Soft Heisenberg Hair

Transcription

1 Soft Heisenberg Hair Daniel Grumiller Institute for Theoretical Physics TU Wien 13 th ICGAC & 15 th KIRA, July 2017 ICGAC = International Conference on Gravitation, Astrophysics, and Cosmology KIRA = Korea-Italy Relativistic Astrophysics Symposium papers: , , , , , ,

2 Daniel Grumiller Soft Heisenberg Hair 2/25

3 Two simple punchlines 1. Heisenberg algebra [X n, P m ] = i δ n, m fundamental not only in quantum mechanics but also in near horizon physics of gravity theories Daniel Grumiller Soft Heisenberg Hair 3/25

4 Two simple punchlines 1. Heisenberg algebra [X n, P m ] = i δ n, m fundamental not only in quantum mechanics but also in near horizon physics of gravity theories 2. Black hole microstates identified as specific soft hair descendants at least in three spacetime dimensions Daniel Grumiller Soft Heisenberg Hair 3/25

5 Two simple punchlines 1. Heisenberg algebra [X n, P m ] = i δ n, m fundamental not only in quantum mechanics but also in near horizon physics of gravity theories 2. Black hole microstates identified as specific soft hair descendants at least in three spacetime dimensions based on work with Hamid Afshar, Shahin Sheikh-Jabbari [IPM Teheran] Martin Ammon [U. Jena] Stephane Detournay, Max Riegler [ULB] Wout Merbis, Stefan Prohazka, Raphaela Wutte [TU Wien] Alfredo Perez, David Tempo, Ricardo Troncoso [CECS Valdivia] Hossein Yavartanoo [ITP Beijing] Daniel Grumiller Soft Heisenberg Hair 3/25

6 Outline Motivation Problems (and possible resolutions) Near horizon boundary conditions and soft hair Proposal for semi-classical BTZ microstates Outlook Daniel Grumiller Soft Heisenberg Hair 4/25

7 Outline Motivation Problems (and possible resolutions) Near horizon boundary conditions and soft hair Proposal for semi-classical BTZ microstates Outlook Daniel Grumiller Soft Heisenberg Hair Motivation 5/25

8 Motivation 1: universality of black hole entropy Bekenstein Hawking S BH = A 4G Daniel Grumiller Soft Heisenberg Hair Motivation 6/25

9 Motivation 1: universality of black hole entropy Bekenstein Hawking plus semi-classical corrections S = S BH q ln S BH + O(1) S BH = A 4G q = number depending on matter currently template for experimental results in quantum gravity Daniel Grumiller Soft Heisenberg Hair Motivation 6/25

10 Motivation 1: universality of black hole entropy Bekenstein Hawking plus semi-classical corrections S = S BH q ln S BH + O(1) S BH = A 4G q = number depending on matter currently template for experimental results in quantum gravity Believing in (semi-)classical Einstein gravity result above universal Daniel Grumiller Soft Heisenberg Hair Motivation 6/25

11 Motivation 1: universality of black hole entropy Bekenstein Hawking plus semi-classical corrections S = S BH q ln S BH + O(1) S BH = A 4G q = number depending on matter currently template for experimental results in quantum gravity Believing in (semi-)classical Einstein gravity result above universal Any purported quantum theory of gravity must reproduce results for S [at least any theory of quantum gravity claiming to reproduce (semi-)classical Einstein gravity in limit of small Newton constant] Daniel Grumiller Soft Heisenberg Hair Motivation 6/25

12 Motivation 1: universality of black hole entropy Bekenstein Hawking plus semi-classical corrections S = S BH q ln S BH + O(1) S BH = A 4G q = number depending on matter currently template for experimental results in quantum gravity Believing in (semi-)classical Einstein gravity result above universal Any purported quantum theory of gravity must reproduce results for S Examples collected e.g. in Sen 12 Daniel Grumiller Soft Heisenberg Hair Motivation 6/25

13 Motivation 1: universality of black hole entropy Bekenstein Hawking plus semi-classical corrections S = S BH q ln S BH + O(1) S BH = A 4G q = number depending on matter currently template for experimental results in quantum gravity Believing in (semi-)classical Einstein gravity result above universal Any purported quantum theory of gravity must reproduce results for S Examples collected e.g. in Sen 12 Perhaps no need for full knowledge of quantum gravity to account microscopically for black hole entropy (of sufficiently large black holes) Daniel Grumiller Soft Heisenberg Hair Motivation 6/25

14 Motivation 2: microscopic counting of generic black hole entropy Idea: count microstates from symmetries of dual field theory Daniel Grumiller Soft Heisenberg Hair Motivation 7/25

15 Motivation 2: microscopic counting of generic black hole entropy Idea: count microstates from symmetries of dual field theory For black holes with AdS 3 factor: microstate counting from CFT 2 symmetries (Strominger, Carlip,...) using Cardy formula ( S Cardy = 2π c + /6 + ) c /6 = A 4G = S BH c: left/right central charges of CFT 2 ± : left/right energies of state whose entropy is counted Daniel Grumiller Soft Heisenberg Hair Motivation 7/25

16 Motivation 2: microscopic counting of generic black hole entropy Idea: count microstates from symmetries of dual field theory For black holes with AdS 3 factor: microstate counting from CFT 2 symmetries (Strominger, Carlip,...) using Cardy formula ( S Cardy = 2π c + /6 + ) c /6 = A 4G = S BH Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT, warped CFT,...) warped CFT: Detournay, Hartman, Hofman 12 Galilean CFT: Bagchi, Detournay, Fareghbal, Simon 13; Barnich 13 Daniel Grumiller Soft Heisenberg Hair Motivation 7/25

17 Motivation 2: microscopic counting of generic black hole entropy Idea: count microstates from symmetries of dual field theory For black holes with AdS 3 factor: microstate counting from CFT 2 symmetries (Strominger, Carlip,...) using Cardy formula ( S Cardy = 2π c + /6 + ) c /6 = A 4G = S BH Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT, warped CFT,...) Microstate countings so far: mostly for (near-)extremal black holes (infinite throat geometries), e.g. Kerr/CFT Kerr/CFT: Guica, Hartman, Song, Strominger 09; Compere 12 Daniel Grumiller Soft Heisenberg Hair Motivation 7/25

