The Generalised Second Law and the unity of physics

Fay Dowker Imperial College

Stockholm August 2015

In memory, Jacob Bekenstein 1947-2015

Bardeen, Carter, Hawking 1973The Laws of Black Hole Mechanics

Thermo Black Hole

Zeroth Law

is constant throughout system in equilibrium

is constant on horizon of stationary black hole

1st Law

2nd Law Hawking’s Area Theorem

3rd Law cannot be reduced to zero in a finite process

cannot be reduced to zero in a finite process

dM =

1

8⇡G dA� workdE = TdS � work

T

dS � 0 dA � 0

T

is the surface gravity of the horizon of a stationary black hole, is the surface area of the horizon A

Remarkable, but there was more..

Bekenstein, Hawking

The greatest unity yet achieved in physics

The Laws of Black Hole Thermodynamics

proposed by Bekenstein

To perfect this unity we need (at least) 1. The stat mech of black hole entropy and 2. Proof of the GSL

• A black hole has a temperature and an entropy

TH = ~

2⇡

, SBH = 2⇡

A

l

2p

where l

2p = 8⇡G~

• Zeroth Law: T is constant over the horizon of a stationary black hole

• First Law: dM = TdS � work

• Generalized Second Law (GSL): dStot = dSBH + dSext � 0

Further inspiring unity : Causal Horizon Thermodynamics

• It has been suggested that all causal horizons obey the laws of thermodynamics, where a causal horizon is the boundary of the causal past of a future-infinite timelike trajectory: Black hole horizons, DeSitter horizons (Gibbons and Hawking) and Rindler horizons (Jacobson).

• Every causal horizon looks locally like a Rindler horizon and the thermodynamics of horizons springs from this and the structure of quantum states of field theories in Minkowski space, their nonlocal, entangled structure (2 mode squeezed states) and the fact that the usual Minkowski vacuum is a thermal state for the Hamiltonian that generates boosts (Bisognano&Wichmann, Sewell, Unruh,…)

• We should think fundamentally about horizon entropy not merely black hole entropy (Jacobson & Parentani).

The Second Law!!“ The law that entropy always increases—the second law of thermodynamics—holds the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations—then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation—well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.” !A. Eddington, The Nature of the Physical World, Gifford Lectures 1926-27

Perhaps this is an exaggeration, but probably the exhibition of a perpetuum mobile within a theory or model would be a severe disincentive to continue working on it. !More persuasive perhaps is the subtlety by which attempts at counterexamples fail. The box lowering example of Bekenstein, as analysed by Unruh and Wald is a case in point. !!!

The Second Law as a guideIt isn’t enough to identify potential microstates giving rise to the area relation for black holes, the GSL also needs to be proved (Sorkin). Turning this around — we can use the GSL to guide us towards finding those microstates: the two tasks go hand in hand. !Sorkin proposed two proofs of the GSL, a sketch of a proof that would hold in full quantum gravity and a more fully worked out proof in a semiclassical approximation. Both rely on certain powerful results about entropy and relative entropy. !Definition: The relative entropy between two quantum states and is⇢ �

S(⇢ k�) = Tr(⇢ log ⇢� ⇢ log �)!!Theorem: If and evolve in time according to a completely positive linear, trace-preserving map, then is non-increasing. (Lindblad)

⇢t �tS(⇢t k�t)

(Uhlmann)

Theorem: If ⇢ and � are states on an algebra of observables A and ⇢t and �t

are their restrictions to a subalgebra At ✓ A then

S(⇢t k�t) S(⇢ k�)

The existence of an equilibrium state

S(⇢ k�) = Tr(⇢ log ⇢� ⇢ log �)

So, if is preserved by the evolution, we have a quantity that just depends on and is non-decreasing with time. !Sorkin’s semiclassical GSL argument proceeds thus:

� ⇢

Consider a quasi-stationary situation in which a large black hole is well-approximated by a Schwarzschild metric at any stage. Matter is quantum.

