered to remain flat, but non-trivial off-diagonalterms x u,i( ) are allowed to obtain gyratonic pp-waves
d= - + +ds dx dx dudr x u du x u dudx2 , 2 , . 17iji j
ii2 2( ) ( ) ( )
Physically, this geometry represents a spinning null beam of pure radiation, first calledgyraton in [FF05]. By flatness of the wave surface, the Petrov type is at least III and in case ofvacuum solutions at least III(a) and N if n=2 (see tables 1, 2). In [Bon70] the metric (17)was matched to an ‘interior’ non-vacuum region, where the spinning source of thegravitational waves was given phenomenologically by an energy-momentum tensor of theform =Tuu and =T jui i. In the surrounding vacuum region, the metric functions and i
are restricted by the vacuum Einstein equations following from (5) for =R 0ui and =R 0uu :
d d d d d= - + + =0, 0, 18mnm i n
mnmn
mnm n u
kl mnk m l n, ,
12 , , , , ,( ) ( )[ ] [ ] [ ]
where in addition the ‘Lorenz’ gauge d = 0mnm n, can be applied. In [FF05, FIZ05] explicit
solutions have been calculated, all displaying fall-off-like inverse powers of x if >n 2 anda logarithmic behaviour if n=2. Moreover, the weak field approximation in the presence of agyratonic source, reflected in the only non-trivial energy-momentum tensor components Tuuand Tui, gives physical meaning to the metric functions in the entire spacetime. In particular, determines the mass-energy density and i correspond to the angular momentum densityof the source.
In four dimensions ( =n 2), the off-diagonal terms in (17) can be removed locally by acoordinate transformation which, however, obscures the global (topological) properties of thespacetime, cf. e.g. [FIZ05, PSŠ14]. In higher dimensions ( >n 2, the off-diagonal terms in(17) can anyway not be removed even locally due to the lack of coordinate freedom.
The spinning character of the four-dimensional gyratonic pp-wave (17) was emphasizedin [FIZ05, PSŠ14] employing transverse polar coordinates r j=x cos , r j=y sin with theidentifications r j= - -J sin1
1 , r j= -J cos21 which gives
r r j r j r j j= + - + +ds d d dudr u du J u dud2 , , 2 , , . 192 2 2 2 2( ) ( ) ( )
This can also be understood as the specific form of the full Brinkmann metric (1) with r=x1 ,j=x2 and r r j= +dh d d2 2 2 2. An important explicit example is given by the axially
symmetric gyraton accompanied by a pp-wave generated by a monopole, namely
r r j m r c j= + - - +ds d d dudr u du u dud2 2 ln , 202 2 2 2 2( ) ( ) ( )
where m u( ) determines the mass density in the Aichelburg–Sexl-like logarithmic term, andthe angular momentum density is determined by c u( ).
3. Geodesic completeness
In this section we discuss geodesic completeness of the full pp-wave metric (1) and its variousspecial cases in the extended, i.e. smooth case before turning to the impulsive limit insubsequent sections. We will see that all these questions can be reduced to a purely Rie-mannian question about the completeness of trajectories on N under a specific force field.
The explicit form of the system of geodesic equations for a curveg =s x s u s r s, ,i( ) ( ( ) ( ) ( )), see (3), is
= -G - - - -x x x h x u h u¨ 2 , 21i Njki j k ik
k j j kj ik
k u k, ,12 , ,
2˙ ˙ ( ) ˙ ˙ ( ) ˙ ( )( )
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
7
=u 0, 22( )
= - - - - - -r x x g ux g u¨ . 23i ji j ri
i j j i jj ri
i u i u, , , ,12 ,
12 ,
2( )( )˙ ˙ ( ( ) ) ˙ ˙ ˙ ( )( ∣∣ )
We observe (again) that the equation for u is trivial and that the equation for r decouplesfrom the rest of the system and can simply be integrated once the x-equations are solved.Finally, the x-equations are the equations of motion on the Riemannian manifold N under anexternal force term depending on time and velocity. Hence, the basic result on the form of thegeodesics of the Brinkmann metric (1) is the following (see [CFS03], proposition 3.1):
Proposition 3.1: (Form of the geodesics). Let g = - x u r a a M, , : ,i( ) ( ) be a curve onMwith constant ‘energy’ g g=gE g ,( ˙ ˙ ) which assumes the data
g g= =x u r x u r0 , , , 0 , , . 24i i0 0 0 0 0 0( ) ( ) ˙ ( ) ( ˙ ˙ ˙ ) ( )
Then, g is a geodesic if the following conditions hold true:
(a) u is affine, i.e. = +u s u su0 0( ) ˙ for all Î -s a a,( ),(b) xi solves
= - - - -D x h x u h u2 , 25xi ik
k j j kj ik
k u k, , 012 , , 0
2˙ ( ) ˙ ˙ ( ) ˙ ( )˙
where D denotes the covariant derivative of N h,( ), and(c) r is given by
ò s s s s
s s s s
= - - -
-
gr s ru
E h x x u H x u
u x u x
1
2, ,
2 , d . 26
s
ii
00 0
02
0
( )˙
( ( ˙ ( ) ˙ ( )) ˙ ( ( ) ( ))
˙ ( ( ) ( )) ˙ ( )) ( )
Here, we have used the explicit form of Eγ in the last condition. The key fact is now thatcompleteness essentially depends on completeness of the solutions to (25). Note that in mostcases we will use a rescaling to achieve the form (4) of the geodesics, which amounts tosetting =u 00 and =u 10˙ in equations (25) and (26).
Corollary 3.2: (Basic condition for completeness). The spacetime (1) is complete if allinextendible solutions of (25) are complete.
Although there are also some results in case of an incomplete spatial manifold (see thediscussion after corollary 3.4 below) we assume for the moment (N, h) to be complete. Thequestion of completeness of solutions of equations such as (25) has, if only in special cases,been addressed in the ‘classical’ literature. To begin with, we observe that in the special caseof NPWs (16) the equation of motion (25) reduces to
= D x x s1
2, 27x x˙ ( ) ( )˙
(∇x denoting the gradient on (N, h)), i.e. to the equation of motion on N under the influence ofa time-dependent potential. First results on the completeness of NPWs, mainly restricted tothe case of autonomous , i.e. = x( ) independent of u, were derived in [CFS03,section 3]. In fact, it follows from, for example, [AM78], theorem 3.7.15 that an NPW (16) iscomplete if is autonomous and controlled by a positively complete function at infinity, i.e.if there exists some arbitrarily fixed Îx N¯ and some positive constant R such that
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
8
R = - Îx V d x x x N d x x, for all with , , 28( ) ( ( ¯)) ( ¯) ( )where d denotes the Riemannian distance function on (N, h) and ¥ V : 0,[ ) is C2 with
/ò - = +¥¥
x e V xd0
( ) for one (hence any) >e V x( ) and all x. Consequently,autonomous NPWs are complete if grows at most quadratically at spatial infinity, i.e. if
R$ Î >x N , 0¯ such that
R = x C d x x x d x x, for all with , , 292( ) ( ¯) ( ¯) ( )for some constant >C 0. Of course, this result extends immediately to sandwich5 NPWs,which grow at most quadratically at spatial infinity. Additionally, the case of plane NPWs iseasily settled, that is, (16) with (N, h) flat and quadratic non-autonomous , i.e.
=x u h A u x x, , , 30( ) ( ( ) ) ( )where A is (at least) a continuous map from into the space of real symmetric
´n n( )-matrices. Here, completeness follows from the global existence of solutions to linearODEs generalizing the case of plane waves (15).