18 Motivation 2: microscopic counting of generic black hole entropy Idea: count microstates from symmetries of dual field theory For black holes with AdS 3 factor: microstate counting from CFT 2 symmetries (Strominger, Carlip,...) using Cardy formula ( S Cardy = 2π c + /6 + ) c /6 = A 4G = S BH Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT, warped CFT,...) Microstate countings so far: mostly for (near-)extremal black holes (infinite throat geometries), e.g. Kerr/CFT Main idea of this talk: consider near horizon symmetries for non-extremal horizons Daniel Grumiller Soft Heisenberg Hair Motivation 7/25

19 Motivation 2: microscopic counting of generic black hole entropy Idea: count microstates from symmetries of dual field theory For black holes with AdS 3 factor: microstate counting from CFT 2 symmetries (Strominger, Carlip,...) using Cardy formula ( S Cardy = 2π c + /6 + ) c /6 = A 4G = S BH Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT, warped CFT,...) Microstate countings so far: mostly for (near-)extremal black holes (infinite throat geometries), e.g. Kerr/CFT Main idea of this talk: consider near horizon symmetries for non-extremal horizons Hope: near horizon symmetries allow for Cardyology Daniel Grumiller Soft Heisenberg Hair Motivation 7/25

20 Motivation 3: semi-classical microstates for generic black holes additional motivation: information loss Besides counting microstates one would like to construct them explicitly Daniel Grumiller Soft Heisenberg Hair Motivation 8/25

21 Motivation 3: semi-classical microstates for generic black holes additional motivation: information loss Besides counting microstates one would like to construct them explicitly if complete set of microstates known: may conclude that black holes behave just like any other thermodynamical system Daniel Grumiller Soft Heisenberg Hair Motivation 8/25

22 Motivation 3: semi-classical microstates for generic black holes additional motivation: information loss Besides counting microstates one would like to construct them explicitly if complete set of microstates known: may conclude that black holes behave just like any other thermodynamical system information loss: for all practical purposes, but not in principle Daniel Grumiller Soft Heisenberg Hair Motivation 8/25

23 Motivation 3: semi-classical microstates for generic black holes additional motivation: information loss Besides counting microstates one would like to construct them explicitly if complete set of microstates known: may conclude that black holes behave just like any other thermodynamical system information loss: for all practical purposes, but not in principle explicit constructions in string theory for (near-)extremal black holes Daniel Grumiller Soft Heisenberg Hair Motivation 8/25

24 Motivation 3: semi-classical microstates for generic black holes additional motivation: information loss Besides counting microstates one would like to construct them explicitly if complete set of microstates known: may conclude that black holes behave just like any other thermodynamical system information loss: for all practical purposes, but not in principle explicit constructions in string theory for (near-)extremal black holes in constructions so far need lot of input of UV completion Daniel Grumiller Soft Heisenberg Hair Motivation 8/25

25 Motivation 3: semi-classical microstates for generic black holes additional motivation: information loss Besides counting microstates one would like to construct them explicitly if complete set of microstates known: may conclude that black holes behave just like any other thermodynamical system information loss: for all practical purposes, but not in principle explicit constructions in string theory for (near-)extremal black holes in constructions so far need lot of input of UV completion string theory constructions so far agree with semi-classical result for entropy but fail to address its universality Daniel Grumiller Soft Heisenberg Hair Motivation 8/25

26 Motivation 3: semi-classical microstates for generic black holes additional motivation: information loss Besides counting microstates one would like to construct them explicitly if complete set of microstates known: may conclude that black holes behave just like any other thermodynamical system information loss: for all practical purposes, but not in principle explicit constructions in string theory for (near-)extremal black holes in constructions so far need lot of input of UV completion string theory constructions so far agree with semi-classical result for entropy but fail to address its universality Perhaps no need for full knowledge of quantum gravity to construct microstates (of sufficiently large non-extremal black holes) [at least for some observer, not necessarily an asymptotic one] Daniel Grumiller Soft Heisenberg Hair Motivation 8/25

27 Synthesis of the three motivations: soft hair Soft hair := zero energy excitations with non-trivial boundary charges Daniel Grumiller Soft Heisenberg Hair Motivation 9/25

28 Synthesis of the three motivations: soft hair Soft hair := zero energy excitations with non-trivial boundary charges Notion/name soft hair : Hawking, Perry, Strominger 16 Daniel Grumiller Soft Heisenberg Hair Motivation 9/25

29 Synthesis of the three motivations: soft hair Soft hair := zero energy excitations with non-trivial boundary charges Notion/name soft hair : Hawking, Perry, Strominger 16 Name motivated by Wheeler s folklore black holes have no hair Daniel Grumiller Soft Heisenberg Hair Motivation 9/25

30 Synthesis of the three motivations: soft hair Soft hair := zero energy excitations with non-trivial boundary charges Notion/name soft hair : Hawking, Perry, Strominger 16 Name motivated by Wheeler s folklore black holes have no hair General relativity with (asymptotic) boundaries: (locally) diffeomorphic geometries may be physically inequivalent Famous example: BTZ black hole is locally AdS 3, but canonical boundary charges (e.g. mass, angular momentum) differ Bañados, Henneaux, Teitelboim, Zanelli 93 Daniel Grumiller Soft Heisenberg Hair Motivation 9/25

31 Synthesis of the three motivations: soft hair Soft hair := zero energy excitations with non-trivial boundary charges Notion/name soft hair : Hawking, Perry, Strominger 16 Name motivated by Wheeler s folklore black holes have no hair General relativity with (asymptotic) boundaries: (locally) diffeomorphic geometries may be physically inequivalent Near horizon symmetry algebras (see below) realize soft hair idea Donnay, Giribet, Gonzalez, Pino 16 Afshar, Detournay, Grumiller, Merbis, Perez, Tempo, Troncoso 16 Daniel Grumiller Soft Heisenberg Hair Motivation 9/25

32 Synthesis of the three motivations: soft hair Soft hair := zero energy excitations with non-trivial boundary charges Notion/name soft hair : Hawking, Perry, Strominger 16 Name motivated by Wheeler s folklore black holes have no hair General relativity with (asymptotic) boundaries: (locally) diffeomorphic geometries may be physically inequivalent Near horizon symmetry algebras (see below) realize soft hair idea Soft hair is semi-classical concept Daniel Grumiller Soft Heisenberg Hair Motivation 9/25

