Scalar Hairy Black Holes, Solitons and Scalar Hairy Black Holes, Solitons and Isolated Horizons Isolated Horizons

Scalar Hairy Black Holes, Solitons and Scalar Hairy Black Holes, Solitons and Isolated Horizons Isolated Horizons

bybyMarcelo Salgado Marcelo Salgado

Instituto de Ciencias Nucleares,UNAMInstituto de Ciencias Nucleares,UNAM(in collaboration with U. Nucamendi and A. Corichi)(in collaboration with U. Nucamendi and A. Corichi)

Einstein Centenary Conference, Paris, July, 2005

Einstein Centenary Conference, Paris, July, 2005

Preliminaries and MotivationsPreliminaries and MotivationsPreliminaries and MotivationsPreliminaries and Motivations

No-hair conjecture: According to Wheeler´s folklorical statement: No-hair conjecture: According to Wheeler´s folklorical statement: ““Black Holes have no –hair: BH are completely characterized by their mass, Black Holes have no –hair: BH are completely characterized by their mass,

charge and angular momentum”charge and angular momentum”

Hair: Quantities (other than the ones related to the conserved charges at Hair: Quantities (other than the ones related to the conserved charges at spatial infinity ) needed to characterize completely a stationary black spatial infinity ) needed to characterize completely a stationary black hole within a theory.hole within a theory.

(sometimes one usually demands that for hair to exist, the theory must admit (sometimes one usually demands that for hair to exist, the theory must admit also the hairless solution as a particular case: that isalso the hairless solution as a particular case: that is

A hairless black hole solution characterized completely by its charges at A hairless black hole solution characterized completely by its charges at and even that the hairy-solution be stable)and even that the hairy-solution be stable)

In order to prove the no-hair conjecture several theorems has been In order to prove the no-hair conjecture several theorems has been established: established:

Black Hole Uniqueness theorems: Black Hole Uniqueness theorems: In Einstein-Maxwell (EM) theory (Israel,Carter, Wald, Robinson): all the In Einstein-Maxwell (EM) theory (Israel,Carter, Wald, Robinson): all the

BH solutions within EM theory are stationary and axially symmetric and BH solutions within EM theory are stationary and axially symmetric and contained within the Kerr-Newman familycontained within the Kerr-Newman family

i0

0i

Other theorems show that BH are hairless in a variety of theories Other theorems show that BH are hairless in a variety of theories coupling different classical fields to Einstein gravity (Chase, coupling different classical fields to Einstein gravity (Chase, Bekenstein, Hartle, Teitelboim). A key ingredient in the no-hair Bekenstein, Hartle, Teitelboim). A key ingredient in the no-hair theorem proofs relies on the assumption of asymptotic flatness (AF) and theorem proofs relies on the assumption of asymptotic flatness (AF) and on the nature of the energy momentum-tensor.on the nature of the energy momentum-tensor.

However, in recent years conterexamples to the no-hair conjecture were However, in recent years conterexamples to the no-hair conjecture were found in several theories with non-Abelian gauge fields: found in several theories with non-Abelian gauge fields:

Einstein-Yang-MillsEinstein-Yang-Mills Einstein-Yang-Mills-HiggsEinstein-Yang-Mills-Higgs Einstein-Yang-Mills-Dilaton, etc.Einstein-Yang-Mills-Dilaton, etc.

For the issue at hand (scalar-field hair), however, no such counterexamples For the issue at hand (scalar-field hair), however, no such counterexamples had been found for an explicit . In fact for matter had been found for an explicit . In fact for matter composed by a single scalar-field, no-hair theorems were proved composed by a single scalar-field, no-hair theorems were proved (Sudarsky, Bekenstein).(Sudarsky, Bekenstein).