More substantial results on non-autonomous NPWs have been given in [CRS12] basedon recent results on the completeness of trajectories of equations like (25) and (27) in[CRS13] (for more general and somewhat sharper results see [Min15]). Since we will also usethese statements in our discussion of the general case (1) we recall in the following the keynotions and theorems. We say that a (time-dependent) tensor field X on the projection
p ´ N N: grows at most linearly in N along finite times if for all >T 0 there existsÎx N¯ and constants >A C, 0T T such that
+ " Î ´ -X A d x x C x s N T T, , , 31x s T T,∣ ∣ ( ¯) ( ) [ ] ( )( )
with ∣ ∣ and d the norm and the distance function of h, respectively. Analogously, we definethe notions of at most quadratic growth along finite times and boundedness along finite times,where in the special case of functions we use the estimate (31) without norm. Now given asmooth 1, 1( )-tensor field F and a smooth vector field X on π we consider the second-orderODE
g g= +g g gD s F s X 32s s s s, ,˙ ( ) ˙ ( ) ( )˙ ( ( ) ) ( ( ) )
and the special case when X is derived from a potential, i.e.
g g g= - g gD s F s V s s, 33s s x,˙ ( ) ˙ ( ) ( ( ) ) ( )˙ ( ( ) )
with V a smooth function on N×. Then, we have
Theorem 3.3 (Theorems 1, 2 in [CRS13]). Let (N, h) be a connected, complete Riemannianmanifold. If the self-adjoint part S of F is bounded in N along finite times then
(1) all inextendible solutions of (32) are complete provided X grows at most linearly in Nalong finite times, and
(2) all inextendible solutions of (33) are complete provided that-V and ¶¶V
sgrow at most
quadratically in N along finite times.
Observe that one may also apply theorem 3.3(1) to equation (33) in which case one has toassume that Vx grows at most linearly along finite times. Provided that we are in the non-autonomous case, this condition is logically independent of the condition of theorem 3.3(2).
5 We call a spacetime (1) a sandwich wave if and i vanish outside some bounded u-interval.
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
9
Hence, in the case of NPWs, which amounts to setting F=0 and = - = X Vx x ,one obtains different types of results based on either of these conditions, see[CRS12, CRS13]. Explicitly we have
Corollary 3.4 (Completeness of NPWs and classical pp-waves). NPW spacetimes (16) and,in particular, classical pp-wave spacetimes (13) with complete wave surface N are completeprovided that either
(1) x grows at most linearly along finite times, or
(2) and ¶¶s
grows at most quadratically along finite times, or
(3) bx u u, 0( ) ( ) and a b -¶¶
x u u u x u, ,u 0 0( ) ( )( ( ) ( )) for some continuous real
functions a0, b0 and all Î ´x u N,( ) .
Condition (3) is, however, not derived from theorem 3.3 but due to [CRS12], Cor. 3.3and, again, logically independent of the other conditions. A physically interesting con-sequence of condition (2), which actually generalizes the above results on autonomousand sandwich NPWs of quadratic growth, is that it provides stability of completeness ofplane waves within the class of NPWs with quadratic behaviour of (see [CRS12],Rem. 3.5).
Observe that physically reasonable models of classical gravitational pp-waves, asdiscussed below equation (14), possess a non-complete wave surface and hence corollary3.4 does not apply in this case. However, the geodesics will still be ‘complete at infinity’since the asymptotic conditions of corollary 3.4 hold true for the multipole as well asfor the logarithmic terms in (14). However, the geodesics could leave the exterior region‘at the inside’ proceeding to the matter region. This behaviour clearly has to be consideredas physically reasonable. Also, mathematically, completeness of the trajectories of (32),(33) on incomplete Riemannian manifolds is subject to very strong conditions, see forexample, [Gor70]: a sufficient condition, for example, is that is proper and boundedfrom below, which certainly does not hold in our case. Note that this applies to wavesurfaces of the form =N 0n⧹{ } as well as to those of the form n with a (closed) ballremoved. In the latter case one would, of course, match the solution to some non-vacuuminterior region inside the ball. The situation is, of course, completely analogous in the caseof NPWs.
Turning now to the general case, i.e. to the quest for completeness of the full pp-wavegeometry (1) we more extensively make use of the power of theorem 3.3. Indeed
= - -F hji ik
k j j k, ,( ) is no longer vanishing, but still its self-adjoint part satisfies S=0 so
that theorem 3.3 puts no restriction onk j, . On the other hand, = - -X h 2ikk u k
1
2 , ,( ) andwe can no longer write X as the gradient of a potential. So we cannot use condition (2) andhave to exclusively resort to theorem 3.3(1). In this way we obtain
Corollary 3.5 (Completeness of the Brinkmann metric). The full pp-wave spacetime (1) iscomplete if N is complete and x and hik
k u, grow at most linearly along finite times.
Finally, we come to discuss completeness of gyratonic pp-waves (17). In this case, thewave surface is flat and so we only have to deal with the asymptotics of the metric functions.
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
10
Corollary 3.6 (Completeness of gyratons). Any gyratonic pp-wave (17) with =N n and x,
as well as k u, growing at most linearly along finite times is complete.
Now, the asymptotics of the explicit gyratonic pp-waves of [FF05, FIZ05] (see section 2,p.7) imply that k, and k u, even decay or only grow logarithmically for large x. However,physically reasonable models are, again, singular on the axis (e.g. (20)) or should be matchedto some interior matter region so that the wave surface is n without a point or n with a ballremoved and hence incomplete. So, again, we obtain for such ‘gravitational’ gyratons only‘completeness at infinity’, but the geodesics could leave the exterior region ‘at the inside’proceeding into the matter region. Once again, this behaviour is to be considered as physicallyperfectly reasonable.
4. The impulsive limit
In this section we turn our focus to impulsive versions of the Brinkmann metric (1). Gen-erally, impulsive gravitational waves model short, but violent pulses of gravitational or otherradiation. In particular, in his seminal work [Pen72], Penrose has considered impulsive pp-waves, that is, spacetimes of the form (13) with
d=x u H x u, , 34( ) ( ) ( ) ( )where δ denotes the Dirac function and H is a function of the spatial variables only. Sincethen, various methods of constructing impulsive gravitational waves with or withoutcosmological constant have been introduced; for an overview see, for example, [GP09,Chapter 20]. In particular, impulsive gravitational waves have been found to arise asultrarelativistic limits of Kerr–Newman and other static spacetimes which make theminteresting models for quantum scattering in general relativistic spacetimes.
More generally, impulsive NPWs (iNPWs), i.e. (16) with (34), have been considered in[SS12, SS15]. In all these models, which are impulsive versions of special cases of (1) withthe off-diagonal terms i vanishing, the field equations put no restriction on the u-behaviourof the profile function (see section 2). Hence, the most straightforward approach toimpulsive waves in this class of solutions is, indeed, to view them as impulsive limits ofsandwich waves with an ever shorter, but stronger profile function which precisely leadsto (34).
However, here we are mainly interested in impulsive versions of the full pp-wavespacetimes (1) which, in particular, includes impulsive versions of gyratonic pp-waves (17).Here, the situation is more subtle, as detailed in [FF05, FIZ05], where such geometries havebeen considered along with their extended versions. A more detailed discussion of four-dimensional geometries with a flat transverse space in the form (19) was given recently in[PSŠ14, section 7.] Since this discussion also applies to the general case and leads to ourmodel of the impulsive full pp-wave metric, we briefly recall it here. To begin with, weintroduce the convenient quantity
w r jr jr
º ru
J u, ,
, ,
2, 35
,( )( )
( )
such that the vacuum field equations take the form (with ! denoting the flat Laplacian)
w w wr
= = = +j r jJ0, 0, 42
, 36u, ,2
2 , ( )
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
11
implying w w= u( ) which corresponds to a rigid rotation. Relation (35) immediately gives
w r c j= +J u u, , 372( ) ( ) ( )where c ju,( ) is an arbitrary p2 -periodic function in j. Taking (37) and a suitable ansatz for,
w r w c j r j= + +u u u u2 , , , , 382 20( ) ( ) ( ) ( ) ( )
the remaining field equation in (36) becomes
r j c w c= S S º -j jj- u, with , 2 . 39u0
2, ,( ) ( ) ( )
Removing the rigid rotation by the natural global gauge w = 0, and using the splitting
r j r j c c j c j c j= = + Fu u u u, , , , , , 40H J0 0( ) ˜ ( ) ( ) ( ) ˜ ( ) ( ) ( ) ( )
we obtain j c j cS = ju u, 2 J u, ,( ) ˜ ( ) ( ) and equation (39) takes the form
r j cr
c j c= ju u,2
. 41H J u0 2 , ,˜ ( ) ( ) ˜ ( ) ( ) ( )
If S = 0, i.e. c j = const.˜ ( ) , equation (41) reduces to = 00 and there is norestriction on the u-dependence of 0 and J. In particular, the energy profile c uH ( ) and theangular momentum density profile c uJ ( ) can be taken independently of each other. In[PSŠ14], it was demonstrated that the curvature is proportional to cH and cJ u, which leads toimpulsive waves by setting c uH ( ) to be proportional to the Dirac δ, but using a box-likeprofile forc uJ ( ).