33 Synthesis of the three motivations: soft hair Soft hair := zero energy excitations with non-trivial boundary charges Notion/name soft hair : Hawking, Perry, Strominger 16 Name motivated by Wheeler s folklore black holes have no hair General relativity with (asymptotic) boundaries: (locally) diffeomorphic geometries may be physically inequivalent Near horizon symmetry algebras (see below) realize soft hair idea Soft hair is semi-classical concept Soft hairy black holes: same energy as black holes but distinguished through their soft hairy charges Daniel Grumiller Soft Heisenberg Hair Motivation 9/25

34 Synthesis of the three motivations: soft hair Soft hair := zero energy excitations with non-trivial boundary charges Notion/name soft hair : Hawking, Perry, Strominger 16 Name motivated by Wheeler s folklore black holes have no hair General relativity with (asymptotic) boundaries: (locally) diffeomorphic geometries may be physically inequivalent Near horizon symmetry algebras (see below) realize soft hair idea Soft hair is semi-classical concept Soft hairy black holes: same energy as black holes but distinguished through their soft hairy charges Hope: soft hair could address black hole entropy puzzles and microstates in a semi-classical framework Daniel Grumiller Soft Heisenberg Hair Motivation 9/25

35 Outline Motivation Problems (and possible resolutions) Near horizon boundary conditions and soft hair Proposal for semi-classical BTZ microstates Outlook Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 10/25

36 Problem 1: TMI Note: this problem may be obvious even to laypersons Suppose we buy suggestion by Hawking 15 that soft hair may resolve information loss problem Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 11/25

37 Problem 1: TMI Note: this problem may be obvious even to laypersons Suppose we buy suggestion by Hawking 15 that soft hair may resolve information loss problem In particular, assume soft hair responsible for different microstates Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 11/25

38 Problem 1: TMI Note: this problem may be obvious even to laypersons Suppose we buy suggestion by Hawking 15 that soft hair may resolve information loss problem In particular, assume soft hair responsible for different microstates Problem: naively get infinite soft hair degeneracy, thus infinite entropy Too Much Information! Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 11/25

39 Problem 1: TMI Note: this problem may be obvious even to laypersons Suppose we buy suggestion by Hawking 15 that soft hair may resolve information loss problem In particular, assume soft hair responsible for different microstates Problem: naively get infinite soft hair degeneracy, thus infinite entropy Too Much Information! Possible resolution: provide cut-off on soft hair spectrum Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 11/25

40 Problem 1: TMI Note: this problem may be obvious even to laypersons Suppose we buy suggestion by Hawking 15 that soft hair may resolve information loss problem In particular, assume soft hair responsible for different microstates Problem: naively get infinite soft hair degeneracy, thus infinite entropy Too Much Information! Possible resolution: provide cut-off on soft hair spectrum Problem: if cut-off imposed in ad-hoc way can get any result for entropy Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 11/25

41 Problem 1: TMI Note: this problem may be obvious even to laypersons Suppose we buy suggestion by Hawking 15 that soft hair may resolve information loss problem In particular, assume soft hair responsible for different microstates Problem: naively get infinite soft hair degeneracy, thus infinite entropy Too Much Information! Possible resolution: provide cut-off on soft hair spectrum Problem: if cut-off imposed in ad-hoc way can get any result for entropy Possible resolution: provide cut-off on soft hair spectrum in a controlled and unique way Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 11/25

42 Problem 1: TMI Note: this problem may be obvious even to laypersons Suppose we buy suggestion by Hawking 15 that soft hair may resolve information loss problem In particular, assume soft hair responsible for different microstates Problem: naively get infinite soft hair degeneracy, thus infinite entropy Too Much Information! Possible resolution: provide cut-off on soft hair spectrum Problem: if cut-off imposed in ad-hoc way can get any result for entropy Possible resolution: provide cut-off on soft hair spectrum in a controlled and unique way Problem: how? Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 11/25

43 Problem 2: TLI Shaving off soft hair: Mirbabayi, Porrati 16; Bousso, Porrati 17; Donnelly, Giddings 17 Factorization theorems of S-matrix for infrared divergences Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 12/25

44 Problem 2: TLI Shaving off soft hair: Mirbabayi, Porrati 16; Bousso, Porrati 17; Donnelly, Giddings 17 Factorization theorems of S-matrix for infrared divergences Appropriately dress hard in- and out-states Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 12/25

45 Problem 2: TLI Shaving off soft hair: Mirbabayi, Porrati 16; Bousso, Porrati 17; Donnelly, Giddings 17 Factorization theorems of S-matrix for infrared divergences Appropriately dress hard in- and out-states Conclusion: soft-quanta part of S-matrix essentially trivial Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 12/25

46 Problem 2: TLI Shaving off soft hair: Mirbabayi, Porrati 16; Bousso, Porrati 17; Donnelly, Giddings 17 Factorization theorems of S-matrix for infrared divergences Appropriately dress hard in- and out-states Conclusion: soft-quanta part of S-matrix essentially trivial Information paradox formulated in terms of dressed hard states Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 12/25

47 Problem 2: TLI Shaving off soft hair: Mirbabayi, Porrati 16; Bousso, Porrati 17; Donnelly, Giddings 17 Factorization theorems of S-matrix for infrared divergences Appropriately dress hard in- and out-states Conclusion: soft-quanta part of S-matrix essentially trivial Information paradox formulated in terms of dressed hard states No dependence on soft quanta Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 12/25

48 Problem 2: TLI Shaving off soft hair: Mirbabayi, Porrati 16; Bousso, Porrati 17; Donnelly, Giddings 17 Factorization theorems of S-matrix for infrared divergences Appropriately dress hard in- and out-states Conclusion: soft-quanta part of S-matrix essentially trivial Information paradox formulated in terms of dressed hard states No dependence on soft quanta Problem: for asymptotic observer Too Little Information (namely none) from soft hair states Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 12/25

49 Problem 2: TLI Shaving off soft hair: Mirbabayi, Porrati 16; Bousso, Porrati 17; Donnelly, Giddings 17 Factorization theorems of S-matrix for infrared divergences Appropriately dress hard in- and out-states Conclusion: soft-quanta part of S-matrix essentially trivial Information paradox formulated in terms of dressed hard states No dependence on soft quanta Problem: for asymptotic observer Too Little Information (namely none) from soft hair states Possible resolution: do not consider asymptotic observer Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 12/25

