On the end point of black hole evaporation

On the end point of black hole evaporation

Daniel Grumiller

Institut f¨ur Theoretische Physik, Universit¨at Leipzig, Augustplatz 10, D-04109 Leipzig, Germany

supported by an Erwin-Schr¨odinger fellowship of the Austrian Science Foundation (FWF), project J2330-N08

This poster is based upon the work

D. Grumiller, Long time black hole evaporation with bounded Hawking flux, JCAP 0405 (2004) 005; gr-qc/0307005.

I. Physical input

• Consider for simplicity only spherically symmetric black holes (BHs) without charge (Schwarzschild)

• Assume that formation process has settled down

• Assume that asymptotic flat region exists during the whole evaporation process

• Assume the existence of matter degrees of freedom without introducing matter explicitly (to support the Hawking quanta)

• Assume asymptotic matter flux induced by evaporation of the BH to be bounded during the whole evaporation process

• Assume that the gauge symmetries of gravity may be deformed due to quantum backreactions

IV. Physical output

• For a long time no essential difference to semi-classical result; thus also the life time is approximately as predicted semi-classically, t ≈ M

• Specific heat remains negative, but vanishes at the end point of evaporation (attractor)

• Violates Hawking’s area theorem

• In addition to the “thunderbolt” there may be a cold remnant. The total energy of remnant and “thunderbolt” is M

/2

0.5 1 1.5 2 2.5 3 3.5

2 4 6 8 10 12 14

r

M

Apparent horizon as a function of the Bondi mass

Without Planck scale contributions the curve would be flat (r

= 2M

)

II. Mathematical realization

• Reformulate Schwarzschild Black Hole in D=4 as dilaton gravity in D=2 in first order formulation as a specific Poisson-σ model (PSM) with Pois- son tensor P

depending on 3 target space coordinates X

,

S

=

Z

M₂

dX

∧ A

+ 1

2 P

A

∧ A

,

where the gauge field 1-forms A

comprise connection and Zweibeine.

• Asymptotic flatness restricts to the class of “Minkowski ground state models”

• Simplify technical issues by restricting to a 3-parameter family of models (generalizations are possible but do not change the main conclusions)

• Relabel the retarded time such that the asymptotic matter flux is not only bounded but constant (“isothermality”)

• Restrict the allowed deformations of the gauge symmetries to consistent ones in the sense of Barnich and Henneaux and exploit Izawas result that most general consistent deformation of 2D BF theory is a PSM

III. Main results

• Isothermality implies evolution equation for deformation parameters

• Dynamical study yields a unique attractor solution: Minkowski space

• Indications for the emission of a final “thunderbolt” with total energy of M

/2

• Indications for a first order phase transition from a linear to a constant dilaton vacuum: “Evaporation of the celestial 2-sphere”

• Phase transition consistent with results from Aichelburg-Sexl boost

• Phase transition has nice interpretation on associated Poisson manifold: transition from a region of rank 2 of P

before the “thunderbolt” to a region of rank 0 afterwards

• Phase transition conjectured by semi-classical considerations by Zaslavskii

• Straightforward generalizations: Reissner-Nordstr¨om, dS, AdS

• Conceivable generalizations: higherdimensional scalar-tensor theories (e.g. in D=4)

Possible implications for the information paradox

Explanation of the four regions

Region IV Attractor solution with constant dilaton

Here the attractor solution is encountered: the metric is Minkowski space and the dilaton is constant. At retarded time u = u_f the “thunderbolt” is emitted and the first order phase transition to the constant dilaton vacuum takes place.

Region III Dynamical deformation with constant flux

By assumption at retarded time u = 0 the evaporation process due to the Hawk- ing effect commences. Dynamical deformations as described above determine an evolution towards the attractor solution and a decreasing apparent horizon.

Region II Schwarzschild before evaporation

In that region classical fluxes from the formation process are relevant. They are neglected in this study.