However, in the case when S ¹ 0 there occurs a coupling of the profile functions.Indeed, the supports of c uH ( ) and c uJ u, ( ) have to coincide since otherwise both sides of (41)have to vanish individually, leading to the vanishing of c jj,˜ ( ) and hence Σ. In particular, thebox profile in the angular momentum density c uJ ( ) leads to two Dirac deltas in the energydensity.
Moreover, we can, of course, combine such a coupled solution with specific homo-geneous solutions. Hence, it is most natural and physically relevant to prescribe a generalbox-like profile for the angular momentum density and a delta-like profile for the energydensity of the form
d J= =a bx u H x u x u a x u, , , , 42i i L,( ) ( ) ( ) ( ) ( ) ( ) ( )
where we define (see figure 1)
d ad bd J= + - = Q - Q -a b u u u L uL
u u L, and1
. 43L, ( ) ( ) ( ) ( ) ( ( ) ( )) ( )
Here, α, β, and >L 0 are some constants, δ denotes the Dirac measure and Θ is theHeaviside function. This ansatz covers the coupled case (a b= = -L L1 , 1 ) as well as all
Figure 1. Illustration of the box-like profile with accompanying δ-spikes.
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
12
the models studied in [YZF07] (p-gyraton: a b= =0 , AS-gyraton: b= =a 0i , a-gyraton:a > 0, b = 0, b-gyraton: a = 0, b > 0), which all arise from specific combinations ofhomogeneous solutions.
So the impulsive full pp-wave metric we will consider in the rest of our work is explicitlygiven by
d J= - + +a bds h dx dx dudr H x u du a x u dudx2 2 . 44iji j
i Li2
,2( ) ( ) ( ) ( ) ( )
Of course, off the wave zone (given by Îu L0,[ ]) the spacetime is just the product ofthe Riemannian wave surface (N, h) with flat 2. From now on we will assume (N, h) to becomplete and call = ´M N0
2 the background of the impulsive wave (44), which is thencomplete as well. From (7)–(12) we immediately observe that the components of boost-weight -1 and of -2, namely,
JY = m h a , 45T n ii kl
i k l L31
,i ( )( ) [ ] ∣∣
JY = - --
m m m a h h a h a , 46ii
jj
kk
k j i nmn
ij k m n ik j m n L3 ,1
1 , ,ijk ( )˜ ( ) ( )( ) ( ) ( ) [ ] ∣∣ [ ] ∣∣ [ ] ∣∣
d d J
d d J
Y = - + +
- - + +
a b
a b
-
-
- -
- -
m m H a h a a
h h H a h a a , 47
ii
jj
ij i j L Lmn
m i n j L
n ijkl
kl k l L Lmn
m k n l L
412 , , , ,
2
1 12 , , , ,
2
ij 1 1
1 1 )(( ) ( )
( ) ( ) ∣∣ ( ∣∣ ) [ ] [ ]
∣∣ ∣∣ [ ] [ ]
are only non-trivial in the wave zone while, in general, the rest of the spacetime correspondsto the type-D background.
5. Completeness of the impulsive limit
First, we review previous results on the completeness of impulsive gravitational waves. In thesimplest case of classical impulsive pp-waves, i.e. (13) with (34), the spacetime is flatMinkowski space off the single wave surface =u 0{ } where the curvature is concentrated.Consequently, the geodesics for impulsive pp-waves have been derived in the physics lit-erature (see, for example, [FPV88]) by matching the geodesics of the background on eitherside of the wave in a heuristic manner—the geodesic equation contains nonlinear terms, ill-defined in distribution theory. This approach, in particular, leaves it open whether the geo-desics cross the wave surface at all.
In [KS99a, KS99b] this question has been answered in the affirmative using a reg-ularization approach within the theory of nonlinear distributional geometry ([GKOS01,Chapter 4]) based on algebras of generalized functions ([Col85]). In this way, a com-pleteness result for all impulsive pp-waves, i.e. for all smooth profile functions H, wasachieved although this aspect was not emphasized in the original works. Observe that thiscontrasts with the completeness results in the extended case (corollary 3.4) where thespatial asymptotics of enter decisively. Moreover, this approach in a limiting processestablishes that the geodesics in the entire spacetime are, indeed, the straight line geodesicsof the background which are refracted by the impulse to become broken and possiblydiscontinuous.
More generally, in [SS12, SS15] geodesics in iNPWs, i.e. (16) with (34), wereinvestigated. Again, using a regularization approach it was proven that if the wave surfaceN is complete, then the iNPW is geodesically complete irrespective of the behaviour of theprofile function H. Again, this is in contrast to the extended case where the completeness
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
13
depends crucially on the spatial asymptotic behaviour of the profile function (seesection 3). Moreover, the geodesics in the limit again are geodesics of the background,which are refracted by the impulse (see also [FIZ05] for a heuristic argument). Moreprecisely, using a fixed-point argument it was shown in [SS12] that in the regularizediNPW
d= - +e eds h dx dx dudr u H x du2 , 48iji j2 2( ) ( ) ( )
with de a standard mollifier6, geodesics are complete in the following sense: for each pointÎp M ‘in front’ of the impulsive wave, i.e. <u 0 and each tangent direction Îv T Mp , we
consider the geodesic ge of (48) starting in p into direction v. If ge reaches the regularizedwave zone given by eu{∣ ∣ }, then there is an > 00 such that for all <0 0 thegeodesic ge passes through the regularized wave zone and continues as a (complete) geodesicof the background ‘behind’ the impulsive wave. This result has been rephrased in thelanguage of nonlinear distributional geometry in [SS15], which allows one to omit thereference to the initial data in the final completeness statement.
As is well known, classical impulsive pp-waves, and more generally non-expanding aswell as expanding impulsive gravitational waves propagating in constant curvaturebackgrounds, have also been described by a continuous form of the metric (see, forexample, [GP09, Chapter 20]). Actually, these metrics are locally Lipschitz continuous andhence the geodesics equations possess locally bounded, but possibly discontinuous, right-hand sides. Employing the solution concepts of Carathéodory and Filippov ([Fil88]),respectively, these systems of ODEs have been recently investigated leading to the fol-lowing results: the geodesics are complete and of C1-regularity in classical pp-waves([LSŠ14]), non-expanding ([PSSŠ15]) and expanding ([PSSŠ16]) impulsive waves pro-pagating on Minkowski, de Sitter, and anti-de Sitter backgrounds. However, so far nocontinuous form of the impulsive full pp-wave metric or merely of the gyratonic pp-wavemetric has been found.
Geodesic completeness for non-expanding impulsive gravitational waves in (anti-)deSitter space has also been proven in the distributional picture in [SSLP16] using a regular-ization approach and a fixed-point argument in a spirit similar to the present article. Finally, aproof of geodesic completeness of impulsive gyratonic pp-waves has been sketched in[PSŠ14], section VIII.
In the following we provide our main result, which establishes completeness of theimpulsive full pp-wave metric (44) using a regularization approach.
To begin with, we consider the regularized metric
d J= - + +a bs h x x u r H x u u a x u u xd d d 2d d d 2 d d , 49ij
i ji L
i2,
2( ) ( ) ( ) ( ) ( )
where we have regularized the profile functions replacing δ by a standard mollifier
⎜ ⎟⎛⎝
⎞⎠d
ef
e=e x
x150( ) ( )
with f a smooth function supported in -1, 1[ ] with unit integral. Moreover, we haveregularized the Heaviside function by the primitive of de, i.e. replacing Θ by
ò dQ º-
x t td . 51x
1( ) ( ) ( )
6 For a precise definition, see (50), below.
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
14
More explicitly, we set
d ad bd J= + - = Q - Q -a be
e ee
e eu u u L uL
u u L, and1
. 52L, ( ) ( ) ( ) ( ) ( ( ) ( )) ( )
In the following we will prove that any geodesic in the regularized impulsive full pp-wave metric (49) that reaches the wave zone given by e eÎ - +u L,{ [ ]} will passthrough it, provided that e is small enough. This will lead to our main result on thecompleteness of the impulsive full pp-wave metric (44) which we state at the end of thissection. In the final section 6, we will relate these complete geodesics to the geodesics ofthe background.