50 Problem 2: TLI Shaving off soft hair: Mirbabayi, Porrati 16; Bousso, Porrati 17; Donnelly, Giddings 17 Factorization theorems of S-matrix for infrared divergences Appropriately dress hard in- and out-states Conclusion: soft-quanta part of S-matrix essentially trivial Information paradox formulated in terms of dressed hard states No dependence on soft quanta Problem: for asymptotic observer Too Little Information (namely none) from soft hair states Possible resolution: do not consider asymptotic observer Problem: how? Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 12/25

51 Starting point Resolving the how -questions easier in simpler models Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 13/25

52 Starting point Resolving the how -questions easier in simpler models Consider as toy model Einstein gravity in three dimensions with negative cc Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 13/25

53 Starting point Resolving the how -questions easier in simpler models Consider as toy model Einstein gravity in three dimensions with negative cc Same conceptual problems as in higher dimension, but technically more manageable Daniel Grumiller Soft Heisenberg Hair Problems (and possible resolutions) 13/25

54 Outline Motivation Problems (and possible resolutions) Near horizon boundary conditions and soft hair Proposal for semi-classical BTZ microstates Outlook Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 14/25

55 Properties of Einstein gravity in 2+1 dimensions with negative cc (AdS 3 ) Second order bulk action: I EH = 1 16πG d 3 x g (R + 2 ) l 2 G: Newton constant in 2+1 dimensions; l: AdS radius Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 15/25

56 Properties of Einstein gravity in 2+1 dimensions with negative cc (AdS 3 ) Second order bulk action: I EH = 1 16πG d 3 x g (R + 2 ) l 2 G: Newton constant in 2+1 dimensions; l: AdS radius No local physical degrees of freedom (dof) Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 15/25

57 Properties of Einstein gravity in 2+1 dimensions with negative cc (AdS 3 ) Second order bulk action: I EH = 1 16πG d 3 x g (R + 2 ) l 2 G: Newton constant in 2+1 dimensions; l: AdS radius No local physical degrees of freedom (dof) Depending on boundary conditions (bc s): boundary physical dof Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 15/25

58 Properties of Einstein gravity in 2+1 dimensions with negative cc (AdS 3 ) Second order bulk action: I EH = 1 16πG d 3 x g (R + 2 ) l 2 G: Newton constant in 2+1 dimensions; l: AdS radius No local physical degrees of freedom (dof) Depending on boundary conditions (bc s): boundary physical dof Brown Henneaux bc s: physical phase space of some CFT 2 Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 15/25

59 Properties of Einstein gravity in 2+1 dimensions with negative cc (AdS 3 ) Second order bulk action: I EH = 1 16πG d 3 x g (R + 2 ) l 2 G: Newton constant in 2+1 dimensions; l: AdS radius No local physical degrees of freedom (dof) Depending on boundary conditions (bc s): boundary physical dof Brown Henneaux bc s: physical phase space of some CFT 2 Brown Henneaux central charge of AdS 3 /CFT 2 : c = 3l/(2G) Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 15/25

60 Properties of Einstein gravity in 2+1 dimensions with negative cc (AdS 3 ) Second order bulk action: I EH = 1 16πG d 3 x g (R + 2 ) l 2 G: Newton constant in 2+1 dimensions; l: AdS radius No local physical degrees of freedom (dof) Depending on boundary conditions (bc s): boundary physical dof Brown Henneaux bc s: physical phase space of some CFT 2 Brown Henneaux central charge of AdS 3 /CFT 2 : c = 3l/(2G) Spectrum of physical states includes BTZ black holes ds 2 = (r2 r+)(r 2 2 r ) 2 r 2 l 2 dt 2 r 2 l 2 dr 2 ( + (r 2 r+ 2 )(r2 r 2 dϕ r +r )+r2 lr 2 ) 2 dt Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 15/25

61 Properties of Einstein gravity in 2+1 dimensions with negative cc (AdS 3 ) Second order bulk action: I EH = 1 16πG d 3 x g (R + 2 ) l 2 G: Newton constant in 2+1 dimensions; l: AdS radius No local physical degrees of freedom (dof) Depending on boundary conditions (bc s): boundary physical dof Brown Henneaux bc s: physical phase space of some CFT 2 Brown Henneaux central charge of AdS 3 /CFT 2 : c = 3l/(2G) Spectrum of physical states includes BTZ black holes ds 2 = (r2 r+)(r 2 2 r ) 2 r 2 l 2 dt 2 r 2 l 2 dr 2 ( + (r 2 r+ 2 )(r2 r 2 dϕ r +r )+r2 lr 2 BTZ BH entropy given by Bekenstein Hawking S BH = A 4G = 2πr + 4G ) 2 dt Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 15/25

62 Properties of Einstein gravity in 2+1 dimensions with negative cc (AdS 3 ) Second order bulk action: I EH = 1 16πG d 3 x g (R + 2 ) l 2 G: Newton constant in 2+1 dimensions; l: AdS radius No local physical degrees of freedom (dof) Depending on boundary conditions (bc s): boundary physical dof Brown Henneaux bc s: physical phase space of some CFT 2 Brown Henneaux central charge of AdS 3 /CFT 2 : c = 3l/(2G) Spectrum of physical states includes BTZ black holes ds 2 = (r2 r+)(r 2 2 r ) 2 r 2 l 2 dt 2 r 2 l 2 dr 2 ( + (r 2 r+ 2 )(r2 r 2 dϕ r +r )+r2 lr 2 BTZ BH entropy given by Bekenstein Hawking and Cardy formula S BH = A 4G = 2πr ( + 4G = 2π c + /6 + ) c /6 ± = (r + ± r ) 2 /(16lG) lm ± J (M: mass, J: angular momentum) Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 15/25 ) 2 dt

63 Near horizon boundary conditions See Afshar, Detournay, DG, Merbis, Perez, Tempo, Troncoso 16 for details Any non-extremal horizon is approximately Rindler near the horizon Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 16/25

64 Near horizon boundary conditions See Afshar, Detournay, DG, Merbis, Perez, Tempo, Troncoso 16 for details Any non-extremal horizon is approximately Rindler near the horizon Near horizon line-element with Rindler acceleration a: ds 2 = 2aρ dv dv dρ + γ 2 dϕ Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 16/25

