Aron C. Wall

Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy

The covariant holographic entropy conjecture of AdS/CFT relates the entropy of a boundary region R to the area of an extremal surface in the bulk spacetime. This extremal surface can be obtained by a maximin construction, allowing many new results to be proven. On manifolds obeying the null curvature condition, these extremal surfaces: (i) always lie outside the causal wedge of R, (ii) have less area than the bifurcation surface of the causal wedge, (iii) move away from the boundary as R grows, and (iv) obey strong subadditivity and monogamy of mutual information. These results suggest that the information in R allows the bulk to be reconstructed all the way up to the extremal area surface. The maximin surfaces are shown to exist on spacetimes without horizons, and on black hole spacetimes with Kasner-like singularities.

Entanglement entropy of electromagnetic edge modes

The vacuum entanglement entropy of Maxwell theory, when evaluated by standard methods, contains an unexpected term with no known statistical interpretation. We resolve this two-decades old puzzle by showing that this term is the entanglement entropy of edge modes: classical solutions determined by the electric field normal to the entangling surface. We explain how the heat kernel regularization applied to this term leads to the negative divergent expression found by Kabat. This calculation also resolves a recent puzzle concerning the logarithmic divergences of gauge fields in 3+1 dimensions.

Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality

In this Letter we prove a simple theorem in quantum information theory, which implies that bulk operators in the anti-de Sitter/conformal field theory (AdS/CFT) correspondence can be reconstructed as CFT operators in a spatial subregion A, provided that they lie in its entanglement wedge. This is an improvement on existing reconstruction methods, which have at most succeeded in the smaller causal wedge. The proof is a combination of the recent work of Jafferis, Lewkowycz, Maldacena, and Suh on the quantum relative entropy of a CFT subregion with earlier ideas interpreting the correspondence as a quantum error correcting code.

Lorentz violation and perpetual motion

We show that any Lorentz-violating theory with two or more propagation speeds is in conflict with the generalized second law of black hole thermodynamics. We do this by identifying a classical energy-extraction method, analogous to the Penrose process, which would decrease the black hole entropy. Although the usual definitions of black hole entropy are ambiguous in this context, we require only very mild assumptions about its dependence on the mass. This extends the result found by Dubovsky and Sibiryakov, which uses the Hawking effect and applies only if the fields with different propagation speeds interact just through gravity. We also point out instabilities that could interfere with their black hole perpetuum mobile, but argue that these can be neglected if the black hole mass is sufficiently large.

Proof of the quantum null energy condition

We prove the quantum null energy condition (QNEC), a lower bound on the stress tensor in terms of the second variation in a null direction of the entropy of a region. The QNEC arose previously as a consequence of the quantum focusing conjecture, a proposal about quantum gravity. The QNEC itself does not involve gravity, so a proof within quantum field theory is possible. Our proof is somewhat nontrivial, suggesting that there may be alternative formulations of quantum field theory that make the QNEC more manifest. Our proof applies to free and super-renormalizable bosonic field theories, and to any points that lie on stationary null surfaces. An example is Minkowski space, where any point p and null vector ka define a null plane N (a Rindler horizon). Given any codimension-2 surface Σ that contains p and lies on N, one can consider the von Neumann entropy Sout of the quantum state restricted to one side of Σ. A second variation Sout′′ can be defined by deforming Σ along N, in a small neighborhood of p with area A. The QNEC states that ⟨Tkk(p)⟩≥ℏ2πlimA→0Sout′′/A.

Second Law Violations in Lovelock Gravity for Black Hole Mergers

We study the classical second law of black hole thermodynamics, for Lovelock theories (other than general relativity), in arbitrary dimensions. Using the standard formula for black hole entropy, we construct scenarios involving the merger of two black holes in which the entropy instantaneously decreases. Our construction involves a Kaluza-Klein compactification down to a dimension in which one of the Lovelock terms is topological. We discuss some open issues in the definition of the second law which might be used to remove this entropy decrease.

Proof of the generalized second law for rapidly evolving Rindler horizons

The generalized second law is proven for rapidly evolving semiclassical Rindler horizons at each instant of time, for arbitrary interacting quantum fields minimally coupled to general relativity. The proof requires the background spacetime to have both boost and null translation symmetry. Possible extensions to more general horizons and matter-gravity couplings are discussed.

The generalized second law implies a quantum singularity theorem

The generalized second law can be used to prove a singularity theorem, by generalizing the notion of a trapped surface to quantum situations. Like Penrose’s original singularity theorem, it implies that spacetime is null-geodesically incomplete inside black holes, and to the past of spatially infinite Friedmann–Robertson–Walker cosmologies. If space is finite instead, the generalized second law requires that there only be a finite amount of entropy producing processes in the past, unless there is a reversal of the arrow of time. In asymptotically flat spacetime, the generalized second law also rules out traversable wormholes, negative masses, and other forms of faster-than-light travel between asymptotic regions, as well as closed timelike curves. Furthermore it is impossible to form baby universes which eventually become independent of the mother universe, or to restart inflation. Since the semiclassical approximation is used only in regions with low curvature, it is argued that the results may hold in full quantum gravity. The introduction describes the second law and its time-reverse, in ordinary and generalized thermodynamics, using either the fine-grained or the coarse-grained entropy. (The fine-grained version is used in all results except those relating to the arrow of time.)

Black Hole Thermodynamics and Lorentz Symmetry

Recent developments point to a breakdown in the generalized second law of thermodynamics for theories with Lorentz symmetry violation. It appears possible to construct a perpetual motion machine of the second kind in such theories, using a black hole to catalyze the conversion of heat to work. Here we describe and extend the arguments leading to that conclusion. We suggest the inference that local Lorentz symmetry may be an emergent property of the macroscopic world with origins in a microscopic second law of causal horizon thermodynamics.

Generalized second law at linear order for actions that are functions of Lovelock densities

In this article we consider the second law of black holes (and other causal horizons) in theories where the gravitational action is an arbitrary function of the Lovelock densities. We show that there exists an entropy which increases locally, for linearized perturbations to regular Killing horizons. In addition to a classical increase theorem, we also prove a generalized second law for semiclassical, minimally-coupled matter fields.