To begin with, we give the explicit form of the geodesic equations for the metric (49).Observe that in the present case the u-equation is also trivial and hence we may use arescaling as in (4) to write any geodesic not parallel to, or contained in the impulsive wavesurface =u 0{ }7 as
g =e e es x s s r s, , . 53i( ) ( ( ) ( )) ( )Now we obtain, cf. (21)
J J d
d d d
= - - -
- - -
ee
e e ee
a be
e
ee
a be
a be
-- -
r a x x g a a H x
g a H H
¨
, 54
L i ji j ri
L i j j i jj
rii L L i u
, , , ,
,12 , ,
12 , ,1 1( )( )
˙ ˙ ( ( ) ) ˙
( ) ( )
( ∣∣ )
J d d= -G - - - -e e ee e
a be
-- -x x x h a a x h a H¨ 2 , 55i Njki j k
Lik
k j j kj ik
k L L k, ,12 , , ,1 1˙ ˙ ( ) ˙ ( ) ( )( )
where J=eeg h ari ik
k L.As in the extended case, the r-equation can simply be integrated once the x-equations are
solved and completeness of the geodesics is determined by completeness of the solutions tothe spatial equations (see proposition 3.1.) The latter again take the form of the equations ofmotion on the Riemannian manifold (N, h), now with an external force term depending ontime, velocity and the regularization parameter e. More explicitly, we may rewriteequation (55) in the form, cf. (32)
= +ee
ee
e e eD x s F x s X , 56x x s s x s s, ,˙ ( ) ˙ ( ) ( )˙ ( ( ) ) ( ( ) )
where D denotes the connection on (N, h) and we have set J= - -e eF h a aji
Lik
k j j k, ,( )and d d= - -e e
a be
-- -X h a H2i ikk L L k
1
2 , , ,1 1( ).Now, for fixed e, the solutions will be complete by corollary 3.5 provided H and ai
show a suitable asymptotic behaviour. But here we aim at a result for general H and ai andso we have to take a different approach. Indeed, for fixed e by ODE theory we have a localsolution ex for any initial condition taken at say the ‘left’ boundary of the wave zone
e= -u (see figure 2). However, the time of existence of such a solution will, in general,depend upon the regularization parameter e and could shrink to zero if e 0. We willprove that this is not the case. More precisely, applying a fixed-point argument we willshow that such solutions ex have a uniform (in e) lower bound η on their time of existencewhich will (at least for small e) allow them to cross the regularization region of the first δ-spike, i.e. eu∣ ∣ . Once they reach e=u they are subject to equations (55) with only theJe-terms being non-trivial. In other words, we have to deal with (56) with eX vanishing. Inthis situation we may now apply the completeness results established in section 3. Moreprecisely, we appeal to theorem 3.3(1) whose assumptions hold anyway in our case and we
7 We will deal with these (simple) geodesics separately.
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
15
obtain that the solution ex will continue at least until it reaches the regularization region ofthe second δ-spike at e= -u L . However, there we can reapply our fixed-point argumentto secure that ex reaches e= +u L and hence leaves the entire wave zone to enter thebackground region ‘behind’ the wave.
We will now state and prove the fixed-point argument. To simplify notations we will,instead of dealing with equation (55) directly, consider the following model initial valueproblem:
d= + + Qe e e e e e e ex F x x F x F x xL
¨ , ,1
, 571 2 3( ˙ ) ( ) ( ˙ ) ( )
e e- = - =ee
eex x x x, . 580 0( ) ˙ ( ) ˙ ( )
Additionally, we will write e eÎx 0,1( ) ( ] or briefly e ex( ) to denote nets (sequences). Now, we have
Proposition 5.1. Let Î ¥F F C, ,n n1 3
2( ), Î ¥F C ,n n2 ( ), let Îx x, n
0 0˙ , let eex0( ) ,
eex0( ˙ ) in n such that ex x0 0 and ex x0 0˙ ˙ for e 0 and let >b c, 0. Define
= Î -I x x x b:n1 0{ ∣ ∣ }, = Î - + ¥ I x x x c K F:n
I2 0 2 , 1{ ∣ ˙ ˙ ∣ } and = ´I I I3 1 2,where K is a bound on the L1-norm of de e( ) . Furthermore, set
⎛⎝⎜
⎞⎠⎟h =
b
C
c
C
Lmin 1, , ,
2, 59
1 2( )
where = + + + +¥ ¥ ¥ C x F K F F2 I IK
L I1 0 1 , 2 , 3 ,3 1 3∣ ˙ ∣ and
= + +¥ ¥ C F F1 IK
L I2 1 , 3 ,3 3. Finally, let e¢0 be such that h-ex x0 0∣ ∣ and
h-ex x0 0∣ ˙ ˙ ∣ for all e e< ¢0 0. Then, the initial value problem (57), (58) has a uniquesolution ex on e h e= - -eI ,[ ] with Íe e e ex I x I I, 3( ( ) ˙ ( )) .
Proof. We aim at applying Weissinger’s fixed-point theorem ([Wei52]) to the solutionoperator
ò òe
s s s d s s s s s
+ +
+ + + Q
ee e
e ee e
- -
A x t x x t
F x x F x F x x s, , d d
60
t s0 0
1 2 3
( )( ) ≔ ˙ ( )
( ( ( ) ˙ ( )) ( ( )) ( ) ( ( ) ˙ ( )) ( ))( )
on the complete metric space
e h e e h eÎ - - - - ÍeX x C x x I, : , , , 6113≔ { ([ ]) ( ˙)([ ]) } ( )
where we use the norm = + x x xC1 ∣ ∣ ∣ ˙∣.To begin with, we show that Aε maps Xε to itself. Indeed, for Î ex X we have
h h
h d h
h
- - + - + +
+ + Q
ee e
e e
¥
¥ ¥ ¥
A x t x x x x x x F
FL
F
C b
1
, and 62
I
I L I
0 0 0 0 0 02
1 ,
2 ,2
3 ,
1
3
11
3
∣ ( )( ) ∣ ∣ ∣ (∣ ˙ ˙ ∣ ∣ ˙ ∣)
( )
⎜ ⎟⎛⎝
⎞⎠
h- - + + +
+
ee
¥ ¥ ¥
¥
tA x t x x x F
K
LF K F
c K F
d
d. 63
I I I
I
0 0 0 1 , 3 , 2 ,
2 ,
3 3 1
1
( )( ) ˙ ∣ ˙ ˙ ∣
( )
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
16
Moreover for Î ex y X, , we have
h h
h
h
h
h
- - + --
+ -
- --
+ --
+ --
e e
e e
-
-
-
-
A x t A y t F I x yn
F I K x yn
F IK
Lx y
n
tA x t
tA y t F I x y
n
F I K x yn
F IK
Lx y
n
Lip ,2
Lip ,2 1
Lip ,2
,
d
d
d
dLip ,
2 1
Lip ,2 2
Lip ,2 1
,
64
n nC
n n
C
n
n nC
n
n
C
n
1 3
2
2 1
2 1
3 3
2
1 3
2 1
2 1
2 2
3 3
2 1
1
1
1
1
∣ ( )( ) ( )( )∣ ( )( )!
( ) ∣ ∣( )!
( )( )!
( )( ) ( )( ) ( )( )!
( ) ∣ ∣( )!
( )( )!
( )
where F ILip ,i j( ) denotes a Lipschitz constant for Fi on Ij. So we have
h- -e e A x A y C
nx y
265n n
C
n
C
21 1( ) ( )
( )!( )
and since å hn2
n2
( )!converges, we obtain a unique fixed point of Aε on Xε hence a net of unique
solutions xε of (57), (58) defined on e h e- -,[ ], which together with its derivatives isuniformly bounded in ε. ,
We will now detail the procedure envisaged prior to proposition 5.1 to obtain our mainresult. Fix a point
=p x u r M, , inp p p( )
lying say ‘before’ the wave zone8, i.e. <u 0p , and a vector v in TpM. In the following we will,most of the time, simplify notations by omitting the index from the x-component as we havedone in the model initial value problem (57), (58) and write, for example, xp instead of x ip.Now, without loss of generality, we may assume e to be so small that p also lies ‘before’ theregularized wave zone, i.e. in e< -u{ } and hence in a the region of eM g,( ) which coincideswith the background spacetime = ´M N0
2 ‘before’ the impulse. Now we consider thegeodesic g =s x s u s r s, ,( ) ( ( ) ( ) ( )) starting at p in direction v in the background spacetimeM0 which we will from now on call our ‘seed geodesic’. By virtue of the geodesic equationsin the background M0
= = =D x u r0, ¨ 0, ¨ 0, 66x ˙ ( )˙
we see again that u(s) is affine and hence it suffices to consider the case of strictly increasingu(s). Indeed, otherwise the ‘seed geodesic’ will never reach the (regularized) wave zone,being either confined to the null surface P up( ) (see section 2), or even moving away from thewave zone and hence in any case being (forward) complete. So we may, without loss ofgenerality, write the seed geodesic in the form (4), i.e. g =s x s s r s, ,( ) ( ( ) ( )) or brieflyas g =s x s r s,( ) ( ( ) ( )).
