Entropy of Quasiblack holes and entropy of black holes in membrane approach

Transcription

1 Entropy of Quasiblack holes and entropy of black holes in membrane approach José P. S. Lemos Centro Multidisciplinar de Astrofísica, CENTRA, Lisbon, Portugal Oleg B. Zaslavskii Department of Physics and Technology, Kharkov V.N. Karazin National University, Kharkov, Ukraine PRD , PLB 2011

2 Entropy of black hole versus entropy without horizon: membrane paradigme K. Thorne, Damour, etc. Lemos and O.Z., PRD 2011 Streched horizon, vacuum inside, shell Timelike boundary vs. lighlike. How S approaches BH A/4? Schwarzschild (Parikh, theses 1999)

3 00 Einstein equation, 3+1 splitting York 1986 (spherical symmetry) O. Z (static, without symmetry)

4 Internal boundary 1) or 2) 2) Physical noundary on inner side of shell S = S m + S mb I = dσβε S Surface gravity

5 Nonextremal case Black hole: S=A/4 or S=A/4 + S(matter) Without horizon: S(matter) Let R = r (1 + δ ) δ 0 + What happens to entropy? Why S(matter) approaches A/4? A/4 due to horizon, light-like surface. Now space-like Whether and how continuity achieved? Quasiblack hole ε 0 i N = ε f ( x ) Extremal case, puzzles. Matter system without horizon

6 Membrane: vacuum inside Shell near gravitational radius Without horizon: S(matter) Quasiblack hole Physical (nonvacuum) system on threshold of collapse QBH: i N = ε f ( x ) Everywhere inside ε 0 What happens to entropy? Why S(matter) approaches A/4? A/4 due to horizon, light-like surface. Now space-like Whether and how continuity achieved? Black hole: S=A/4 or S=A/4 + S(matter) Let R = r (1 + δ ) + Nonextremal case δ 0 Extremal case, puzzles. Matter system without horizon

7 A/4 due to horizon, light-like surface. Now space-like Whether and how continuity achieved? Answer expected to be model-independent Quasiblack hole System on threshold of horizon formation But horizon does not form True BH: N = f ( r), f ( r ) = 0 N Lapse function 0 QBH: i N = ε f ( x ) ε 0 but f > 0

8 QBH: definition and general properties Gravitational radius is approached by sequence of static configurations Horizon almost forms but does not form 1) Methodical tool: mass and entropy of black hole from system without horizon 2) Real physical systems (extremal case) Remote obsrver at infinity no difference from true BH Vicinity of quasihorizon crucial difference

9 Usual situation: size approaches gravitational radius, system collapses Special cases when gravitational radius is approached by sequence of static configurations Majumdar Papapetrou systems p = 0, Compact objects: Bonnor stars ρ = ρ e 18 Sphere of neutral hydrogen lost 10 of its electrons Self-gravitating magnetic monopole Threshold of formation of event horizon. Quasihorizon Massive charged extremal shells Different physical systems share common features: geometry of spacetime behavior of tidal forces

10 Lemos and E. Weinberg ds = Brdt () + Ardr () + r ( dθ + sin θφ d ) Lue and E. Weinberg 2000

11 (i) V(r) attains minimum at General approach ds = Brdt () + Ardr () + r ( dθ + sin θφ d ) r * 0 1 A = V V r ε * ( ) = = 1 (ii) ε 0 (iii) In limit ε 0 Consequences: (a) infinite redshift regular configuration * V ( r ) 0 Br () 0 * for all r r (b) infinite tidal forces for free-falling observer Limit ε 0 Singular (degenerate) or regular? Properties of spacetimes -?

12 Extremal RN outside Minkowski inside (shell) Outer metric Classical model of electron A. V. Vilenkin, P. I. Fomin m 2 m ds = 1 dt + 1 dr + rdω r r Inner metric 2 ds 1 m = dt + dr + rdω r r > r 0 r < r 0 Inside. Two alternatives 1) Using time t as good coordinate. Then g 00 0 in the entire region 0 r But Riemann tensor =0 there! r Degenerate behavior Surface = becomes light-like in limit r0 m + 0 r r 0

