Three dimensional black holes, microstates & the Heisenberg algebra

Transcription

1 Three dimensional black holes, microstates & the Heisenberg algebra Wout Merbis Institute for Theoretical Physics, TU Wien September 28, 2016 Based on [ ] and work to appear with Hamid Afshar, Stephane Detournay, Daniel Grumiller, Alfredo Perez, David Tempo & Ricardo Troncoso

2 Introduction Introduction: Black Hole information loss I + Black holes emit Hawking radiation (black body radiation) r = 0 I W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

3 Introduction Introduction: Black Hole information loss I + Black holes emit Hawking radiation (black body radiation) If a pure state is thrown into the black hole, it cannot be retrieved from the outgoing radiation = U r = 0 I W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

4 Introduction Introduction: Black Hole information loss I + Black holes emit Hawking radiation (black body radiation) If a pure state is thrown into the black hole, it cannot be retrieved from the outgoing radiation = U r = 0 I Assumptions Initial and final vacuum state is unique Black hole have (nearly) no hair W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

5 Introduction Introduction: Black Hole information loss I + Black holes emit Hawking radiation (black body radiation) If a pure state is thrown into the black hole, it cannot be retrieved from the outgoing radiation = U r = 0 I Assumptions Initial and final vacuum state is unique Black hole have (nearly) no hair [Hawking, Perry, Strominger, 16] W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

6 Introduction Introduction: Asymptotic Symmetries I + i + Conformal S 2 s Infinite dimensional symmetry group for asymptotically flat spacetimes (BMS) [Bondi, van der Burg, Metzner; Sachs, 62] i 0 I i W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

7 Introduction Introduction: Asymptotic Symmetries I + i + Conformal S 2 s i 0 Infinite dimensional symmetry group for asymptotically flat spacetimes (BMS) [Bondi, van der Burg, Metzner; Sachs, 62] Leads to infinitely many conservation laws (Weinberg s soft theorems) [Weinberg, 65] I i W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

8 Introduction Introduction: Asymptotic Symmetries I + i + Conformal S 2 s Infinite dimensional symmetry group for asymptotically flat spacetimes (BMS) [Bondi, van der Burg, Metzner; Sachs, 62] i 0 Leads to infinitely many conservation laws (Weinberg s soft theorems) [Weinberg, 65] I Infinitely many soft charges defined at asymptotic infinity [Strominger, 13] i W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

9 Introduction Introduction: Asymptotic Symmetries I + I i + Conformal S 2 s i 0 Infinite dimensional symmetry group for asymptotically flat spacetimes (BMS) [Bondi, van der Burg, Metzner; Sachs, 62] Leads to infinitely many conservation laws (Weinberg s soft theorems) [Weinberg, 65] Infinitely many soft charges defined at asymptotic infinity [Strominger, 13] i Black holes may carry these soft charges (i.e. soft hair) Consistency of BH evaporation with the conservation of these symmetries could correlate early and late time Hawking radiation W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

10 Gravitons in flatland Three dimensional gravity We ll work in 2+1 dimensions Gravity is topological: no propagating gravitons All solutions are locally equivalent W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

11 Gravitons in flatland Three dimensional gravity We ll work in 2+1 dimensions Interesting global solutions BTZ black holes [Bañados, Teitelboim, Zanelli 92] Flat space cosmologies [Cornalba, Costa 02] W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

12 Gravitons in flatland Three dimensional gravity We ll work in 2+1 dimensions Interesting global solutions BTZ black holes [Bañados, Teitelboim, Zanelli 92] Flat space cosmologies [Cornalba, Costa 02] Physical state-space is determined by boundary conditions W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

13 Gravitons in flatland Three dimensional gravity We ll work in 2+1 dimensions Interesting global solutions BTZ black holes [Bañados, Teitelboim, Zanelli 92] Flat space cosmologies [Cornalba, Costa 02] Physical state-space is determined by boundary conditions Imposing suitable boundary conditions leads to the asymptotic symmetry algebra of boundary condition preserving gauge transformations. All inequivalent bulk solutions fall into representations of this asymptotic symmetry algebra. W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