Now γ will reach the wave zone of the impulsive wave, i.e. s=0 in finite time and it isconvenient to introduce the data of γ at this instance as
8 The entire argument is precisely the same in the ‘time-reflected’ case when p is assumed to lie ‘behind’ the wavezone, i.e. in >u L{ }.
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
17
g g= =x r x r0 , 0, , 0 , 1, . 670 0 0 0( ) ( ) ˙ ( ) ( ˙ ˙ ) ( )
Now, we start to think of the ‘seed geodesic’ γ also as being a geodesic in the regularizedspacetime (49). In fact, it will reach the regularized wave zone at e= -u with data
g e e g e- = =e e e ex r x r, , , , 1, . 680 0 0 0( ) ( ) ˙ ( ) ( ˙ ˙ ) ( )
Using this data we solve the initial value problem for the geodesics in the regularizedspacetime (49), that is, we consider the system (54), (55) with data (68). Now, by smoothnessof the ‘seed geodesic’ γ the data (68) converges to the data (67), in particular:
e ex x x xand , 690 0 0 0˙ ˙ ( )
and we may apply proposition 5.1 to obtain a solution ex of (55), (68) on e-¥,( ), providede h 2. Hence, we also obtain a solution er of (54) with data (68), hence a geodesic ge which
coincides with the ‘seed geodesic’ γ up to e= -s and exists until it leaves the regularizedfirst δ-spike at e=s and we denote the corresponding data by
g e e g e= =ee e
ee ex r x r, , , , 1, . 701 1 1 1( ) ( ) ˙ ( ) ( ˙ ˙ ) ( )
As discussed earlier in e e-L,[ ], the geodesic equation (55) reduces to
= -G - -e e e ex x xL
h a a x¨1
, 71i Njki j k ik
k j j kj
, ,˙ ˙ ( ) ˙ ( )( )
whose right-hand side is actually independent of ε. However, we have to solve (71) with thee-dependent data (70). Anyway, by theorem 3.3(1) we obtain a solution ex which extends ourprior solution from e=s up to e= -s L , and since by proposition 5.1 the data ex1 and ex1are uniformly bounded (in e), the solutions will be uniformly bounded as well. In particular,this applies to the data at e= -s L :
e e- = - =e e e e x L x x L x, . 722 2( ) ( ) ( )
We now wish to reapply proposition 5.1 on the interval e e h- - +L L,[ ] and so we needthe data (72) to even converge. This however, follows from continuous dependence ofsolutions to ODEs once we have established that the data (70) converges, which we do next.
Lemma 5.2. Let e ex( ) , given by proposition 5.1. Then
(i) e- =e e
eÎ -
x t x Osupt ,
0∣ ( ) ∣ ( )[ ]
,
(ii) e -ex x F x0 2 0˙ ( ) ˙ ( ) as e 0.
Proof. (i) On e e- ,[ ], the solution ex is given by proposition 5.1 and can be expressed by
ò ò s s s d s
s s q s s
= + +
+ +
+- -
x t x x t
F x x F x
F x x s
,
, d d . 73
t s0 0
1 2
3
( ) ( )
( ( ) ( )) ( ( )) ( )
( ( ) ( )) ( ) ( )
Consequently,
ee e e
- - + - ++ + +
ee e
¥ ¥ ¥ x t x x x x x x
F F K K F
2
4 2 4 , 74I I I
0 0 0 0 0 0
21 , 2 ,
23 ,3 1 3
∣ ( ) ∣ ∣ ∣ (∣ ˙ ˙ ∣ ∣ ˙ ∣)( )
where we used the uniform boundedness of e ex( ) and e ex( ˙ ) established in proposition 5.1.
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
18
To obtain (ii) we differentiate (73), insert e=t , and then we estimate
ò
e e
d e
ee
- - - +
+ - +
- ++ - +
ee
e
ee e
e
e ee
¥
-¥
¥
Î -¥
x x F x x x F
F x s F x s s K F
x x F
K F x s F x K F
2
d 2
2
sup 2 ,
I
I
I
sI
0 2 0 0 0 1 ,
2 2 0 3 ,
0 0 1 ,
,2 2 0 3 ,
3
3
3
3
∣ ˙ ( ) ˙ ( )∣ ∣ ˙ ˙ ∣
( ( ( )) ( )) ( )
∣ ˙ ˙ ∣∣ ( ( )) ( )∣
[ ]
where we have used that ò d =e
ee-
s sd 1( ) in the first inequality and (i) to see that the first termin the final line converges to zero as e 0. ,
We will explicitly give the limit in (ii) for our case in section 6, below. For the time beingwe are in the position to reapply proposition 5.1 to obtain a solution g =e e ex r,( ) to (55), (54)with data (72) on the domain e e- +L L,[ ], again provided that e h 2. Moreover, ex isuniformly bounded in e together with its derivative which, in particular, applies to the data at
e= +s L :
e e+ = + =ee
ee x L x x L x, . 753 3( ) ( ) ( )
But now we have reached the background spacetime ‘behind’ the regularized wave zone andthe solutions just obtained can be continued as solutions ex of the background geodesicequations (66) with data (75). By completeness of the background M0 these solutions extendto all positive values of their parameter. Now, inserting this solution into the geodesicequation’s r-component (54), we obtain also a forward complete solution er . Hence, togetherwe have obtained a complete smooth geodesic ge, which coincides with the ‘seed geodesic’ γon e-¥ -,( ) and with a background geodesic for e+u L . Note, however, that in thebackground ‘behind’ the regularized wave zone, ge does not coincide with a single geodesicof the background since the data (75), which we feed into the background geodesicequation (66) at e= +s L , depends on e. Therefore, the global geodesic ge for e+s Lcoincides with a background geodesic starting at L+ε with data ex3 ,
ex3˙ , ande+ =e
er L r3( ) , e+ =eer L r3˙ ( ) ˙ .
Finally, it remains to deal with the geodesics which start at points p with Îu L0,p [ ]. Tobegin with, if up = 0, we start within the first impulsive surface in some specified directionÎv T Mp . If v is tangential to the null hypersurface P(0) then the corresponding geodesic will
be either null or space-like, but will, in any case, stay entirely within P(0) and thus have atrivial u-component (see section 2). However, then an inspection of the geodesicequation (21) reveals that the x-equation coincides with the geodesic equation on the completeRiemannian manifold (N, h) and hence its solution is complete. Feeding this solution into ther-equation (which again simplifies drastically) we obtain completeness. In the case where v istransversal to P(0) there is a ‘seed geodesic’ with data (67) coinciding with p and v and wehave already covered this case. Precisely the same argument applies in the ‘time-reflected’case to all points p with =u Lp , i.e. which lie on the impulsive surface of the second spike.
Finally, for all points p with Îu L0,p ( ) we may assume that e is so small thate eÎ -u L,p ( ), hence that p lies in the ‘intermediate’ region where the geodesic
equations (71) are independent of e. In case Îv T Mp is tangential to P up( ) the geodesicagain stays entirely in the hypersurface P up( ) and is complete by theorem 3.3(1). In the casewhere v is transversal to P up( ), again, by ‘time symmetry’ we have to only discuss the case ofan increasing u-component. So once more by theorem 3.3(1), the geodesic will reach
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
19
e= -s L and we may apply proposition 5.1 since the data at this instant will converge to thedata of the corresponding solution of (71) at s=L.
Summing up, we have proved our main result.
Theorem 5.3. Given a point p in the regularized impulsive full pp-wave spacetime (49) andÎv T Mp , then there exists e0 such that the maximal unique geodesic ge starting in p in
direction v is complete, provided e e0.
