Black hole thermodynamics and spacetime symmetry breaking

Transcription

1 Black hole thermodynamics and spacetime symmetry breaking David Mattingly University of New Hampshire Experimental Search for Quantum Gravity, SISSA, September 2014

2 What do we search for? What does the experimental quantum gravity community look for? Non-locality Extra dimensions QG induced decoherence Dimensional reduction B-mode polarization Symmetry violation Extra dimensions

3 What do we search for? What does the experimental quantum gravity community look for? Non-locality Extra dimensions QG induced decoherence Dimensional reduction B-mode polarization Symmetry violation Extra dimensions

4 Locality An easy multiple choice question Fundamentally, quantum gravity is should be may be should not be is not a local quantum field theory.

5 Many approaches give up quantum gravity as local QFT Why is this a popular choice? In many ways it s the most radical! Fundamentally, quantum gravity should be may be should not be is not a local quantum field theory.

6 Why are we so comfortable with giving up local QFT? GR is perturbatively nonrenormalizable S 1 2 d4 x[ h 2 + κ h 2 h] Finite IR irrelevant operators Divergent operators in far UV Formation of black holes/unitarity issues with ultrahigh energy scattering (c.f. Giddings et. al ) R L Planck Others c.f. Oriti

7 Why are we so comfortable with giving up local QFT? Black hole thermodynamics itself Finiteness of entropy As horizon microstates implies discreteness? Sorkin, hep-th/ As entanglement entropy needs cutoff c.f. Solodukhin Density of states from UV scale invariance S QFT E3 3+z S BH E 2 β functions vanish in UV c.f. Shomer, Firewalls Unitarity Equivalence principle Local QFT near horizon Please pick your favorite two AMPS, Giddings,

8 So what if gravity IS a local QFT? What happens in putative quantum gravity theories where gravity remains a local, renormalizable QFT? Can black hole physics inform how we think about those theories as well?

9 So what if gravity IS a local QFT? N=8 Supergravity General black hole thermodynamics in N=8 SUGRA only recently computed (Chow, Compere ) Approaches to renormalizable QG Asymptotic Safety Work still needs to be done in understanding black hole solutions in ASG. c.f. Koch, Saueressig, Horava- Lifshitz gravity Black hole solutions understood in some cases. Thermodynamics yields interesting interplay with how to implement H-L theory in matter sector.

10 So what if gravity IS a local QFT? N=8 Supergravity General black hole thermodynamics in N=8 SUGRA only recently computed (Chow, Compere ) Approaches to renormalizable QG Asymptotic Safety Work still needs to be done in understanding black hole solutions in ASG. c.f. Koch, Saueressig, Horava- Lifshitz gravity Black hole solutions understood in some cases. Thermodynamics yields interesting interplay with how to implement H-L theory in matter sector.

11 The fundamental questions Can there even be black hole thermodynamics? Does requiring black hole thermodynamics lead to interesting restrictions on parameter space of Horava-Lifshitz gravity? I m not asking or worrying about experimental limits on Horava-Lifshitz gravity

12 Horava-Lifshitz theory Horava-Lifshitz theory Horava: timelike infinity There exists a time function U generating a preferred foliation in spacetime. spacelike infinity

13 Horava-Lifshitz theory Horava-Lifshitz theory timelike infinity There exists a time function U generating a preferred foliation in spacetime. spacelike infinity UV theory has Lifshitz symmetry t b z t, x bx

14 Horava-Lifshitz theory Dynamical Horava-Lifshitz theory Blas, Pujolas,Sibiryakov Dynamical foliation given by time function U. u a : = a U b U b U 3+1 split, due to reduced symmetry more terms in gravitational action...

15 Horava-Lifshitz theory Dynamical Horava-Lifshitz theory gravitational piece N = lapse g ab = spatial metric K ij = extrinsic curvature of U hypersurface R = 3d Ricci scalar a i = acceleration of u a Changes UV divergence structure without introducing ghosts by permitting higher spatial derivatives in propagators without higher time derivatives.

16 Horava-Lifshitz and Einstein-Aether Einstein aether theory: Jacobson, DM gr-qc/ Assume aether is hypersurface orthogonal. u a a U : = b U b U Dynamical, non-projectable HL theory in IR: c 13 = c 1 + c 3

17 Simplest regular massive vacuum solutions What is H-L Schwarzschild black hole? Schwarzschild Infrared solution Vacuum Four dimensional aether/killing vectors aligned at infinity spherically symmetric Static asymptotically flat Trivial M 0 limit Note: Other solutions (rotating, ds/ads asymptotics, 2+1) exis

18 Simplest regular massive vacuum solutions AS AN EXAMPLE we ll use the analytic solutions that exist when c 14 = 0 or c 123 = 0. c 123 = 0 ds 2 = e r dv f r dvdr + r 2 dω 2 u χ = 1 + r UH r s χ = 2 2 c 14 r UH 2 1 c 13 r 2 f r = 1 e r = 1 r 0 r + c c 13 r UH 2 1 c 13 r 2 r UH = r 0 2 One parameter family of solutions, controlled by r 0 u a s a Killing horizon χ 2 = 0 Universal horizon u χ = 0

