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This essay contends that in quantum gravity, some spatial regions do not admit a unitary Hilbert space. Because the gravitational path integral spontaneously breaks CPT symmetry, "states" with negative probability can be identified on either side of trapped surfaces. I argue that these negative norm states are tolerable, by analogy to quantum mechanics. This viewpoint suggests a resolution of the firewall paradox, similar to black hole complementarity. Implications for cosmology are briefly discussed., 8 pages, 3 figs. Submitted to the Gravity Research Foundation 2021 Essay Competition

The AdS/CFT understanding of CFT entanglement is based on HRT surfaces in the dual bulk spacetime. While such surfaces need not exist in sufficiently general spacetimes, the maximin construction demonstrates that they can be found in any smooth asymptotically locally AdS spacetime without horizons or with only Kasner-like singularities. In this work, we introduce restricted maximin surfaces anchored to a particular boundary Cauchy slice $C_\partial$. We show that the result agrees with the original unrestricted maximin prescription when the restricted maximin surface lies in a smooth region of spacetime. We then use this construction to extend the existence theorem for HRT surfaces to generic charged or spinning AdS black holes whose mass-inflation singularities are not Kasner-like. We also discuss related issues in time-independent charged wormholes., 10 pages, 3 figures

When a black hole forms from collapse in a holographic theory, the information in the black hole interior remains encoded in the boundary. We prove that the area of the black hole's apparent horizon is precisely the entropy associated to coarse graining over the information in its interior, subject to knowing the exterior geometry. This is the maximum holographic entanglement entropy that is compatible with all classical measurements conducted outside of the apparent horizon. We identify the boundary dual to this entropy and explain why it obeys a Second Law of Thermodynamics., 4+1 pages, 2 figures; v4: typos fixed in equation (15)

To study quantum gravity in asymptotically flat spacetimes, one would like to understand the algebra of observables at null infinity. Here we show that the Bondi mass cannot be observed in finite retarded time, and so is not contained in the algebra on any finite portion of ${\mathscr{I}}^+$. This follows immediately from recently discovered asymptotic entropy bounds. We verify this explicitly, and we find that attempts to measure a conserved charge at arbitrarily large radius in fixed retarded time are thwarted by quantum fluctuations. We comment on the implications of our results to flat space holography and the BMS charges at ${\mathscr{I}}^+$., Comment: 9 pages, 3 figures. v2 typos fixed and minor additions

We expand our recent work on the outer entropy, a holographic coarse-grained entropy defined by maximizing the boundary entropy while fixing the classical bulk data outside some surface. When the surface is marginally trapped and satisfies certain "minimar" conditions, we prove that the outer entropy is exactly equal to a quarter the area (while for other classes of surfaces, the area gives an upper or lower bound). We explicitly construct the entropy-maximizing interior of a minimar surface, and show that it satisfies the appropriate junction conditions. This provides a statistical explanation for the area-increase law for spacelike holographic screens foliated by minimar surfaces. Our construction also provides an interpretation of the area for a class of non-minimal extremal surfaces. On the boundary side, we define an increasing simple entropy by maximizing the entropy subject to a set of "simple experiments" performed after some time. We show (to all orders in perturbation theory around equilibrium) that the simple entropy is the boundary dual to our bulk construction., Comment: 49 + 10 pages, 15 figures; v2: minor clarifications added

A novel method for deriving energy conditions in stable field theories is described. In a local classical theory with one spatial dimension, a local energy condition always exists. For a relativistic field theory, one obtains the dominant energy condition. In a quantum field theory, there instead exists a quantum energy condition, i.e. a lower bound on the energy density that depends on information-theoretic quantities. Some extensions to higher dimensions are briefly discussed., 7 pages, no figures. v2: fixed typos, added refs, other minor changes, v3: minor corrections, changed title (was "From Global to Local Energy Conditions")

We calculate the vacuum entanglement entropy of Maxwell theory in a class of curved spacetimes by Kaluza-Klein reduction of the theory onto a two-dimensional base manifold. Using two-dimensional duality, we express the geometric entropy of the electromagnetic field as the entropy of a tower of scalar fields, constant electric and magnetic fluxes, and a contact term, whose leading order divergence was discovered by Kabat. The complete contact term takes the form of one negative scalar degree of freedom confined to the entangling surface. We show that the geometric entropy agrees with a statistical definition of entanglement entropy that includes edge modes: classical solutions determined by their boundary values on the entangling surface. This resolves a longstanding puzzle about the statistical interpretation of the contact term in the entanglement entropy. We discuss the implications of this negative term for black hole thermodynamics and the renormalization of Newton's constant., 41 pages, an extended version of arXiv:1412.1895