65 Near horizon boundary conditions See Afshar, Detournay, DG, Merbis, Perez, Tempo, Troncoso 16 for details Any non-extremal horizon is approximately Rindler near the horizon Near horizon line-element with Rindler acceleration a: ds 2 = 2aρ dv dv dρ + γ 2 dϕ Meaning of coordinates: ρ: radial direction (ρ = 0 is horizon) ϕ ϕ + 2π: angular direction (horizon has S 1 topology) v: (advanced) time Rindler acceleration: vev (δa 0) or source (δa = 0)? Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 16/25

66 Near horizon boundary conditions See Afshar, Detournay, DG, Merbis, Perez, Tempo, Troncoso 16 for details Any non-extremal horizon is approximately Rindler near the horizon Near horizon line-element with Rindler acceleration a: ds 2 = 2aρ dv dv dρ + γ 2 dϕ Meaning of coordinates: ρ: radial direction (ρ = 0 is horizon) ϕ ϕ + 2π: angular direction (horizon has S 1 topology) v: (advanced) time Rindler acceleration: vev (δa 0) or source (δa = 0)? Both options possible (see Afshar, Detournay, DG, Oblak 16) Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 16/25

67 Near horizon boundary conditions See Afshar, Detournay, DG, Merbis, Perez, Tempo, Troncoso 16 for details Any non-extremal horizon is approximately Rindler near the horizon Near horizon line-element with Rindler acceleration a: ds 2 = 2aρ dv dv dρ + γ 2 dϕ Meaning of coordinates: ρ: radial direction (ρ = 0 is horizon) ϕ ϕ + 2π: angular direction (horizon has S 1 topology) v: (advanced) time Rindler acceleration: vev (δa 0) or source (δa = 0)? Both options possible (see Afshar, Detournay, DG, Oblak 16) Follow here suggestion by Donnay, Giribet, Gonzalez, Pino 15 δa = 0 a = source/state-inependent/chemical potential Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 16/25

68 Near horizon boundary conditions See Afshar, Detournay, DG, Merbis, Perez, Tempo, Troncoso 16 for details Any non-extremal horizon is approximately Rindler near the horizon Near horizon line-element with Rindler acceleration a: ds 2 = 2aρ dv dv dρ + γ 2 dϕ Meaning of coordinates: ρ: radial direction (ρ = 0 is horizon) ϕ ϕ + 2π: angular direction (horizon has S 1 topology) v: (advanced) time Rindler acceleration: vev (δa 0) or source (δa = 0)? Both options possible (see Afshar, Detournay, DG, Oblak 16) Follow here suggestion by Donnay, Giribet, Gonzalez, Pino 15 δa = 0 a = source/state-inependent/chemical potential Consequence: all states in theory have same (Unruh-)temperature T U = a 2π Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 16/25

69 Near horizon boundary conditions See Afshar, Detournay, DG, Merbis, Perez, Tempo, Troncoso 16 for details Any non-extremal horizon is approximately Rindler near the horizon Near horizon line-element with Rindler acceleration a: ds 2 = 2aρ dv dv dρ + γ 2 dϕ Meaning of coordinates: ρ: radial direction (ρ = 0 is horizon) ϕ ϕ + 2π: angular direction (horizon has S 1 topology) v: (advanced) time Rindler acceleration: vev (δa 0) or source (δa = 0)? Both options possible (see Afshar, Detournay, DG, Oblak 16) Follow here suggestion by Donnay, Giribet, Gonzalez, Pino 15 δa = 0 a = source/state-inependent/chemical potential Consequence: all states in theory have same (Unruh-)temperature T U = a 2π This is somewhat unusual, but convenient for our purposes! Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 16/25

70 Explicit form of our boundary conditions in metric formulation Note: everything much simpler in Chern Simons formulation! Boundary conditions as near horizon expansion of metric g tt = a 2 r 2 + O ( r 3) g ϕϕ = γ 2 + ( γ 2 l 2 ω 2) r 2 g tϕ = aωr 2 + O ( r 3) l 2 + O ( r 3) g rr = 1 + O ( r 2) g rt = O ( r 2) g rϕ = O ( r 2) Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 17/25

71 Explicit form of our boundary conditions in metric formulation Note: everything much simpler in Chern Simons formulation! Boundary conditions as near horizon expansion of metric g tt = a 2 r 2 + O ( r 3) g ϕϕ = γ 2 + ( γ 2 l 2 ω 2) r 2 g tϕ = aωr 2 + O ( r 3) l 2 + O ( r 3) g rr = 1 + O ( r 2) g rt = O ( r 2) g rϕ = O ( r 2) Boundary conditions as asymptotic expansion of metric g tt = 1 4 a2 r l2 a 2 + O ( ) 1 r g ϕϕ = ( γ 2 l 2 ω 2) r 2 4l ( 2 γ 2 + l 2 ω 2) + O ( ) 1 r g tϕ = 1 4 aωr2 1 2 aωl2 + O ( ) 1 r g rr = l2 r 2 + O( 1 r 3 ) g rt = O ( 1 r ) g rϕ = O ( 1 r ). Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 17/25

72 Explicit form of our boundary conditions in metric formulation Note: everything much simpler in Chern Simons formulation! Boundary conditions as near horizon expansion of metric g tt = a 2 r 2 + O ( r 3) g ϕϕ = γ 2 + ( γ 2 l 2 ω 2) r 2 g tϕ = aωr 2 + O ( r 3) l 2 + O ( r 3) g rr = 1 + O ( r 2) g rt = O ( r 2) g rϕ = O ( r 2) Boundary conditions in Chern Simons formulation A ± = b 1 ( ± d+a ± ) b ± with fixed sl 2 group element b ± = exp and 1-form (J ± = γ/l ± ω) ( ± r ) 2l (L 1 L 1 ) a ± = L 0 ( ±J ± dϕ a dt ) δj ± 0 δa = 0 Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 17/25

73 Consequences of our near horizon boundary conditions To reduce clutter consider henceforth constant Rindler acceleration, a = const. Two towers of canonical boundary charges J ± (ϕ) Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 18/25

74 Consequences of our near horizon boundary conditions To reduce clutter consider henceforth constant Rindler acceleration, a = const. Two towers of canonical boundary charges J ± (ϕ) Asymptotic symmetry algebra (ASA) generated by those charges [J ± n, J ± m] inδ n+m, 0 [J + n, J m] = 0 Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 18/25