19 Is there a black hole? Is there any surface of constant r that acts as a trapped surface for propagating excitations above the vacuum, matter or otherwise? Need to specify a matter Lagrangian e.g. L φ = s φ 2 2 g φ ab a φ b φ ± ( 2 ) n φ 2 g φ ab = g ab s φ 2 1 u a u b s φ is speed of low energy mode, n is an integer, related to UV Lifshitz scaling 2k 0 4n 2 Flat space dispersion relation: ω 2 = s φ 2 k 2 ± k4n k 0 4n 2

20 Is there a black hole? Flat space dispersion relation: ω 2 = s φ 2 k 2 ± k4n k 0 4n 2 We phenomenologically test such modified dispersion with +, - sign. Is there a difference between these two cases from the perspective of BH thermo?

21 Is there a black hole? Natural to start with a minus sign as then all propagating modes have a finite speed, the naïve guess as to what you need for a horizon. Flat space dispersion relation: ω 2 = s φ 2 k 2 k4n k 0 4n 2 t x

22 Is there a black hole? ω 2 = s φ 2 k 2 k4n k 0 4n 2 A rainbow causal horizon

23 Deviations from thermal spectrum ω 2 = s φ 2 k 2 k4n k 0 4n 2 Calculate spectrum via mode conversion Corley/Jacobson hep-th/ Thermal spectrum only reproduced to a high degree for very low frequency with respect to k 0 outgoing radiation. Different fields with different s φ have different T More importantly

24 First law ω 2 = s φ 2 k 2 k4n k 0 4n 2 There are first law problems! Via Noether at infinity and Killing horizon (Foster, gr-qc/ ) δm Standard TdS form Non-standard contribution proportional to c s Generically, Noether approach on fixed r hypersurfaces does not yield a thermodynamics where S A

25 First law Demanding a first law that has S A kills the rainbow horizon situation and hence ω 2 = s φ 2 k 2 k4n k 0 4n 2 does not yield full black hole thermodynamics

26 First law again However, first law on the universal horizon is a standard first law (Bhattacharyya, DM , ) c 123 = 0: δm = 1 c 13 κ UH 8πG δa κ UH = 1 2 aχ b a χ b If you want a standard/holographic first law for black hole thermodynamics, you have to use the universal horizon!

27 Is there a black hole part deux? Universal horizon IS also the causal horizon for ω 2 = s φ 2 k 2 + k4n k 0 4n 2

28 Is there a black hole part deux? Wang et. al

29 Radiation from universal horizon Tunneling approach Requirements Vacuum: assume the infalling vacuum No matter/aether Cerenkov radiation so c 123 = 0 or c 14 = 0 (Technically convenient but likely not necessary) Lifshitz coefficient yields chemical potential preserves thermality I e ω μ T UH μ = c ae 2 k 0 2N, T UH = κ UH 4πc ae S = 1 c 13 c ae A UH 2G ae Berglund, Bhattacharyya, DM:

30 Killing horizon reprocessing UH KH UH KH Low energy High energy (Cropp, Liberati, Mohd, Visser, ) Thermal spectrum modified at ω k 0 by scattering off Killing/IR horizon. New greybody factor. Final low energy spectrum uncalculated thermal with T = κ KH 2π?

31 Moral of the story Some evidence universal black hole thermodynamics is not destroyed when you violate spacetime symmetries in EFT s but ONLY if you include superluminal UV corrections!* Different low energy speeds alone destroys BH thermo when multiple fields considered Subluminal higher order dispersion correction destroys single field BH thermo *Of course, whether you want to keep BH thermo is up to you

32 Where we stand on universal horizon thermodynamics (spherical symmetry) 0. The surface gravity is constant on a stationary horizon. Yes, but it s a bit of a cheat in spherical symmetry. 1. First law. δe = TδS Yes. We have thermal radiation and a first law. 2. Second law. δa 0. Yes. However, the GSL has conceptual issues when interactions are turned on. 3. Cannot reach vanishing surface gravity in a finite number of processes. Nobody s looked!

33 The second law and interactions Problem: if we have two interacting scalar fields they will generically have different IR speeds s φ1 > s φ2 = c UH H 1 KH/H 2 Ergoregion of φ 2 accessible to φ 1 1. Take a system of φ 1 and φ 2 in a pure state. 2. Let it fall into ergoregion and split. φ 2 φ 1 3. Can arrange this so that φ 2 has negative Killing energy. φ 1, φ 2 4. Can arrange that no increase in entropy of outgoing φ 1 (stays pure). 5. Negative Killing energy goes into hole, S hole decreases, S outside stays the same. φ 1, φ 2 6. Violation of GSL. Jacobson, Wall:

34 Questions that need answers 1. What to do about the second law? 2. What is the general/axisymmetric solution space for HL/AE theories? 3. Is GR as a thermodynamic limit unique? 4. Can one be more robust in calculating radiation from the UH? 5. Can we get more general analytic solutions? 6. What are the solutions with a UH and Lifshitz asymptotics (Lifshitz holography)?