14 Three dimensional gravity Gravitons in flatland We ll work in 2+1 dimensions Interesting global solutions Physical state-space is determined by boundary conditions Asymptotically AdS spacetimes AdS 3 Gravity Vir Vir [Brown, Henneaux 86] r ϕ t [L m, L n ] = (m n)l m+n + c 12 m(m2 1)δ m+n,0 [ L m, L n ] = (m n) L m+n + c 12 m(m2 1)δ m+n,0 W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

15 Three dimensional gravity Gravitons in flatland i + We ll work in 2+1 dimensions Interesting global solutions Physical state-space is determined by boundary conditions Asymptotically flat spacetimes I + Conformal S 1 s i 0 FS 3 Gravity BMS 3 [Ashtekar, Bičák, Schmidt 96, Barnich, Compère 06] [L m, L n ] = (m n)l m+n [L m, M n ] = (m n)m m+n + c M 12 m(m2 1)δ m+n,0 I i W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

16 Near horizon boundary conditions A new set of boundary conditions for 3D gravity r = 0 r = r h r Choose boundary conditions such that all solutions have a (regular) horizon (Rindler space) W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

17 Near horizon boundary conditions A new set of boundary conditions for 3D gravity ρ = 0 ρ = c v Choose boundary conditions such that all solutions have a (regular) horizon (Rindler space) Metric to leading order given by ds 2 = 2aρdv 2 + 2dvdρ + γ(ϕ) 2 dϕ 2 ( 2ω(ϕ)a 1 dϕdρ) +... where a = 2π/T is the Rindler acceleration and γ(ϕ) and ω(ϕ) are arbitrary state-dependent functions W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

18 Near horizon boundary conditions A new set of boundary conditions for 3D gravity ρ = 0 ρ = c v Choose boundary conditions such that all solutions have a (regular) horizon (Rindler space) Metric to leading order given by ds 2 = 2aρdv 2 + 2dvdρ + γ(ϕ) 2 dϕ 2 ( 2ω(ϕ)a 1 dϕdρ) +... where a = 2π/T is the Rindler acceleration and γ(ϕ) and ω(ϕ) are arbitrary state-dependent functions The near horizon symmetry algebra is spanned by the Fourier modes J n and K n of γ and ω [Afshar, Detournay, Grumiller, WM, Perez, Tempo, Troncoso 16] [J n, K m ] = kn δ n+m,0, [J n, J m ] = 0 = [K n, K m ]. This is essentially the Heisenberg algebra with two Casimirs J 0 and K 0. W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

19 Soft Heisenberg hair Consequences of the near horizon symmetry algebra The near horizon symmetry algebra is spanned by the Fourier modes J n and K n of γ and ω [Afshar, Detournay, Grumiller, WM, Perez, Tempo, Troncoso 16] [J n, K m ] = kn δ n+m,0, [J n, J m ] = 0 = [K n, K m ]. This is essentially the Heisenberg algebra with two Casimirs J 0 and K 0. W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

20 Soft Heisenberg hair Consequences of the near horizon symmetry algebra The near horizon symmetry algebra is spanned by the Fourier modes J n and K n of γ and ω [Afshar, Detournay, Grumiller, WM, Perez, Tempo, Troncoso 16] [J n, K m ] = kn δ n+m,0, [J n, J m ] = 0 = [K n, K m ]. This is essentially the Heisenberg algebra with two Casimirs J 0 and K 0. Soft Hair The infinite number of generators J n and K n commute with the Hamiltonian and hence all descendents N M ψ = (J ni ) m i (K ki ) l i E i=1 i=1 have the same energy as E W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