75 Consequences of our near horizon boundary conditions To reduce clutter consider henceforth constant Rindler acceleration, a = const. Two towers of canonical boundary charges J ± (ϕ) Asymptotic symmetry algebra (ASA) generated by those charges [J ± n, J ± m] inδ n+m, 0 [J + n, J m] = 0 Two u(1) current algebras like free boson in 2d! Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 18/25

76 Consequences of our near horizon boundary conditions To reduce clutter consider henceforth constant Rindler acceleration, a = const. Two towers of canonical boundary charges J ± (ϕ) Asymptotic symmetry algebra (ASA) generated by those charges [J ± n, J ± m] inδ n+m, 0 [J + n, J m] = 0 Two u(1) current algebras like free boson in 2d! ASA isomorphic to infinite copies of Heisenberg algebras Map P 0 = J J 0 P n = i kn (J n + + J n ) if n 0 yields Heisenberg algebra (with Casimirs X 0, P 0 ) X n = J + n J n [X n, X m ]= [P n, P m ] = [X 0, P n ] = [P 0, X n ] = 0 [X n, P m ]= iδ n,m if n 0 Map explains word Heisenberg in title and provides first punchline Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 18/25

77 Consequences of our near horizon boundary conditions To reduce clutter consider henceforth constant Rindler acceleration, a = const. Two towers of canonical boundary charges J ± (ϕ) Asymptotic symmetry algebra (ASA) generated by those charges [J ± n, J ± m] inδ n+m, 0 [J + n, J m] = 0 Two u(1) current algebras like free boson in 2d! ASA isomorphic to infinite copies of Heisenberg algebras For real J 0 all states in theory regular and have horizon Whole spectrum (subject to reality) compatible with regularity! Could be used as defining property of our bc s Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 18/25

78 Consequences of our near horizon boundary conditions To reduce clutter consider henceforth constant Rindler acceleration, a = const. Two towers of canonical boundary charges J ± (ϕ) Asymptotic symmetry algebra (ASA) generated by those charges [J ± n, J ± m] inδ n+m, 0 [J + n, J m] = 0 Two u(1) current algebras like free boson in 2d! ASA isomorphic to infinite copies of Heisenberg algebras For real J 0 all states in theory regular and have horizon Near horizon Hamiltonian H J J 0 commutes with all J ± n Near horizon Hamiltonian defined as diffeo charge generated by unit translations v in (advanced) time direction Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 18/25

79 Consequences of our near horizon boundary conditions To reduce clutter consider henceforth constant Rindler acceleration, a = const. Two towers of canonical boundary charges J ± (ϕ) Asymptotic symmetry algebra (ASA) generated by those charges [J ± n, J ± m] inδ n+m, 0 [J + n, J m] = 0 Two u(1) current algebras like free boson in 2d! ASA isomorphic to infinite copies of Heisenberg algebras For real J 0 all states in theory regular and have horizon Near horizon Hamiltonian H J J 0 commutes with all J ± n Consequence: soft hair! H ψ = E ψ H ψ = E ψ where state ψ is state ψ dressed arbitrarily with soft hair ψ = ψ Explains word soft hair in title J + J n + n ± i n i i Z+ Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 18/25

80 Consequences of our near horizon boundary conditions To reduce clutter consider henceforth constant Rindler acceleration, a = const. Two towers of canonical boundary charges J ± (ϕ) Asymptotic symmetry algebra (ASA) generated by those charges [J ± n, J ± m] inδ n+m, 0 [J + n, J m] = 0 Two u(1) current algebras like free boson in 2d! ASA isomorphic to infinite copies of Heisenberg algebras For real J 0 all states in theory regular and have horizon Near horizon Hamiltonian H J J 0 commutes with all J n ± Consequence: soft hair! Entropy formula remarkably simple S = 2π ( J J 0 ) = T 1 H also remarkably universal: generalizes to flat space, higher spins, higher derivatives! Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 18/25

81 Consequences of our near horizon boundary conditions To reduce clutter consider henceforth constant Rindler acceleration, a = const. Two towers of canonical boundary charges J ± (ϕ) Asymptotic symmetry algebra (ASA) generated by those charges [J ± n, J ± m] inδ n+m, 0 [J + n, J m] = 0 Two u(1) current algebras like free boson in 2d! ASA isomorphic to infinite copies of Heisenberg algebras For real J 0 all states in theory regular and have horizon Near horizon Hamiltonian H J J 0 commutes with all J n ± Consequence: soft hair! Entropy formula remarkably simple S = 2π ( J J 0 ) = T 1 H Simple first law dh = T ds and trivial specific heat Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 18/25

82 Consequences of our near horizon boundary conditions To reduce clutter consider henceforth constant Rindler acceleration, a = const. Two towers of canonical boundary charges J ± (ϕ) Asymptotic symmetry algebra (ASA) generated by those charges [J ± n, J ± m] inδ n+m, 0 [J + n, J m] = 0 Two u(1) current algebras like free boson in 2d! ASA isomorphic to infinite copies of Heisenberg algebras For real J 0 all states in theory regular and have horizon Near horizon Hamiltonian H J J 0 commutes with all J n ± Consequence: soft hair! Entropy formula remarkably simple S = 2π ( J J 0 ) = T 1 H Simple first law dh = T ds and trivial specific heat Relations to asymptotic Virasoro charges L ± and sources µ ± L J 2 + J µ µj a Twisted Sugawra construction emerges! (yields Brown Henneaux c) Daniel Grumiller Soft Heisenberg Hair Near horizon boundary conditions and soft hair 18/25

83 Outline Motivation Problems (and possible resolutions) Near horizon boundary conditions and soft hair Proposal for semi-classical BTZ microstates Outlook Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 19/25

84 Assumptions For technical details see Afshar, DG, Sheikh-Jabbari, Yavartanoo Central charges quantized in integers Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 20/25

85 Assumptions For technical details see Afshar, DG, Sheikh-Jabbari, Yavartanoo Central charges quantized in integers Needed due to relations like J cn W 0 n Note non-local relation W e 2 J Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 20/25

86 Assumptions For technical details see Afshar, DG, Sheikh-Jabbari, Yavartanoo Central charges quantized in integers Needed due to relations like J cn W 0 n Justifiable e.g. through Chern Simons level quantization c = 6k Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 20/25