⌃t1

⌃t2 Surfaces of constant t, compatible with Killing vec tor expres s i ng stationarity

Horizon

The state of the external matter is given by ⇢t on ⌃t

2⇡A2 � Tr(⇢2 log ⇢2) � 2⇡A1 � Tr(⇢1 log ⇢1)We want:

Autonomous evolution and Hartle Hawking state

⌃t1

⌃t2 Surfaces of constant t, compatible with Killing vec tor expres s i ng stationarity

The state of the external matter is given by

Horizon

⇢t on ⌃t

⇢t evolves autonomously due to the causal nature of the horizon There exists an equilibrium state, the Hartle Hawking state, The theorem thus gives us a non-decreasing function of

⇢HH = Z�1e��H

⇢�S(⇢ k ⇢HH) = �Tr(⇢ log ⇢)� � Tr(⇢H)� � logZ

= Sext � � < Eext > �� logZ

Conservation of energy and 1st Law: � < Eext >= ��MBH = �T �SBH

So that: �(Sext + SBH) � 0

Note: formally divergent quantities here

The lacuna (Wall)

⌃t1

⌃t2Horizon

This H is the Hamiltonian on surfaces that pass through the bifurcation 2-surface, not the required for the argument, the Hamiltonian on the t=constant surfaces.

⇢HH = Z�1e��H

Hext

Can Sorkin’s conceptually elegant proof be completed? Is the Hartle-Hawking state thermal on “nice slices”? Is there a more subtle resolution involving a boundary contribution at the horizon? What about charged, rotating black holes: replace by ?

Wall has produced an alternative proof using an algebra of observables defined on the horizon.

H H � ⌦J � �Q

Using the GSL to rule things out Aron Wall: 1) Introduces the concept of a “quantum trapped surface” on which the generalised

entropy is decreasing on future-outgoing null normals. Uses this to prove that the energy condition in the Penrose/Hawking singularity theorems can be replaced by the GSL. Roughly, if the GSL holds (and spacetime is globally hyperbolic) and a quantum trapped surface exists then the spacetime is singular. GSL rules out bouncing universes, nonsingular black holes, traversable wormholes, and viable baby universes (in which the metric is everywhere Lorentzian and nonsingular).

2) Shows in a 1+1 dimensional model that the GSL is violated for any horizon other than the causal horizon (e.g. dynamical, or trapping). He explicitly shows that the validity of the GSL rests on the teleological — nonlocal — nature of the causal horizon.

!Bianchi, de Lorenzo and Smerlak: look at entanglement entropy production during gravitational collapse and evaporation in a concrete model of nonsingular black holes with no event horizon. They find a (not very surprising) violation of the GSL in the “purifying phase”. The Planck star model of Haggard, Rovelli&Vidotto can’t be treated so concretely due to the unknown nature of part of the metric but, if that metric is non-singular, it will be covered by Wall’s theorem and also violate the GSL. !!

Summary

• Proving the GSL is a crucial part of understanding black hole entropy (not just the Area-Entropy relation) and causal horizon entropy more generally.

• Since black hole entropy holds the key to quantum gravity, the GSL can be a guide towards quantum gravity

• Sorkin: the GSL arises from the non-decreasing nature of minus the relative entropy between any external state and a state that is constant under the autonomous evolution outside the horizon.

• Sorkin’s semiclassical proof of the GSL has a lacuna (Wall) — how to complete the proof? Is the Hartle-Hawking state thermal on “nice slices”? Can we show that the non-increasing quantity guaranteed to exist by the theorem is (proportional to) the appropriate free energy. (What else could it physically be?)

• Though many approaches/models claim an Area-Entropy relation for horizons, the GSL can point the way. Work by Wall shows this: the GSL can rule out nonsingular black holes and bouncing black holes. (Can we exhibit a perpetuum mobile in any of these cases?) Wall provides evidence that the GSL supports the event horizon over the dynamical or trapping horizons.

• If the GSL holds in full quantum gravity it would be a beautiful unification and it would mean that the discovery of Hawking radiation was the historic moment which opened the door to that achieving that unity.

Some References!

• Jacobson and Parentanti, “Horizon Entropy” gr-qc/0302099

• Sorkin, “Towards a Proof of Entropy Increase in the Presence of Quantum Black Holes” Phys Rev Lett 56 (1986) 1885

• Sorkin, “The Statistical Mechanics of Black Hole Thermodynamics” gr-qc/9705006

• Sorkin, “Ten Theses on Black Hole Entropy” hep-th/0504037

• Wall, “Ten Proofs of the GSL” 0901.3865

• Wall, “The Generalized Second Law implies a Quantum Singularity Theorem” 1010.5513

• Wall, “Testing the Generalized Second Law in 1+1 dimensional Conformal Vacua: An Argument for the Causal Horizon” 1105.3520

• Bianchi, de Lorenzo & Smerlak, “Entanglement entropy production in gravitational collapse: covariant regularization and solvable models” 1507.01567