Basically, Sudarsky´s proof assumes AF boundary conditions and the Basically, Sudarsky´s proof assumes AF boundary conditions and the validity of the Weak-Energy Condition (WEC) which constrains the validity of the Weak-Energy Condition (WEC) which constrains the scalar-field potential to be non-negative: scalar-field potential to be non-negative:

I´ll return to the no-scalar-hair theorems later.I´ll return to the no-scalar-hair theorems later.0)( V

)(V

...Finally, the main motivation that led us to analyse the ...Finally, the main motivation that led us to analyse the possible existence of scalar-hair in AF spacetimes, came by possible existence of scalar-hair in AF spacetimes, came by the recent discovery of scalar-hair in AdS spacetimes the recent discovery of scalar-hair in AdS spacetimes (T.Tori, K. Maeda, M. Narita, PRD vol.64 2001) and the (T.Tori, K. Maeda, M. Narita, PRD vol.64 2001) and the suggestion that by adjusting the scalar-potential parameters suggestion that by adjusting the scalar-potential parameters one could obtain a possible AF hairy solution (D. Sudarsky, one could obtain a possible AF hairy solution (D. Sudarsky, J.A. González, PRD. Vol. 67 2003).J.A. González, PRD. Vol. 67 2003).

Plan of the talkPlan of the talkPlan of the talkPlan of the talk

The theory leading to hairy BH solutions and The theory leading to hairy BH solutions and solitons.solitons.

Presentation of solutionsPresentation of solutions

Stability analysisStability analysis

1st Part: Hairy Black Holes and Solitons in an Einstein-Higgs theory

(see PRD vol. 68 (2003) 0404026, U.Nucamendi, and M.S)

Plan of the talkPlan of the talkPlan of the talkPlan of the talk

Isolated-horizon preliminaries and mass formulaeIsolated-horizon preliminaries and mass formulae

Numerical results and empirical formulaNumerical results and empirical formula

ConclusionsConclusions

2nd Part: Analysis of the above black hole solutions within the isolated-horizon framework

(see A.Corichi, U.Nucamendi,& MS: gr-qc/0504126)

1st part1st partThe theory and the spacetimeThe theory and the spacetime

Einstein-scalar field equations and Lagrangian:Einstein-scalar field equations and Lagrangian:

Units where are employed. The gravitational and scalar Units where are employed. The gravitational and scalar field equations obtained from the Lagrangian arefield equations obtained from the Lagrangian are

Where the scalar-field potential is given as followsWhere the scalar-field potential is given as follows

10 cG

)(V

The scalar-field potential The scalar-field potential We choose the following asymmetric scalar-field potential leading to We choose the following asymmetric scalar-field potential leading to

the desired asymptotically flat solutions:the desired asymptotically flat solutions:

Where , and are constants. For this class of potentials one Where , and are constants. For this class of potentials one can see that for , corresponds to a local minimum, can see that for , corresponds to a local minimum,

is the global minimum and is a local is the global minimum and is a local maximummaximum

0 i

a

02 21 a

2 a1 a

1 a

2 a

a

The scalar-field potential and numerical resultsThe scalar-field potential and numerical results The key ingredients in the form of the potential for nontrivial (hairy The key ingredients in the form of the potential for nontrivial (hairy

BH/solitonic) solutions to exist are:BH/solitonic) solutions to exist are: is not non-negative (in this way one can avoid the no-hair is not non-negative (in this way one can avoid the no-hair

theorems for scalar fields: WEC is violated)theorems for scalar fields: WEC is violated) has a root and a local minimum at the same place (this allows to has a root and a local minimum at the same place (this allows to

obtain genuine asymptotically flat solutions).obtain genuine asymptotically flat solutions).

1 a

2 a

a

)(V

)(V

We will focus on a metric describing spherical and static spacetimes:We will focus on a metric describing spherical and static spacetimes:

And the metric potential as well as the scalar field will be functions of the And the metric potential as well as the scalar field will be functions of the -coordinate solely. The resulting field equations are -coordinate solely. The resulting field equations are

We shall attempt to solve the above system of equations using a numerical analisys.We shall attempt to solve the above system of equations using a numerical analisys.

r

Boundary conditions and numerical Boundary conditions and numerical methodologymethodology

For BH configurations we demandFor BH configurations we demand regularity on the event horizon regularity on the event horizon located at and for the solitons (scalarons) located at and for the solitons (scalarons) regularity at the origin . This implies the following conditions for regularity at the origin . This implies the following conditions for the fieldsthe fields

Where the values and will be determined so that the Where the values and will be determined so that the asymptotically flat conditions are verified (see below). For the asymptotically flat conditions are verified (see below). For the scalaron, regularity at the origin results by taking in the scalaron, regularity at the origin results by taking in the above regularity conditions. As mentioned, in addition to the above regularity conditions. As mentioned, in addition to the regularity conditions, we impose asymptotically flat conditions on the regularity conditions, we impose asymptotically flat conditions on the spacetime (for BH and scalarons). These imply the following spacetime (for BH and scalarons). These imply the following conditions on fields when conditions on fields when

hr

0r

h h

0r 0hr

0/)(21)( hhh rrmrN

Boundary conditions and numerical Boundary conditions and numerical methodology (cont.)methodology (cont.)

