Scalar Hairy Black Holes, Solitons and Isolated Horizons

1. Einstein Centenary Conference, Paris, July, 2005 Scalar Hairy Black Holes, Solitons and Isolated Horizons by Marcelo Salgado Instituto de Ciencias Nucleares,UNAM (in collaboration with U. Nucamendi and A. Corichi)

2. Preliminaries and Motivations • No-hair conjecture: According to Wheeler´s folklorical statement: “Black Holes have no –hair: BH are completely characterized by their mass, charge and angular momentum” Hair: Quantities (other than the ones related to the conserved charges at spatial infinity ) needed to characterize completely a stationary black hole within a theory. (sometimes one usually demands that for hair to exist, the theory must admit also the hairless solution as a particular case: that is A hairless black hole solution characterized completely by its charges at and even that the hairy-solution be stable) In order to prove the no-hair conjecture several theorems has been established: Black Hole Uniqueness theorems: In Einstein-Maxwell (EM) theory (Israel,Carter, Wald, Robinson): all the BH solutions within EM theory are stationary and axially symmetric and contained within the Kerr-Newman family

3. Other theorems show that BH are hairless in a variety of theories coupling different classical fields to Einstein gravity (Chase, Bekenstein, Hartle, Teitelboim). A key ingredient in the no-hair theorem proofs relies on the assumption of asymptotic flatness (AF) and on the nature of the energy momentum-tensor. • However, in recent years conterexamples to the no-hair conjecture were found in several theories with non-Abelian gauge fields: • Einstein-Yang-Mills • Einstein-Yang-Mills-Higgs • Einstein-Yang-Mills-Dilaton, etc. For the issue at hand (scalar-field hair), however, no such counterexamples had been found for an explicit . In fact for matter composed by a single scalar-field, no-hair theorems were proved (Sudarsky, Bekenstein). Basically, Sudarsky´s proof assumes AF boundary conditions and the validity of the Weak-Energy Condition (WEC) which constrains the scalar-field potential to be non-negative: I´ll return to the no-scalar-hair theorems later.

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

5. Plan of the talk 1st Part: Hairy Black Holes and Solitons in an Einstein-Higgs theory (see PRD vol. 68 (2003) 0404026, U.Nucamendi, and M.S) • The theory leading to hairy BH solutions and solitons. • Presentation of solutions • Stability analysis

6. Plan of the talk 2nd Part: Analysis of the above black hole solutions within the isolated-horizon framework (see A.Corichi, U.Nucamendi,& MS: gr-qc/0504126) • Isolated-horizon preliminaries and mass formulae • Numerical results and empirical formula • Conclusions

7. 1st partThe theory and the spacetime • Einstein-scalar field equations and Lagrangian: • Units where are employed. The gravitational and scalar field equations obtained from the Lagrangian are • Where the scalar-field potential is given as follows

8. The scalar-field potential • We choose the following asymmetric scalar-field potential leading to the desired asymptotically flat solutions: Where , and are constants. For this class of potentials one can see that for , corresponds to a local minimum, is the global minimum and is a local maximum

9. The scalar-field potential and numerical results The key ingredients in the form of the potential for nontrivial (hairy BH/solitonic) solutions to exist are: • is not non-negative (in this way one can avoid the no-hair theorems for scalar fields: WEC is violated) • has a root and a local minimum at the same place (this allows to obtain genuine asymptotically flat solutions).

10. 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 -coordinate solely. The resulting field equations are We shall attempt to solve the above system of equations using a numerical analisys.

11. Boundary conditions and numerical methodology • For BH configurations we demandregularity on the event horizon located at and for the solitons (scalarons) regularity at the origin . This implies the following conditions for the fields Where the values and will be determined so that the asymptotically flat conditions are verified (see below). For the scalaron, regularity at the origin results by taking in the above regularity conditions. As mentioned, in addition to the regularity conditions, we impose asymptotically flat conditions on the spacetime (for BH and scalarons). These imply the following conditions on fields when

12. Boundary conditions and numerical methodology (cont.) • Where is the ADM-mass associated with a given BH configuration and is the value of the scalar field at which correspond to a local minimum and a root of (see below). A BH configuration will be parametrized by the arbitrary free parameter 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 parameter which is fixed so that the above AF conditions are fullfilled. In this way .As mentioned above the scalarons is included as the particular case when . • Using the above system of differential equations together with the regularity and asymptotic conditions, we have performed a numerical analysis for one class of scalar field potentials.