21 Soft Heisenberg hair Consequences of the near horizon symmetry algebra The near horizon symmetry algebra is spanned by the Fourier modes J n and K n of γ and ω [Afshar, Detournay, Grumiller, WM, Perez, Tempo, Troncoso 16] [J n, K m ] = kn δ n+m,0, [J n, J m ] = 0 = [K n, K m ]. This is essentially the Heisenberg algebra with two Casimirs J 0 and K 0. Soft Hair The infinite number of generators J n and K n commute with the Hamiltonian and hence all descendents N M ψ = (J ni ) m i (K ki ) l i E i=1 i=1 have the same energy as E A puzzle How could having infinitely many symmetry generators, or black holes with an infinite amount of soft hair ever lead to a finite answer for the black hole entropy? S BH = A 4G. W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

22 Black hole entropy The BTZ black hole entropy The mass of a solution is only well-defined as spatial infinity! Map near horizon to the asymptotic region and count different microstates there W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

23 Black hole entropy The BTZ black hole entropy The mass of a solution is only well-defined as spatial infinity! Map near horizon to the asymptotic region and count different microstates there Find exact solution which can be extended to the asymptotic region. W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

24 Black hole entropy The BTZ black hole entropy The mass of a solution is only well-defined as spatial infinity! Map near horizon to the asymptotic region and count different microstates there Find exact solution which can be extended to the asymptotic region. In AdS, the asymptotic Virasoro generators L + n and L n are found to be L ± n = 1 J ± k n pj p ± + inj ± in terms of the near horizon generators J ± n = 1 2 (J n ± K n ). p W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

25 Black hole entropy The BTZ black hole entropy The mass of a solution is only well-defined as spatial infinity! Map near horizon to the asymptotic region and count different microstates there Find exact solution which can be extended to the asymptotic region. In AdS, the asymptotic Virasoro generators L + n and L n are found to be L ± n = 1 J ± k n pj p ± + inj ± in terms of the near horizon generators J ± n = 1 2 (J n ± K n ). p Use the Cardy formula, which gives the asymptotic density of states in a (large c) two-dimensional CFT to compute the entropy S BH = 2π(J J 0 ) = 2πJ 0 = A 4G N. W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

26 Black hole entropy The BTZ black hole entropy The mass of a solution is only well-defined as spatial infinity! Map near horizon to the asymptotic region and count different microstates there Find exact solution which can be extended to the asymptotic region. In AdS, the asymptotic Virasoro generators L + n and L n are found to be L ± n = 1 J ± k n pj p ± + inj ± in terms of the near horizon generators J ± n = 1 2 (J n ± K n ). p Use the Cardy formula, which gives the asymptotic density of states in a (large c) two-dimensional CFT to compute the entropy S BH = 2π(J J 0 ) = 2πJ 0 = A 4G N. Similar arguments work for asymptotically flat spacetimes W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

27 Summary & Outlook Summary & Outlook We have discussed new boundary conditions in 3D gravity and shown that they lead to an infinite amount of soft hair on the black hole horizon W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

28 Summary & Outlook Summary & Outlook We have discussed new boundary conditions in 3D gravity and shown that they lead to an infinite amount of soft hair on the black hole horizon The correct BTZ black hole entropy is recovered after mapping the near horizon region to the asymptotic region W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

29 Summary & Outlook Summary & Outlook We have discussed new boundary conditions in 3D gravity and shown that they lead to an infinite amount of soft hair on the black hole horizon The correct BTZ black hole entropy is recovered after mapping the near horizon region to the asymptotic region Similar constructions work in asymptotically flat spacetimes and the entropy for flat space cosmological solutions can likewise be obtained. W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

30 Summary & Outlook Summary & Outlook We have discussed new boundary conditions in 3D gravity and shown that they lead to an infinite amount of soft hair on the black hole horizon The correct BTZ black hole entropy is recovered after mapping the near horizon region to the asymptotic region Similar constructions work in asymptotically flat spacetimes and the entropy for flat space cosmological solutions can likewise be obtained. Some interesting future research would be Generalize to higher dimensions Study dynamics (Could infalling matter excite the soft hairs? Do the soft hairs leave an imprint on outgoing Hawking radiation?) W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8

31 The End Summary & Outlook Thank you for your attention W. Merbis (ITP, TU Wien) 3D black holes & the Heisenberg algebra September 28, / 8