Where is the ADM-mass associated with a given BH Where is the ADM-mass associated with a given BH configuration and is the value of the scalar field at which configuration and is the value of the scalar field at which

correspond to a local minimum and a root of (see below)correspond to a local minimum and a root of (see below) . . A BH A BH configuration will be parametrized by the arbitrary free parameter configuration will be parametrized by the arbitrary free parameter which specifies the location of the BH horizon. On the other hand, the which specifies the location of the BH horizon. On the other hand, the value is not a free parameter but it is rather a shooting value is not a free parameter but it is rather a shooting parameter which is fixed so that the above AF conditions are fullfilled. parameter which is fixed so that the above AF conditions are fullfilled. In this way .As mentioned above the scalarons is included In this way .As mentioned above the scalarons is included as the particular case when .as the particular case when .

Using the above system of differential equations together with the Using the above system of differential equations together with the regularity and asymptotic conditions, we have performed a numerical regularity and asymptotic conditions, we have performed a numerical analysis for one class of scalar field potentials.analysis for one class of scalar field potentials.

ADMM

0i

)(V

hr

)( hADMADM rMM

)( hhh r

0hr

Asymptotic behavior and the shooting methodAsymptotic behavior and the shooting method

As mentioned before, asymptotically the spacetime behaves like As mentioned before, asymptotically the spacetime behaves like Minkowski spacetime : Minkowski spacetime :

And the scalar field should verify the KG equation expanded And the scalar field should verify the KG equation expanded asymptoticallly around the local minimum asymptoticallly around the local minimum

This implies that This implies that

The shooting method uses as a shooting parameter and The shooting method uses as a shooting parameter and fixes the value so that the constant above is zero. fixes the value so that the constant above is zero.

Therefore the method eliminates the runaway solution. In this way one Therefore the method eliminates the runaway solution. In this way one enforces that the scalar field goes to the local minimum enforces that the scalar field goes to the local minimum asymptotically.asymptotically.

Minkababr

gg

lim

Minkababr

gg

lim

a

re

Br

eA

rr 2121

re

Br

eAa

rr 2121

)( hhh r )( hhh r A

Numerical resultsNumerical results

BH configuration withBH configuration with ...40786.0 , 1.0

,1.0 ,5.0 ,0 21 hhra

Numerical resultsNumerical results

BH configuration withBH configuration with The above mass function converges to the ADM mass The above mass function converges to the ADM mass

...40786.0 , 1.0

,1.0 ,5.0 ,0 21 hhra

/834.3ADMM

Numerical resultsNumerical results

BH configuration withBH configuration with

...40786.0 , 1.0

,1.0 ,5.0 ,0 21 hhra

ttg

rrge

Numerical resultsNumerical results

Soliton configuration withSoliton configuration with ...40594.0 ,0 ,1.0 ,5.0 ,0 021 hra

Numerical resultsNumerical results

Soliton configuration withSoliton configuration with The above mass function converges to the ADM mass The above mass function converges to the ADM mass

/827.3ADMM

...40594.0 ,0 ,1.0 ,5.0 ,0 021 hra

Numerical resultsNumerical results

Soliton configuration withSoliton configuration with

...40594.0 ,0 ,1.0 ,5.0 ,0 021 hra

rrg

ttg

e

Stability AnalysisStability Analysis

A)Heuristic Analysis: when fixing boundary conditions (at A)Heuristic Analysis: when fixing boundary conditions (at the horizon/origin) one would expect that the BH/soliton the horizon/origin) one would expect that the BH/soliton configuration of minimum energy is the most stable within configuration of minimum energy is the most stable within the theory. In the case of BH it turns that the theory. In the case of BH it turns that

Therefore since the total energy of the BH-hairy Therefore since the total energy of the BH-hairy configuration is greater that the corresponding energy of configuration is greater that the corresponding energy of the hairless (Schwarzschild) BH, one expects heuristically the hairless (Schwarzschild) BH, one expects heuristically that the hairy configuration is that the hairy configuration is unstableunstable within the same within the same theory. Same happens with the soliton.theory. Same happens with the soliton.