Region I Minkowski space before formation

For simplicity it is assumed that a shock-wave creates the BH at advanced time v = v₀.

i ₀

i ^- i ⁺

ℑ ^- ℑ ⁺

u=0 u=u _f

v=v ₀ I

II III

IV

Sketchy Carter-Penrose diagram

Pathological features?

The information paradox manifests itself typically in a pathological feature of the Carter-Penrose diagrams related to BH evaporation. Thus, it is worthwhile to study the present Carter-Penrose diagram and to check what, if any, pathological features are encoded there. To this end the consideration of light-like test particles is useful.

• Particles starting at I ⁻ close to i⁻: Scatter off to I ⁺ in region II—

nothing remarkable happens here

• Particles starting at I ⁻ close to i⁰: Scatter off to I ⁺ in region IV—

except for passing the “thunderbolt” nothing remarkable happens here

• Particles starting at v > v₀ (not too close to i⁰): Pass beyond horizon from region II to III and after an encounter with the “thunderbolt” they scatter off to I ⁺ in region IV

• Particles starting at v < v₀ (not too close to i⁻): Finally, something remarkable happens: after a reflection in region I they enter region III by crossing the ingoing matter shock wave; then they follow a curve similar to the apparent horizon, cross the “thunderbolt” and are reflected again in region IV and finally scatter off to I ⁺ in region IV

Thus, the following situation is encountered:

•

No mixing of “hard” and “soft” matter

No test-particle can ever be scattered off to I ⁺ in region III, which is precisely the region dominated by “hard matter” responsible for the quantum backreac- tions deforming the gauge symmetries

•

Inversion of causality

For the last scenario, the later a particle is emitted at I ⁻ the earlier it arrives at I ⁺. Consequently, there is a part of I ⁺ in region IV where test-particles may arrive in two ways.

Cosmological implications

[1] A. M. Green and A. R. Liddle, “Constraints on the density perturbation spec- trum from primordial black holes,” Phys. Rev. D56(1997) 6166–6174.

[2] S. M. Carroll, “Quintessence and the rest of the world,” Phys. Rev. Lett. 81 (1998) 3067–3070; I. Zlatev, L.-M. Wang, and P. J. Steinhardt, “Quintessence, cosmic coincidence, and Λ,” Phys. Rev. Lett. 82 (1999) 896–899.

Semi-classical aspects

[3] O. B. Zaslavsky, “Two-dimensional self-consistent quantum-corrected ge- ometries with a constant dilaton field,” Phys. Lett. B424 (1998) 271, hep-th/9802117.

[4] W. Kummer and D. V. Vassilevich, “Hawking radiation from dilaton gravity in (1+1) dimensions: A pedagogical review,”Annalen Phys. 8 (1999) 801–827.

2D dilaton gravity

[5] D. Grumiller, W. Kummer, and D. V. Vassilevich, “Dilaton gravity in two dimensions,” Phys. Rept. 369 (2002) 327–429,hep-th/0204253.

[6] H. Balasin and D. Grumiller, “The ultrarelativistic limit of 2D dilaton gravity and its energy momentum tensor,” Class. Quant. Grav. 21 (2004) 2859–2872, gr-qc/0312086.

Deformations

[7] K. I. Izawa, “On nonlinear gauge theory from a deformation theory perspec- tive,” Prog. Theor. Phys. 103 (2000) 225–228,hep-th/9910133.

[8] G. Barnich and M. Henneaux, “Consistent couplings between fields with a gauge freedom and deformations of the master equation,” Phys. Lett. B311 (1993) 123–129,hep-th/9304057.

PSMs

[9] P. Schaller and T. Strobl, “Poisson structure induced (topological) field theo- ries,” Mod. Phys. Lett. A9 (1994) 3129–3136, hep-th/9405110; “Poisson sigma models: A generalization of 2-d gravity Yang-Mills systems,” hep-th/9411163.

[10] N. Ikeda, “Two-dimensional gravity and nonlinear gauge theory,” Ann. Phys.

235 (1994) 435–464, hep-th/9312059.