We now briefly discuss the case of profile functions H and ai in the metric (49) pos-sessing poles, making it necessary to remove them from the spacetime, or likewise the casethat the exterior solution (49) is matched to some interior non-vacuum solution for small x andat least some u-interval. Recall from section 2 that such situations occur in physicallyinteresting models and that this leads to an incomplete wave surface N. In such a case ourmethod still applies, but with some restrictions. Indeed, if a ‘seed geodesic’ γ hits the wavezone at =u 0 sufficiently far away from the poles or the matching surface to an interiorsolution we may first apply proposition 5.1 with the constants b and c chosen to be so smallthat the problematic region is omitted. Then, in the ‘intermediate region’ e eÎ -u L,[ ] wecan estimate the solution ex in terms of the data of the ‘seed geodesic’ and the right-hand sideof (71), hence independently of e, which again makes it possible to avoid the problematicregion. This finally applies as well to the second application of proposition 5.1 in the interval
e e- +L L,[ ]. On the other hand, for ‘seed geodesics’ aiming too closely at poles or thematching boundary to an interior solution, completeness cannot be guaranteed, which isactually in complete agreement with physical expectations.
Finally, to end this section we prove an additional boundedness result for the globalgeodesics ge of theorem 5.3. Indeed, local uniform boundedness of the x-component ex (andof its derivative) follows directly from proposition 5.1 and the fact that in the ‘intermediateregion’ e eÎ -u L,[ ] the geodesic equation is actually e-independent. However, we alsoobtain local uniform boundedness of the r-component and hence of ge itself, as follows fromthe next statement.
Figure 2. An illustration of the construction leading to theorem 5.3. The x-componentof the seed geodesic γ is drawn in bold black and the regularized geodesic given byproposition 5.1 is drawn dotted in blue. The seed geodesic provides the initial data forthe geodesic ex at e- which on e e- ,[ ] solves (55) and continues as a solution of (71)until L−ε, where it provides new initial data for a solution ex of (55) until L+ε andthen continues as a solution of the background geodesic equation (66) with data (75).
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
20
Lemma 5.4 (Uniform boundedness of rε). The r-component er of any complete geodesic ge oftheorem 5.3 is locally uniformly bounded in e.
Proof. We first consider the first spike and hence let e eÎ -s ,[ ]. Then
ò òh s s- + + - + +ee e
e
e
e
ee
- -r s r r r r r r r s¨ d d , 760 0 0 0 0 0∣ ( )∣ ∣ ∣ ∣ ∣ (∣˙ ˙ ∣ ∣˙ ∣) ∣ ( )∣ ( )
where for the last term we use equation (54) and estimate
ò ò s s
a b
a b a b r
+ + ´ +
+ + + + +
- -
¢
- ¢ ¢
- ¢
r sK
LDa C
h aK
LDa C DH K C
h aK
LDH K DH
¨ d d 4
4 2
4 2 ,
2 2
12
2
1 22
2
( ) ( )
( )
( ) ( )
where ¢ + + ¥ C x c K F0 2≔ ∣ ˙ ∣ . Since e 1, this is bounded independently of e. Here, thenorms are ¥L -norms over the compact set ÍI n
1 given by proposition 5.1.In the interval e e- +L L,[ ], we may argue inprecisely the same way. Finally, for the
intervals e-¥ -,( ], e e-L,[ ] and e+ ¥L ,[ ) one uses the continuous dependence ofsolutions to ODEs on the initial conditions and the convergence of the data of the ‘seedgeodesics’ e-r ( ) of (67) to =r r0 0( ) . ,
6. Limits
In this final section we investigate the limiting behaviour of the complete regularizedgeodesics ge provided by theorem 5.3 as the regularization parameter e goes to zero.Physically, this amounts to calculating the geodesics of the impulsive full pp-wavemetric (44). In fact, this is only interesting if the geodesics are not parallel to, orcontained in the wave zone Îu L0,[ ]. So let g = x s s r s, ,( ( ) ( )) again be a ‘seed geo-desic’ starting ‘in front’ of the wave zone with increasings. To simplify notations we willalso briefly write g = x r,( ) and denote the data at the first impulsive surface byg = x r0 ,0 0( ) ( ) and g = x r0 ,0 0˙ ( ) ( ˙ ˙ ), respectively. Clearly motivated by the procedurewhich leads to the completeness result in section 5 we define the limiting geodesic g by(see also figure 3)
⎧⎨⎪⎩⎪
gggg
<<+
++
s
s s
s s L
s s L
0 ,
0 ,
,
77˜( ) ≔( ) ( )( ) ( )
( ) ( )( )
where g =+ + +x r,( ) is a geodesic between the spikes, i.e. a solution of
⎜ ⎟⎛⎝
⎞⎠= - - = - -D x
Lh a a x r
La x x h
La a a x
1, ¨
1 1, 78x
i ikk j j k
ji j
i j ikk i j j i
j, , , ,˙ ( ) ˙ ˙ ˙ ( ) ˙ ( )˙ ( ∣∣ )
with initial data (cf. (70))
g g e= =e
ee
e e+ + +
x r x r0 , lim lim , , 790 0
0 01 1( ) ( ) ≔ ( ) ( ) ( )
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
21
g g e= =e
ee
e e+ + +
x r x r0 , lim lim , , 800 0
0 01 1˙ ( ) ( ˙ ˙ ) ≔ ˙ ( ) ( ˙ ˙ ) ( )
where of course ge is the global geodesic of theorem 5.3 associated with the ‘seed’ γ.Furthermore, g =++ ++ ++x r,( ) is a background geodesic ‘behind’ the wave zone, i.e. g++
solves(66) with initial data (cf. (75))
g g e= + =e e
e e++ ++ ++
L x r L x r, lim lim , , 810 0
0 03 3( ) ( ) ≔ ( ) ( ) ( )
g g e= + =e e
e e++ ++ ++
L x r L x r, lim lim , . 820 0
0 03 3˙ ( ) ( ˙ ˙ ) ≔ ˙ ( ) ( ˙ ˙ ) ( )
Now we turn to the explicit calculation of the limits of the data (79) (80), i.e. thebehaviour of the limiting geodesic at the first spike.
Proposition 6.1. Let g =e e ex r,( ) given by theorem 5.3 with ‘seed’ g as above. Then,
⎜ ⎟⎛⎝
⎞⎠a= = - -+ +x x x x h x
a x
LH x,
1
2
2, 83i i ik k
k0 0 0 0 00
, 0( ˙ ) ( ˙ ) ( ) ( ) ( ) ( )
a= ++r r H x
2, 840 0 0( ) ( )
⎜ ⎟⎛⎝⎜
⎛⎝
⎞⎠
⎞⎠⎟a a= + - - +
+
+r r H xx
h xa x
LH x h x a x
Lh x a x a x
2
1
8
2
1
2. 85
j
jjk k
kjk
k
jkk j
0 0 , 00
00
, 0 0 0
2 0 0 0
˙ ˙ ( )˙
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
Figure 3. The limiting behaviour of the complete geodesics ge with two values of theregularization parameter e e< <0 2 1 exemplified. The x-components of the seed andlimiting geodesics are drawn in bold black, the regularized geodesics given byproposition 5.1 are drawn dotted in green ( ex 1) and blue ( ex 2), respectively. Note that
eex ( ) converges to x(0) for e 0 and similarly, e+ e+x L x L( ) ( ).
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
22
Proof. We only sketch these overly technical calculations. First, =+x x0 0 is a directconsequence of lemma 5.2(i). To obtain +x0˙ by lemma 5.2(ii) we only have to read off F x2 0( )from (55).