ildSchwarzsch)()( 00 hADMhADM rMrM

ildSchwarzsch)()( 00 hADMhADM rMrM

Stability AnalysisStability Analysis

B)Rigorous Perturbation Analysis: the heuristic expectations B)Rigorous Perturbation Analysis: the heuristic expectations are confirmed by a linear-radial perturbation analysis:are confirmed by a linear-radial perturbation analysis:

One can easily show that the perturbations decouple, and One can easily show that the perturbations decouple, and that metric perturbations depend on :that metric perturbations depend on :

Therefore, all the analysis reduces to solving the Therefore, all the analysis reduces to solving the perturbation equation for , which turns to beperturbation equation for , which turns to be

),(1 rt

),(1 rt

Stability analysisStability analysis

wherewhereand the effective potential is given byand the effective potential is given by

One can seek mode perturbationsOne can seek mode perturbations So that the above perturbation equation writes as a Schrodinger-So that the above perturbation equation writes as a Schrodinger-

like equationlike equation

Therefore it suffices to find a “bound state” with for the Therefore it suffices to find a “bound state” with for the perturbation to grow unboundedly with time. perturbation to grow unboundedly with time.

tierrt )(),(

*** 2**2** rVrr

02 ),( rt

Stability analysisStability analysis

Stability analysisStability analysis We have solved the Schrodinger-like equation numericallyWe have solved the Schrodinger-like equation numerically

For both the BH and solitons, and found that there exist indeed bound For both the BH and solitons, and found that there exist indeed bound states with andstates with and

*** 2**2** rVrr

BH)...(for 00241.02 solitons)(for ...00243.02

EvolutionEvolution

For a full non-perturbative evolution see For a full non-perturbative evolution see M.Alcubierre, J.A. González & M.S. PRD M.Alcubierre, J.A. González & M.S. PRD vol.70 064016 (2004)vol.70 064016 (2004)

2nd part2nd partBH, solitons and the IH formalismBH, solitons and the IH formalism

Isolated Horizon (IH) formalism (A quick review; Isolated Horizon (IH) formalism (A quick review; for details for details see Ashtekar,Beetle,Fairhurst, CQG vol.16 (1999) L1; idem CQG vol. 17 see Ashtekar,Beetle,Fairhurst, CQG vol.16 (1999) L1; idem CQG vol. 17

(2000) 253 (2000) 253 )) Physical View: The IH of a spacetime containing a BH is a Physical View: The IH of a spacetime containing a BH is a

quasi-local definition (as opposed to the event horizon). quasi-local definition (as opposed to the event horizon). One is interested in situations where the BH is in quasi-One is interested in situations where the BH is in quasi-equilibrium (settle down) even if some radiation process is equilibrium (settle down) even if some radiation process is taking outside the BH (e.g. gravitational radiation). taking outside the BH (e.g. gravitational radiation). Therefore we shall restrict to situations where no incoming Therefore we shall restrict to situations where no incoming radiation is entering into the Hole (…but perhaps outgoing radiation is entering into the Hole (…but perhaps outgoing radiation will reach Scri+ as emitted earlier by the BH (or radiation will reach Scri+ as emitted earlier by the BH (or whatever it formed the BH) when it was not in whatever it formed the BH) when it was not in equilibrium)equilibrium)

An example of a physical situation where a isolated horizon forms but where the Event horizon is not fully formed because matter falls in to the BH at later times

-While the standard BH (formalism) cannot account for establishing the laws of BH mechanics in the intermediary stages but only after the stationary situation is

reached, the IH allows to incorporate locally the BH mechanical laws during the time while the apparent horizon is in equilibrium.

-Moreover, the IH formalism can apply also to describe thermodynamical properties of spacetimes without a BH but with cosmological horizons (e.g. de

Sitter space)

Ashtekar et al.

Finally and not least,

A) The usual definition of ADM mass is globally defined, and therefore when a BH and raditation are present requires taking into account the presence of such radiation which contributes to the mass of the whole spacetime (but like in the cases previously elucidated, such contribution is irrelevant

for an intuitive notion of the BH-horizon mass-i.e., the ADM-mass without taking into account the radiation). How to define then this Horizon mass ?