13 Redshift ω ( ) = ω A( r) A 0 in the whole inner region infinite redshift in quasi-horizon limit

14 2) Let us introduce inside the coordinate T : t = r 0 Tr 0 m Finite intervals of T infinite intervals of t ds = dt + dr + r dω Finite intervals of t vanishing intervals of T Let T be legitimate coordinate ( r m) dt 2 r 0 dt = r0 m Time-like surface. No matching between inside and outside Complementarity and mutual impenetrability No penetration from outside to inside because of infinite tidal forces (naked behavior) No penetration from inside to outside because of infinite redshift

15 Role of surface stresses: mass formula J. Lemos and O. Z. PRD 2008 Contribution of inner region 0 Inside: i N = ε f ( x ) ε 0

16 Surface stresses: nonextremal versus extremal 1) Nonextremal. Main contribution from the term α κ N κ surface gravity i N = ε f ( x ) ε 0 α diverges but α N κ 2) Extremal N const exp( Al) gives finite contribution to mass κ A h 4π α finite α N gives zero contribution to mass

17 Byproduct: Abraham-Lorentz classical electron in GR Original idea: there are no extrenal forces, no non-electromagnetic stresses, self-consistent solution Electromagnetic forces + gravitation (GR) In QBH approach requirement is weakened: In extremal case non-electromagnetic stresses are present but they do not contribute to the mass!

18 Kab extrinsic curvature

19 Dominant contribution: Role of surface stresses Integrate along subset of configurations near would-be horizon

20 Predecessors: F. Pretorius, D. Vollick, and W. Israel, Phys. Rev. D 1998 Spherical shell near the gravitational radius, Euler relation between pressure, entropy density and energy density. Model of three surfaces Now: model-independent approach, spherical symmetry is not required, Euler relation is in general invalid Concept of QBH

21 when δ 0 Otherwise backreaction is unbounded on horizon

22 How to integrate Let R = r (1 + δ ) δ << 1 + In space of parameters we move along this line By substitution into 1 st law, we obtain after integration S=A/4 Constant of integration =0 If the system shrinks, S vanishes.

23 Simplified example: shell in vacuum Minkowski inside, Schwarzschild outside A area E quasilocal Brown and York energy V rr = g = 1 r+ R p gravitational pressure

24 We consider vicinity of horizon only Let R = r (1 + δ ) δ << 1 + Term with p dominates 1 st law After integration the same BH entropy S=A/4 reobtained

25 Entropy in nonextremal case. General features (i) Systems without horizons Bridge between thermal matter configurations and black hole entropy (ii) Entropy comes from surface layer. Details of interior are irrelevant QBH deletes information (iii) Role of huge surface stresses. A/4 in model-independent way (iv) Role of surface layer quantum states on quasihorizon (v) Interplay between matter and space-time geometry

26 I =βe Extremal case Eucludean action approach S tot Free energy and Euclidean action S = S + S tot m bh S bh = A TH 4 T 0 1) Nonextremal case: T 0 =T H S bh = A 4 2) Extremal case T H = 0 S bh =0 Temperature arbitrary (Hawking, Horowitz, Ross 1995) Geometrical explanation: proper distance infinite

27 Classically, with quantum corrections neglected, picture self-consistent Quantum corrections change picture drastically (P. Anderson et al) Quantum stress-energy tensor: T ν µ = (...) T 4 H T 2 g finite Divergences destroy horizon Now = 0 T H Contradiction T 0 0 Another attempt: S = F T But T is fixed =0

28 Direct definition faces difficulties. QBH approach helps Extremal RN outside Lemos and OZ PLB 2011 No horizon, ordinary: system at each stage 1 st law T = 1 T 0 r r +

29 T = T 0 r+ 1 r E = R[ 1 V ( R ) ] E = r = e + p r( matter) 0 when R r + (property of QBH with matter)

30 Simplest example: extremal charged shell in vacuum Integrability condition: T is local temperature To compare with uncharged case T ds 0 = dm T0 temperature at infinity

31 Now, T0 0 but T loc Is finite This resolves the problem with quantum backreaction: ν µ 4 T = (...) T loc + finite

32 More general case Extremal case. Features of entropy. S is model-dependent two functions T, D Memory: dependence on prehistory encoded in the law how temperature at Infinity approaches zero: Unusual thermodynamics Agrees with Israel et al PRD 1996 for particular state

33 General remarks Considering systems which almost collapse but do not form BH (membrane, QBHs) lets us gain insight into properties of true black holes

34 Thank you!