Black Hole Entropy and Thermodynamics of Space=me Ted Jacobson University of Maryland 1
Sadi Carnot How does the maximum efficiency of steam engines depend on the working fluid? Carnot: given the temperatures of the hot and cold reservoirs there is a universal upper limit, realized for reversible engines. Otherwise you could make a perpetual mo=on machine and extract work for free. Réflexions sur la puissance motrice du feu (1824) ("Reflec=ons on the Mo=ve Power of Fire ) ds = δq T Clausius rela=on (1854) Entropy change = heat per thermal energy unit ds total 0 Second law of thermodynamics Heat flows from hot to cold T 1 dq T 2 δq T 1 + δq T 2 > 0 2
Sta=s=cal interpreta=ons of entropy S/k = ln(# microstates) Boltzmann, 1872 S/k = i p i ln p i Gibbs, 1878 S/k = Tr(ρ ln ρ) Von Neumann, 1927 Ideal gas entropy & equa=on of state Molecules occupy a negligible frac=on of the available volume, and interact weakly: S(E,V )=k ln(#states )=k ln(v N f(e)) * A UV cutoff comes from QM. Clausius rela=on & energy conserva=on: ds = δq T = de + pdv T = p T = S V = Nk V = pv = NkT Ideal gas law 3
Einstein equa=on of state - Entropy of causal horizons is propor=onal to area - Clausius rela=on - energy conserva=on Causal structure of space=me is dynamical, and the metric sa=sfies the Einstein equa=on Wheeler to Bekenstein (1971): If I drop a teacup into a black hole, I conceal from all the world the increase of entropy. S outside < 0 Black hole 4
Black hole entropy Bekenstein, 1972 S BH = α(horizon Area) α L 2 Planck L 2 Planck = G/c 3 = (10 33 cm) 2 Generalized second law: (S outside + S BH ) 0 Requires that the BH radiates! GSL holds only if the black hole radiates at the Hawking temperature, T H = κ 2π, κ = surface gravity = 1 2R horizon and has entropy S BH = A 4L 2 P Hawking radiadon: - is a quantum field effect - comes from outside the black hole - is correlated to quantum field fluctua=ons inside the black hole - has thermal wavelength 8π 2 R horizon - has temperature 10-7 K (M sun /M BH ) 5
What is black hole entropy coun=ng? A horizon is locally just like any other place in spacedme! A horizon is observer dependent. 6
Causal Horizon The boundary of the causal past of an observer is their causal horizon. Examples: - observer who remains outside a black hole forever - iner=al observer in an exponen=ally expanding cosmological space=me - uniformly accelera=ng observer in Minkowski space=me These all have entropy and temperature, and the GSL applies to them. So, to understand black hole entropy, we may as well look in Minkowski spacedme! Local causal horizons Sta=onary black hole horizon. Arbitrary equilibrium point p Equivalence principle: An approximate local Minkowski space with Lorentz boost symmetry exists around p Local horizon Boundary of the past of the red line (2- surface) 7
Euclidean space Minkowski space RotaDon symmetry ds 2 = dx 2 + dy 2 = dr 2 + r 2 dθ 2 Lorentz boost symmetry ds 2 = dx 2 dt 2 = dl 2 l 2 dη 2 Davies- Unruh effect Lorentz invariance and energy positivity imply the Minkowsi vacuum is a thermal state when restricted to the wedge: ρ R = Tr L 0 0 exp 2π H Boost Bisognano-Wichmann (1975), Davies (1975), Unruh (1976) L R A uniformly accelerated observer a distance l from the horizon sees the temperature, T local = a /2π = /2π l. Accelera=on and T local diverge as l goes to 0. 8
Vacuum entanglement entropy S = Tr(ρ R ln ρ R ) da dl T 3 local l> (Sorkin 83, Bombelli, Koul, Lee, Sorkin 86) da dl l 3 A/ 2 Horizon thermodynamics Postulate for all such horizons 1. The horizon system is a heat bath, with universal entropy area density. 2. Boost energy flux across the horizon is thermalized at the Unruh temperature. S = α A 3. Energy conserva=on (energy- momentum tensor divergence- free) Implies focusing of light rays by space=me curvature: the causal structure must sa=sfy Einstein field equa=on, with Newton s constant G = 1 4 α 9
Gravity and vacuum entanglement It seems to follow that: Black hole entropy includes - - and may be 100% - - vacuum entanglement Infinite entanglement entropy implies G zero. We have gravity only because the entropy is finite. Conversely: gravity must cut off entanglement when the separa=on is smaller than the Planck length, by virtual black holes. G depends on the species of marer fields. There is no species problem. Corrobora=on from AdS/CFT where the idea can be tested: CFT has infinite entanglement and NO gravity. And using the duality on brane at finite radius, there is a UV cutoff on the brane, hence a finite entanglement entropy and NONZERO G! 10
summary Black hole entropy is a special case of causal horizon entropy. It seems to originate in the spa=al entanglement of the vacuum of quantum fields. The vacuum is thermal when viewed in a causal wedge. Without a UV cutoff, vacuum entanglement entropy is infinite, and there is no gravity. With a UV cutoff the entanglement entropy accounts for the strength of gravity. Whether this is deep guidance in the quest to understand quantum gravity, or just a consistency condi=on that many quantum gravity theories could sa=sfy, remains to be seen 11