87 Assumptions For technical details see Afshar, DG, Sheikh-Jabbari, Yavartanoo Central charges quantized in integers Needed due to relations like J cn W 0 n Justifiable e.g. through Chern Simons level quantization c = 6k 2. Conical deficit ν (0, 1) quantized in integers over c Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 20/25

88 Assumptions For technical details see Afshar, DG, Sheikh-Jabbari, Yavartanoo Central charges quantized in integers Needed due to relations like J cn W 0 n Justifiable e.g. through Chern Simons level quantization c = 6k 2. Conical deficit ν (0, 1) quantized in integers over c Needed due to relations like J c(n+ν) W ν n Note twisted periodicity conditions W ν (ϕ + 2π) = e 2πνi W ν (ϕ) Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 20/25

89 Assumptions For technical details see Afshar, DG, Sheikh-Jabbari, Yavartanoo Central charges quantized in integers Needed due to relations like J cn W 0 n Justifiable e.g. through Chern Simons level quantization c = 6k 2. Conical deficit ν (0, 1) quantized in integers over c Needed due to relations like J c(n+ν) W ν n Justifiable through explicit stringy construction in D1-D5 system Maldacena, Maoz 00; Lunin, Maldacena, Maoz 02 Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 20/25

90 Assumptions For technical details see Afshar, DG, Sheikh-Jabbari, Yavartanoo Central charges quantized in integers Needed due to relations like J cn W 0 n Justifiable e.g. through Chern Simons level quantization c = 6k 2. Conical deficit ν (0, 1) quantized in integers over c Needed due to relations like J c(n+ν) W ν n Justifiable through explicit stringy construction in D1-D5 system 3. Black hole/particle correspondence Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 20/25

91 Assumptions For technical details see Afshar, DG, Sheikh-Jabbari, Yavartanoo Central charges quantized in integers Needed due to relations like J cn W 0 n Justifiable e.g. through Chern Simons level quantization c = 6k 2. Conical deficit ν (0, 1) quantized in integers over c Needed due to relations like J c(n+ν) W ν n Justifiable through explicit stringy construction in D1-D5 system 3. Black hole/particle correspondence Identify states in Hilbert space H BTZ as (composite) states in H CG J nc p J p J n p J p + incj n p p Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 20/25

92 Assumptions For technical details see Afshar, DG, Sheikh-Jabbari, Yavartanoo Central charges quantized in integers Needed due to relations like J cn W 0 n Justifiable e.g. through Chern Simons level quantization c = 6k 2. Conical deficit ν (0, 1) quantized in integers over c Needed due to relations like J c(n+ν) W ν n Justifiable through explicit stringy construction in D1-D5 system 3. Black hole/particle correspondence Identify states in Hilbert space H BTZ as (composite) states in H CG Justification 1: obtain Virasoro at central charge c in H BTZ and H CG Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 20/25

93 Assumptions For technical details see Afshar, DG, Sheikh-Jabbari, Yavartanoo Central charges quantized in integers Needed due to relations like J cn W 0 n Justifiable e.g. through Chern Simons level quantization c = 6k 2. Conical deficit ν (0, 1) quantized in integers over c Needed due to relations like J c(n+ν) W ν n Justifiable through explicit stringy construction in D1-D5 system 3. Black hole/particle correspondence Identify states in Hilbert space H BTZ as (composite) states in H CG Justification 1: obtain Virasoro at central charge c in H BTZ and H CG Justification 2: gives nice result Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 20/25

94 List of all semi-classical BTZ black hole microstates Given a BTZ black hole with mass M and angular momentum J (as measured by asymptotic observer) define parameters ± = 1 ( ) ( ) 2 lm ± J = c 6 J ± 2 0 Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 21/25

95 List of all semi-classical BTZ black hole microstates Given a BTZ black hole with mass M and angular momentum J (as measured by asymptotic observer) define parameters ± = 1 ( ) ( ) 2 lm ± J = c 6 J ± 2 0 Define sets of positive integers {n ± i } obeying n ± i = c ± Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 21/25

96 List of all semi-classical BTZ black hole microstates Given a BTZ black hole with mass M and angular momentum J (as measured by asymptotic observer) define parameters ± = 1 ( ) ( ) 2 lm ± J = c 6 J ± 2 0 Define sets of positive integers {n ± i } obeying n ± i = c ± Label BTZ black hole microstates as B({n ± i }); J ± 0 with sets of positive integers {n ± i } obeying constraint above Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 21/25

97 List of all semi-classical BTZ black hole microstates Given a BTZ black hole with mass M and angular momentum J (as measured by asymptotic observer) define parameters ± = 1 ( ) ( ) 2 lm ± J = c 6 J ± 2 0 Define sets of positive integers {n ± i } obeying n ± i = c ± Label BTZ black hole microstates as B({n ± i }); J ± 0 with sets of positive integers {n ± i } obeying constraint above Define vacuum state 0 by highest weight conditions J ± n 0 = 0 n 0 Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 21/25

98 List of all semi-classical BTZ black hole microstates Given a BTZ black hole with mass M and angular momentum J (as measured by asymptotic observer) define parameters ± = 1 ( ) ( ) 2 lm ± J = c 6 J ± 2 0 Define sets of positive integers {n ± i } obeying n ± i = c ± Label BTZ black hole microstates as B({n ± i }); J ± 0 with sets of positive integers {n ± i } obeying constraint above Define vacuum state 0 by highest weight conditions J ± n 0 = 0 n 0 Full set of semi-classical BTZ black hole microstates given by B({n ± i }); J 0 ± = ( ) J + 0 {n ± i } J n + i n i Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 21/25

99 BTZ black hole entropy from counting all semi-classical microstates We proposed (after some Bohr-type semi-classical quantization conditions) explicit set of BTZ black hole microstates Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 22/25

100 BTZ black hole entropy from counting all semi-classical microstates We proposed (after some Bohr-type semi-classical quantization conditions) explicit set of BTZ black hole microstates Now let us count these microstates Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 22/25

101 BTZ black hole entropy from counting all semi-classical microstates We proposed (after some Bohr-type semi-classical quantization conditions) explicit set of BTZ black hole microstates Now let us count these microstates Straightforward combinatorial problem: partition of integers p(c ± ) Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 22/25

102 BTZ black hole entropy from counting all semi-classical microstates We proposed (after some Bohr-type semi-classical quantization conditions) explicit set of BTZ black hole microstates Now let us count these microstates Straightforward combinatorial problem: partition of integers p(c ± ) Entropy given by Boltzmann s formula S = ln N = ln p(c + ) + ln p(c ) Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 22/25