B) In the stationary situation, the definition of the BH-surface gravity requires the existence of a globally defined Killing field (whose normalization is specified at spatial infinity; this removes the ambiguity in defining ), and so, in situations like those described previously it would be not possible to use the standard formalisms to introduce the notions of the BH mechanical laws due to the absence of such a global Killing field.

…. the IH-formalism repairs and allows to generalize this notions for situations exactly like the above.

Non-rotating isolated horizons

Def. A non-rotating isolated horizon is a sub-manifold of spacetime at which the following conditions are imposed:

1) is a null surface, topologically , which can be foliated by a preferred foliation given by 2-spheres , with 2 null normal vector fields defined on : one (equipped with a class of equivalence , such that any member of the equivalence class differs from any other only by functions (called spherically symmetric) that are constant on .

The other normal , is a null future directed vector field which is

chosen such that holds. There is a class of equivalence

where every member of the class (a pair of such vector fields) is related to any other as

2S

San

a

1aan

aaaa FnFn ,, 1

an

aa n,

F

2) The field equations (Einstein eqs.) together with the matter equationshold at

3) In the present case, we require that scalar fields at be spherically symmetric

A null dyad , can in addition be completed as to form a null tetrad with other two null (complex) vectors

aa n,

1

aa mmSatisfying the following conditions

aa

aa mnm 0

Moreover, the following expansions satisfy the conditions as shown below

24)0

r S of (Areaa mm baba

0

baba nmm

0

abba nm

0)(;

nmm baba

),( aa mm

BH mechanical propertiesBH mechanical properties

Surface gravity: due to the fact that we are considering SSS Surface gravity: due to the fact that we are considering SSS configurations with a timelike-static Killing field , and configurations with a timelike-static Killing field , and

withwith tt /

ttttg ,

It turns out that the BH-surface gravity is given by

)(812

2124

1 2)()(2/1

Vr r

em

r

eg

r

r

r

rrtt

Introducing the quantity r2

The IHF together with the Hamiltonian formalism of gravity and the 1st law of thermodynamics, leads to two compatible mass formulae for the BH horizon (Corichi & Sudarsky PRD Vol.61 (2000) 101501R):

solADM MrMrM )()(

Where the BH ADM mass is obtained from the expression

drrVNr

rMr

rADM

22

)](2

)([4

2)(

And the soliton mass corresponds to )0(ADMM

(from the Hamiltonian formalism)

rdrrMr

)(2

1)(

0

1)

2)

(from the 1st law of BH-thermodynamics) r2

holesblack hairy -Scalar

M

ADMM

2/r

An empirical formulaAn empirical formula

It is remarkable that the following empirical formula It is remarkable that the following empirical formula reproduces the ADM-mass with a precision better that 5%reproduces the ADM-mass with a precision better that 5%

C

DDrCr

C

DMM solADM 2

)()2

( 22

and the values of and depend of the theory:

solM C-For EYM:

2

1 ,)( CnMM solsol

-For the E-Higgs theory presented here

)12.0(2

1 ,827.3 BCM sol

0

0

2

2

C

MCD solwhere

C

DDrCr

C

DMM solADM 2

)()2

( 22

line) (dashed

(solid line: numerical)

Scalar-hairy BH´s and scalaron

C

DDrCr

C

DMM sol

EYM

ADM 2)()

2( 22

line) (dashed

(solid line: numerical)

Colored BH´s and Bartnik-McKinnon soliton

ConclusionsConclusions

We showed scalar-hairy BH and their solitons in an Einstein-We showed scalar-hairy BH and their solitons in an Einstein-Higgs theory with non-positive semi-definite Higgs theory with non-positive semi-definite

Such objects are in fact unstable Such objects are in fact unstable A mass formula relating the ADM, IH and soliton masses that A mass formula relating the ADM, IH and soliton masses that

holds in the EYM case was proved to hold also in the case holds in the EYM case was proved to hold also in the case presented here.presented here.

A remarkable-simple empirical formula captures the essential A remarkable-simple empirical formula captures the essential qualitative behavior of the ADM mass for the hairy BH qualitative behavior of the ADM mass for the hairy BH (specially for small and large values of ) (specially for small and large values of )

It remains a theoretical challenge to understand the origin of It remains a theoretical challenge to understand the origin of such simple formula.such simple formula.

)(V

r