13. Asymptotic behavior and the shooting method As mentioned before, asymptotically the spacetime behaves like Minkowski spacetime : And the scalar field should verify the KG equation expanded asymptoticallly around the local minimum This implies that The shooting method uses as a shooting parameter and fixes the value so that the constant above is zero. Therefore the method eliminates the runaway solution. In this way one enforces that the scalar field goes to the local minimum asymptotically.

14. Numerical results • BH configuration with

15. Numerical results • BH configuration with • The above mass function converges to the ADM mass

16. Numerical results • BH configuration with

17. Numerical results • Soliton configuration with

18. Numerical results • Soliton configuration with • The above mass function converges to the ADM mass

19. Numerical results • Soliton configuration with

20. Stability Analysis • A)Heuristic Analysis: when fixing boundary conditions (at the horizon/origin) one would expect that the BH/soliton configuration of minimum energy is the most stable within the theory. In the case of BH it turns that • Therefore since the total energy of the BH-hairy configuration is greater that the corresponding energy of the hairless (Schwarzschild) BH, one expects heuristically that the hairy configuration is unstable within the same theory. Same happens with the soliton.

21. Stability Analysis B)Rigorous Perturbation Analysis: the heuristic expectations are confirmed by a linear-radial perturbation analysis: • One can easily show that the perturbations decouple, and that metric perturbations depend on : • Therefore, all the analysis reduces to solving the perturbation equation for , which turns to be

22. Stability analysis where and the effective potential is given by One can seek mode perturbations So that the above perturbation equation writes as a Schrodinger-like equation Therefore it suffices to find a “bound state” with for the perturbation to grow unboundedly with time.

23. Stability analysis

24. Stability analysis • We have solved the Schrodinger-like equation numerically For both the BH and solitons, and found that there exist indeed bound states with and

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

26. 2nd partBH, solitons and the IH formalism • Isolated Horizon (IH) formalism (A quick review; for details see Ashtekar,Beetle,Fairhurst, CQG vol.16 (1999) L1; idem CQG vol. 17 (2000) 253 ) • Physical View: The IH of a spacetime containing a BH is a quasi-local definition (as opposed to the event horizon). One is interested in situations where the BH is in quasi-equilibrium (settle down) even if some radiation process is taking outside the BH (e.g. gravitational radiation). Therefore we shall restrict to situations where no incoming radiation is entering into the Hole (…but perhaps outgoing radiation will reach Scri+ as emitted earlier by the BH (or whatever it formed the BH) when it was not in equilibrium)

28. 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 Ashtekar et al. -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)

29. 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.

30. 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

31. A null dyad , can in addition be completed as to form a null tetrad with other two null (complex) vectors Satisfying the following conditions Moreover, the following expansions satisfy the conditions as shown below 2) The field equations (Einstein eqs.) together with the matter equations hold at 3) In the present case, we require that scalar fields at be spherically symmetric

32. BH mechanical properties • Surface gravity: due to the fact that we are considering SSS configurations with a timelike-static Killing field , and with It turns out that the BH-surface gravity is given by Introducing the quantity

33. 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): 1) (from the Hamiltonian formalism) Where the BH ADM mass is obtained from the expression And the soliton mass corresponds to 2) (from the 1st law of BH-thermodynamics)

35. An empirical formula • It is remarkable that the following empirical formula reproduces the ADM-mass with a precision better that 5% where and the values of and depend of the theory: -For EYM: -For the E-Higgs theory presented here

36. Scalar-hairy BH´s and scalaron (solid line: numerical)

37. Colored BH´s and Bartnik-McKinnon soliton (solid line: numerical)

38. Conclusions • We showed scalar-hairy BH and their solitons in an Einstein-Higgs theory with non-positive semi-definite • Such objects are in fact unstable • A mass formula relating the ADM, IH and soliton masses that holds in the EYM case was proved to hold also in the case presented here. • A remarkable-simple empirical formula captures the essential qualitative behavior of the ADM mass for the hairy BH (specially for small and large values of ) • It remains a theoretical challenge to understand the origin of such simple formula.