103 BTZ black hole entropy from counting all semi-classical microstates We proposed (after some Bohr-type semi-classical quantization conditions) explicit set of BTZ black hole microstates Now let us count these microstates (not Cardyology but Hardyology ) Straightforward combinatorial problem: partition of integers p(c ± ) Entropy given by Boltzmann s formula S = ln N = ln p(c + ) + ln p(c ) Solved long ago by Hardy, Ramanujan; asymptotic formula (large N): ln p(n) = 2π N/6 ln N + O(1) Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 22/25

104 BTZ black hole entropy from counting all semi-classical microstates We proposed (after some Bohr-type semi-classical quantization conditions) explicit set of BTZ black hole microstates Now let us count these microstates (not Cardyology but Hardyology ) Straightforward combinatorial problem: partition of integers p(c ± ) Entropy given by Boltzmann s formula S = ln N = ln p(c + ) + ln p(c ) Solved long ago by Hardy, Ramanujan; asymptotic formula (large N): ln p(n) = 2π N/6 ln N + O(1) Our final result for semi-classical BTZ black hole entropy is S = 2π( c + /6 + ) c /6 ln(c + ) ln(c ) + O(1) Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 22/25

105 BTZ black hole entropy from counting all semi-classical microstates We proposed (after some Bohr-type semi-classical quantization conditions) explicit set of BTZ black hole microstates Now let us count these microstates (not Cardyology but Hardyology ) Straightforward combinatorial problem: partition of integers p(c ± ) Entropy given by Boltzmann s formula S = ln N = ln p(c + ) + ln p(c ) Solved long ago by Hardy, Ramanujan; asymptotic formula (large N): ln p(n) = 2π N/6 ln N + O(1) Our final result for semi-classical BTZ black hole entropy is S = 2π( c + /6 + ) c /6 ln(c + ) ln(c ) + O(1) Leading order coincides with Bekenstein Hawking/Cardy formula! Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 22/25

106 BTZ black hole entropy from counting all semi-classical microstates We proposed (after some Bohr-type semi-classical quantization conditions) explicit set of BTZ black hole microstates Now let us count these microstates (not Cardyology but Hardyology ) Straightforward combinatorial problem: partition of integers p(c ± ) Entropy given by Boltzmann s formula S = ln N = ln p(c + ) + ln p(c ) Solved long ago by Hardy, Ramanujan; asymptotic formula (large N): ln p(n) = 2π N/6 ln N + O(1) Our final result for semi-classical BTZ black hole entropy is S = 2π( c + /6 + ) c /6 ln(c + ) ln(c ) + O(1) Leading order coincides with Bekenstein Hawking/Cardy formula! Subleading log corrections also turn out to be correct! Daniel Grumiller Soft Heisenberg Hair Proposal for semi-classical BTZ microstates 22/25

107 Outline Motivation Problems (and possible resolutions) Near horizon boundary conditions and soft hair Proposal for semi-classical BTZ microstates Outlook Daniel Grumiller Soft Heisenberg Hair Outlook 23/25

108 Summary, loose ends and generalizations Summary: We proposed semi-classical set of BTZ black hole microstates Their counting reproduces Bekenstein Hawking entropy Also subleading log corrections to entropy are correct Daniel Grumiller Soft Heisenberg Hair Outlook 24/25

109 Summary, loose ends and generalizations Summary: We proposed semi-classical set of BTZ black hole microstates Their counting reproduces Bekenstein Hawking entropy Also subleading log corrections to entropy are correct Loose ends: Derivation of Bohr-type quantization conditions of c and ν? Daniel Grumiller Soft Heisenberg Hair Outlook 24/25

110 Summary, loose ends and generalizations Summary: We proposed semi-classical set of BTZ black hole microstates Their counting reproduces Bekenstein Hawking entropy Also subleading log corrections to entropy are correct Loose ends: Derivation of Bohr-type quantization conditions of c and ν? Derivation of black hole/particle correspondence? Daniel Grumiller Soft Heisenberg Hair Outlook 24/25

111 Summary, loose ends and generalizations Summary: We proposed semi-classical set of BTZ black hole microstates Their counting reproduces Bekenstein Hawking entropy Also subleading log corrections to entropy are correct Loose ends: Derivation of Bohr-type quantization conditions of c and ν? Derivation of black hole/particle correspondence? Near horizon field theory beyond semi-classical approximation? Daniel Grumiller Soft Heisenberg Hair Outlook 24/25

112 Summary, loose ends and generalizations Summary: We proposed semi-classical set of BTZ black hole microstates Their counting reproduces Bekenstein Hawking entropy Also subleading log corrections to entropy are correct Loose ends: Derivation of Bohr-type quantization conditions of c and ν? Derivation of black hole/particle correspondence? Near horizon field theory beyond semi-classical approximation? Generalizations: Semi-classical microstate construction for cosmological horizons? Daniel Grumiller Soft Heisenberg Hair Outlook 24/25

113 Summary, loose ends and generalizations Summary: We proposed semi-classical set of BTZ black hole microstates Their counting reproduces Bekenstein Hawking entropy Also subleading log corrections to entropy are correct Loose ends: Derivation of Bohr-type quantization conditions of c and ν? Derivation of black hole/particle correspondence? Near horizon field theory beyond semi-classical approximation? Generalizations: Semi-classical microstate construction for cosmological horizons? Soft resolution of information loss problem? Neglecting soft gravitons generates information loss Carney, Chaurette, Neuenfeld, Semenoff 17 Conjectured resolution of information loss problem: include soft gravitons Strominger 17 Daniel Grumiller Soft Heisenberg Hair Outlook 24/25

114 Summary, loose ends and generalizations Summary: We proposed semi-classical set of BTZ black hole microstates Their counting reproduces Bekenstein Hawking entropy Also subleading log corrections to entropy are correct Loose ends: Derivation of Bohr-type quantization conditions of c and ν? Derivation of black hole/particle correspondence? Near horizon field theory beyond semi-classical approximation? Generalizations: Semi-classical microstate construction for cosmological horizons? Soft resolution of information loss problem? Kerr? Daniel Grumiller Soft Heisenberg Hair Outlook 24/25

115 Thanks for your attention! Daniel Grumiller Soft Heisenberg Hair Outlook 25/25