For +r0 and +r0 we use the integral equation for er with the integral equation for ex inserted,together with the identity ò òd d s s =
e
ee e e- -
s sd ds 1
2( ) ( ) and lemma 5.2. We only detail this in
case of r0:
I II
III IV
V VI
ò ò ò ò
ò ò ò ò
ò ò ò ò
e e q s q s
d s q d s
q d s d s
= + + -
+ -
+ +
ee e
e
e
ee e e
e
e
ee e
e
e
ea be
ee
e
ee
e
e
e
ee a b
e
e
e
ea be
- - - -
- - - -
- - - -
- -
r r rL
Dax x sL
ha Dax s
DH x sL
ha s
Lha DH s H s
21
d d1
d d
d d1
d d
1
2d d
1
2d d ,
s s
s s
L L
s s
u
0 0 22
,2
,
, , ,
1 1
( ) ˙ ˙ ˙ ˙
˙
( )
where clearly I–V are eO ( ). For example, III e a + + ¥ DH K x c K F2 I0 2 , 1∣ ∣ ∣ ∣ (∣ ˙ ∣ ). Finally,we estimate VI:
VI
ò òa
s d s s
s a r
- -
- ¢
e
e
ee a b
e
s e ee
- -
Î -
H x H x H x s
H x H x
2
1
2d d
1
2sup 4 ,
s
u0 0 , ,
,0
( ) ∣ ( ( )) ( )∣∣( ( )) ∣
∣ ( ( )) ( )∣∣ ∣[ ]
where we used that ò ò d s s¢ =e
e
e e- -sd d 1
s( ) and s -s e e eÎ - H x H xsup , 0∣ ( ( )) ( )∣[ ] converges to
zero by lemma 5.2.The most difficult case is +r0 , since e er( ˙ ) is not uniformly bounded in e. However, ee er( ˙ ( ))
converges, as can be seen in the following. As above, write Ie = + ¢ +eer r0˙ ( ) ˙
II III IV V VI¢ + ¢ + ¢ + ¢ + ¢. Then, I II e¢ + ¢ = O ( ) and in III¢ one inserts the integral equationfor ex and uses that ò òd d s s =
e
ee e e- -
s sd ds 1
2( ) ( ) . Finally, IV¢ and V¢ can be handled similarly
and for VI¢ one uses integration by parts to obtain ò òd d¢ = -ae
ee
ae
ee- -
H s DHx sd d2 2
˙ , which
can be handled as III¢. ,
We see from the explicit expressions given in proposition 6.1 that the limiting geodesic gdisplays the following behaviour at the first spike: the x-component is continuous with a finitejump in its velocity, since ¹+x x0 0˙ ˙ (in general). On the other hand, the r-component itself isdiscontinuous, suffering a finite jump, and the same is also true for its derivative. Thisbehaviour is correlated with the fact that while er is not uniformly bounded on e e- ,[ ], itsvalue when leaving the regularization strip eer ( ) is nevertheless uniformly bounded in e.
Now, we may analogously calculate the limits at the second spike to obtain
Proposition 6.2. Let g =e e ex r,( ) given by theorem 5.3 with ‘seed’ g as above. Then,
=++ +x x L , 860 ( ) ( )
⎛⎝⎜
⎞⎠⎟b= - -++ + +
++x x L h x L
a x L
LH x L
1
2
2, 87i i ik k
k0 ,( ˙ ) ( ˙ ( )) ( ( )) ( ( )) ( ( )) ( )
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
23
b= +
=
++ + +
++ +
r r L H x L
r r L2
,
88
0
0
( ) ( ( ))
( ) ( )
⎟
⎛⎝⎜
⎛⎝⎜
⎞⎠⎟
⎞⎠
b b+ - -
+ +
++
++
+
+ + + + +
H x L
x Lh x L
a x L
LH x L
h x L a x LL
h x L a x L a x L
2
1
8
2
1
2. 89
j
jjk k
k
jkk
jkk j
, ,
2
( ( )) ( ( )) ( ( )) ( ( )) ( ( ))
( ( )) ( ( ) ( ( )) ( ( )) ( ( )) ( )
Finally, we may prove the actual convergence result, saying that the regularized completegeodesics ge of theorem 5.3 converge to the limiting geodesics γ of (77), consisting ofappropriately matched geodesics of the complete background and the ‘intermediate’ region.
Theorem 6.3. Let g =e e ex r,( ) be the complete geodesic of theorem 5.3 with ‘seed’ g asabove. Then, ge converges to the limiting geodesic g = x r,˜ ( ˜ ˜) of (77) in the following sense:
(1) ex x locally uniformly, ex x˙ ˜ as distribution and uniformly on compact intervals notcontaining =t 0 or =t L.
(2) er r as distribution and in 1 on compact intervals not containing =t 0 or =t L.
Observe that the notions of convergence in the theorem are optimal given the regularityof the limits: x is discontinuous at s=0 and s=L so uniform convergence can only hold onbounded intervals not containing these two points. The same reasoning applies to r and itsderivative.
Proof. Let >T 0; then, on e- -T ,[ ] the regularized geodesic ge is equal to the ‘seedgeodesic’ g since both solve the same initial value problem.
Now, on e T,[ ] with e-T L (the first spike), ge and g+ solve the same ODE, i.e. (78),
but with different initial conditions. By continuous dependence of the solutions of ODEs oninitial data we have for eÎt T,[ ]
g g g g g e g e g e g e- - - -e e e e+ + + +t t t t emax , max , , 90LT(∣ ( ) ( )∣ ∣ ˙ ( ) ˙ ( )∣) (∣ ( ) ( )∣ ∣ ˙ ( ) ˙ ( )∣) ( )
where L is a Lipschitz constant for the (e-independent) right-hand side of (78) (on somesuitable bounded set). Now, we can insert g+ 0( ) to obtain
g e g e g e g g g e- - + - e e+ + + +0 0 0, 91∣ ( ) ( )∣ ∣ ( ) ( )∣ ∣ ( ) ( )∣ ( )
since g g e= e e+
0 lim 0( ) ( ) and g+ is continuous. Analogously, we insert g+ 0˙ ( ) to obtain
g e g e g e g g g e- - + - e e+ + + +0 0 0, 92∣ ˙ ( ) ˙ ( )∣ ∣ ˙ ( ) ˙ ( )∣ ∣ ˙ ( ) ˙ ( )∣ ( )
since g g e= e e+
0 lim 0˙ ( ) ˙ ( ) and by the continuity of g+˙ . This gives uniform convergence ofge e( ) on any compact interval not containing t=0 (in its interior).
On e e- ,[ ], lemma 5.2 yields the convergence of e ex( ) to x0 and to establish the globaldistributional convergence of e er( ) it remains only to consider e er( ) on e e- ,[ ]. So let
f Î ( ) and by lemma 5.4 there is a >C 0 such that er t C∣ ( )∣ and thus
ò f e f- e
ee
-¥ r s r s s s Cd 2 0. 93( ( ) ˜( )) ( ) ( )
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
24
Finally, the second spike, i.e. e e- +L L,[ ], and behind the wave zone, i.e.e+ ¥L ,( ), can be handled analogously. ,
7. Conclusion
In this contribution we have provided completeness results both for the extended as well asfor the impulsive case of full pp-waves. This class of geometries allows for an arbitrary n-dimensional Riemannian manifold N as a wave surface and for non-trivial off-diagonal termsin the metric (encoding the internal spin of the source), hence includes (as special cases)classical pp-waves, NPWs, and gyratons alike. In the extended case, we have generalized theresults on NPWs by providing a sufficient criterion for completeness in terms of the spatialasymptotics of the metric functions with a certain (local) uniformity with respect to propertime. In the impulsive case we have employed a regularization approach to prove that allthese geometries are complete (provided the spatial profile functions are smooth). Thisconfirms earlier results saying that the effect of the spatial asymptotics of the metric functionson completeness is wiped out in the impulsive limit. Finally, we have explicitly derived thegeodesics in the impulsive case in terms of a matching of corresponding background geo-desics. This result, in particular, allows us to derive the particle motion in the field of specificultrarelativistic particles possessing an internal spin, opening the door to applications inquantum scattering and high-energy physics.
Acknowledgements
We thank Jirí Podolský for numerous discussions and for generously sharing his experience.RŠ was supported by the grants GAČR P203/12/0118, UNCE204020/2012 and theMobility grant of Charles University. CS and RS were supported by FWF grants P25326 andP28770.
References
[AM78] Abraham R and Marsden J E 1978 Foundations of Mechanics 2nd edn (Reading, MA:Benjamin/Cummings ) Advanced Book Program, revised and enlarged, with the assistance ofT Raţiu and R Cushman
[AS71] Aichelburg P C and Sexl R U 1971 On the gravitational field of a massless particle J. Gen.Rel. Grav. 2 303–12
[BEE96] Beem J K, Ehrlich P E and Easley K L 1996 Global Lorentzian Geometry of Monographsand Volume 202 Textbooks in Pure and Applied Mathematics 2nd end (New York: MarcelDekker)
[Bon70] Bonnor W B 1970 Spinning null fluid in general relativity Int. J. Theor. Phys. 3 257–66[Bri25] Brinkmann H W 1925 Einstein spaces which are mapped conformally on each other Math.
Ann. 94 119–45[CFS03] Candela A M, Flores J L and Sánchez M 2003 On general plane fronted waves Geodesics
Gen. Rel. Gravit. 35 631–49[CFS04] Candela A M, Flores J L and Sánchez M 2004 Geodesic connectedness in plane wave type
spacetimes. A variational approach. Dyn. Sys. Appl. 4 458–64[CG12] Chruściel P T and Grant J D E 2012 On Lorentzian causality with continuous metrics Class.
Quantum Grav. 29 32 145001[Col85] Colombeau J-F 1985 Elementary introduction to new generalized functions, volume 113 of
North-Holland Mathematics Studies. Notes on Pure Mathematics (Amsterdam: North-Holland)p 103
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
25
[CRS12] Candela A M, Romero A and Sánchez M 2012 Remarks on the completeness of trajectories ofaccelerated particles in Riemannian manifolds and plane waves Proc. Int. Meeting onDifferential Geometry ((November 15–17, 2010, University of Córdoba)) pp 27–38
[CRS13] Candela A M, Romero A and Sánchez M 2013 Completeness of the trajectories of particlescoupled to a general force field Arch. Ration. Mech. Anal. 208 255–74
[EE92a] Ehrlich P E and Emch G G 1992 The conjugacy index and simple astigmatic focusing, InGeometry and nonlinear partial differential equations (Fayetteville, AR, 1990), volume 127 ofContemp. Math (Providence, RI: AMS) pp 27–39
[EE92b] Ehrlich P E and Emch G G 1992 Gravitational waves and causality Rev. Math. Phys. 4163–221
[EE93] Ehrlich P E and Emch G G 1993 Geodesic and causal behavior of gravitational plane waves:astigmatic conjugacy differential geometry: geometry in mathematical physics and relatedtopics Proc. Sympos. Pure Math. vol 54 (Los Angeles, CA, 1990) (Providence, RI: AMS)pp 203–9
[FF05] Frolov V P and Fursaev D V 2005 Gravitational field of a spinning radiation beam pulse inhigher dimensions Phys. Rev. D 71 104034
[Fil88] Filippov A F 1988 Differential Equations with Discontinuous Righthand Sides (Mathematicsand its Applications (Soviet Series) vol 18) (Dordrecht: Kluwer) translated from the Russian
[FIZ05] Frolov V P, Israel W and Zelnikov A 2005 Gravitational field of relativistic gyratons Phys.Rev. D(3) 72 11 084031
[FPV88] Ferrari V, Pendenza P and Veneziano G 1988 Beam-like gravitational waves and theirgeodesics. Gen. Rel. Gravit. 20 1185–91
[FS03] Flores J L and Sánchez M 2003 Causality and conjugate points in general plane waves Class.Quantum Grav. 20 2275–91
[FS06] Flores J L and Sánchez M 2006 On the Geometry of pp-wave type Spacetimes. In Analytical andNumerical Approaches to Mathematical Relativity, vol 692 of Lecture Notes in Phys. (Berlin:Springer) pp 79–98
[FZ06] Frolov V P and Zelnikov A 2006 Gravitational field of charged gyratons Class. Quantum Grav.23 2119–28
[GKOS01] Grosser M, Kunzinger M, Oberguggenberger M and Steinbauer R 2001 Geometric Theoryof Generalized Functions with Applications to General Relativity volume 537 of Mathematicsand its Applications (Dordrecht: Kluwer)
[Gor70] Gordon W B 1970 On the completeness of Hamiltonian vector fields Proc. Amer. Math. Soc.26 329–31
[GP09] Griffiths J B and Podolský J 2009 Cambridge Monographs on Mathematical Physics ExactSpace-times in Einstein’s General Relativity (Cambridge: Cambridge University Press)
[KS99a] Kunzinger M and Steinbauer R 1999 A note on the Penrose junction conditions Class.Quantum Grav. 16 1255–64
[KS99b] Kunzinger M and Steinbauer R 1999 A rigorous solution concept for geodesic and geodesicdeviation equations in impulsive gravitational waves. J. Math. Phys. 40 1479–89
[KSSV15] Kunzinger M, Steinbauer R, Stojković M and Vickers J A 2015 Hawkingʼs singularitytheorem for C1,1-metrics Class. Quantum Grav. 32 075012
[Kun61] Kundt W 1961 The plane-fronted gravitational waves Z. Physik 163 77–86[Kun62] Kundt W 1962 Exact solutions of the fields equations: twist-free pure radiation fields Proc. R.
Soc. Ser. A 270 pp 328–34[LSŠ14] Lecke A, Steinbauer R and Švarc R 2014 The regularity of geodesics in impulsive pp-waves
Gen. Rel. Gravit. 46 8 Art. 1648[Min15] Minguzzi E 2015 Completeness of first and second order ODE flows and of Euler–Lagrange
equations J. Geom. Phys. 97 156–65[OPP13] Ortaggio M, Pravda V and Pravdová A 2013 Algebraic classification of higher dimensional
spacetimes based on null alignment Class. Quantum Grav. 30 013001[Pen65a] Penrose R 1965 Gravitational collapse and space-time singularities Phys. Rev. Lett. 14
pp 57–9[Pen65b] Penrose R 1965 A remarkable property of plane waves in general relativity Rev. Modern
Phys. 37 215–20[Pen72] Penrose R 1972 The geometry of impulsive gravitational waves General Relativity (papers in
honour of J. L. Synge) (Oxford: Clarendon Press) pp 101–115[Per59] Peres A 1959 Some gravitational waves Phys. Rev. Lett. 3 571–2
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
26
[PG98] Podolský J and Griffiths J B 1998 Boosted static multipole particles as sources of impulsivegravitational waves Phys. Rev. D 58 5 124024
[PŠ13] Podolský J and Švarc R 2013 Explicit algebraic classification of Kundt geometries in anydimension Class. Quantum Grav. 30 125007
[PSŠ14] Podolský J, Steinbauer R and Švarc R 2014 Gyratonic pp waves and their impulsive limitPhys. Rev. D 90 044050
[PSSŠ15] Podolský J, Sämann C, Steinbauer R and Švarc R 2015 The global existence, uniqueness andC1-regularity of geodesics in nonexpanding impulsive gravitational waves Class. QuantumGrav. 32 23 025003
[PSSŠ16] Podolský J, Sämann C, Steinbauer R and Švarc R 2016 The global uniqueness andC1-regularity of geodesics in expanding impulsive gravitational waves Class. Quantum Grav.33 195010
[PV98] Podolský J and Veselý K 1998 Chaos in pp-wave spacetimes. Phys. Rev. D 58 081501[PV99] Podolský J and Veselý K 1999 Smearing of chaos in sandwich pp-waves Class. Quantum
Grav. 16 3599–618[Säm16] Sämann C 2016 Global hyperbolicity for spacetimes with continuous metrics Ann. Henri
Poincaré 17 1429–55[Sbi15] Sbierski J 2015 The C0-inextendibility of the Schwarzschild spacetime and the spacelike
diameter in Lorentzian Geometry J. Diff. Geom. to appear arXiv:1507.00601 [gr-qc][SS12] Sämann C and Steinbauer R 2012 On the completeness of impulsive gravitational wave
spacetimes Class. Quantum Grav. 29 11 245011[SS15] Sämann C and Steinbauer R 2015 Geodesic completeness of generalized space-times In
Pseudo-differential operators and generalized functions, Oper. Theory Adv. Appl. vol 245(Cham: Birkhäuser/Springer) pp 243–253
[SSLP16] Sämann C, Steinbauer R, Lecke A and Podolský J 2016 Geodesics in nonexpandingimpulsive gravitational waves with L, Part I Class. Quantum Grav. 33 115002
[Wei52] Weissinger J 1952 Zur Theorie und Anwendung des Iterationsverfahrens Math. Nachr. 8193–212
[YZF07] Yoshino H, Zelnikov A and Frolov V P 2007 Apparent horizon formation in the head-oncollision of gyratons Phys. Rev. D 75 21 124005
Class. Quantum Grav. 33 (2016) 215006 C Sämann et al
27