Duality is an indispensable tool for describing the strong-coupling dynamics of gauge theories. However, its actual realization is often quite subtle: quantities such as the partition function can transform covariantly, with degrees of freedom rearranged in a nonlocal fashion. We study this phenomenon in the context of the electromagnetic duality of abelian $p$-forms. A careful calculation of the duality anomaly on an arbitrary $D$-dimensional manifold shows that the effective actions agree exactly in odd $D$, while in even $D$ they differ by a term proportional to the Euler number. Despite this anomaly, the trace of the stress tensor agrees between the dual theories. We also compute the change in the vacuum entanglement entropy under duality, relating this entanglement anomaly to the duality of an "edge mode" theory in two fewer dimensions. Previous work on this subject has led to conflicting results; we explain and resolve these discrepancies., 16 pages + appendices

We show the entanglement entropy in certain quantum field theories to contain state-dependent divergences. Both perturbative and holographic examples are exhibited. However, quantities such as the relative entropy and the generalized entropy of black holes remain finite, due to cancellation of divergences. We classify all possible state-dependent divergences that can appear in both perturbatively renormalizeable and holographic covariant $d\le 6$ quantum field theories., Comment: 20 pages, 1 figure

After turning on an interaction that couples the two boundaries of an eternal BTZ black hole, we find a quantum matter stress tensor with negative average null energy, whose gravitational backreaction renders the Einstein-Rosen bridge traversable. Such a traversable wormhole has an interesting interpretation in the context of ER=EPR, which we suggest might be related to quantum teleportation. However, it cannot be used to violate causality. We also discuss the implications for the energy and holographic entropy in the dual CFT description., 19 pages, 4 figures, 1 appendix

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., Comment: 7 pages, 1 figure; v2: corrected the proof of the reconstruction theorem, added new references, acknowledgements, and minor clarifications; v3: added an appendix and minor clarifications, title changed to match PRL version

We propose a universal inequality that unifies the Bousso bound with the classical focussing theorem. Given a surface $\sigma$ that need not lie on a horizon, we define a finite generalized entropy $S_\text{gen}$ as the area of $\sigma$ in Planck units, plus the von Neumann entropy of its exterior. Given a null congruence $N$ orthogonal to $\sigma$, the rate of change of $S_\text{gen}$ per unit area defines a quantum expansion. We conjecture that the quantum expansion cannot increase along $N$. This extends the notion of universal focussing to cases where quantum matter may violate the null energy condition. Integrating the conjecture yields a precise version of the Strominger-Thompson Quantum Bousso Bound. Applied to locally parallel light-rays, the conjecture implies a Quantum Null Energy Condition: a lower bound on the stress tensor in terms of the second derivative of the von Neumann entropy. We sketch a proof of this novel relation in quantum field theory., Comment: 45 pages, 6 figures

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., 4 pages; v2: Version to appear in Foundations of Physics. Potential counterexamples addressed, argument given applying to LV theories where all speeds (or horizons) coincide, and editing for clarity

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 Focussing 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 superrenormalizable 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 $k^a$ define a null plane $N$ (a Rindler horizon). Given any codimension-2 surface $\Sigma$ that contains $p$ and lies on $N$, one can consider the von Neumann entropy $S_\text{out}$ of the quantum state restricted to one side of $\Sigma$. A second variation $S_\text{out}^{\prime\prime}$ can be defined by deforming $\Sigma$ along $N$, in a small neighborhood of $p$ with area $\cal A$. The QNEC states that $\langle T_{kk}(p) \rangle \ge \frac{\hbar}{2\pi} \lim_{{\cal A}\to 0}S_\text{out}^{ \prime\prime}/{\cal A}$., Comment: 32 pages, 3 figures. v2: references and minor typos

Standard methods for calculating the black hole entropy beyond general relativity are ambiguous when the horizon is non stationary. We fix these ambiguities in all quadratic curvature gravity theories, by demanding that the entropy be increasing at every time, for linear perturbations to a stationary black hole. Our result matches with the entropy formula found previously in holographic entanglement entropy calculations. We explicitly calculate the entropy increase for Vaidya-like solutions in Ricci-tensor gravity to show that (unlike the Wald entropy) the holographic entropy obeys a Second Law., 6 pages, minor changes and references added

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., 6 + epsilon pages. v2: Simplified derivation, added references. Matches PRL version

We study and construct spacetimes, dubbed planar AdS-dS-wormholes, satisfying the null energy condition and having two asymptotically AdS boundaries connected through a (non-traversable) inflating wormhole. As for other wormholes, it is natural to expect dual descriptions in terms of two disconnected CFTs in appropriate entangled states. But for our cases certain expected bulk entangling surfaces used by the Hubeny-Rangamani-Takayanagi (HRT) prescription to compute CFT entropy do not exist. In particular, no real codimension-2 extremal surface can run from one end of the wormhole to the other. According to HRT, the mutual information between any two finite-sized subregions (one in each CFT) must then vanish at leading order in large $N$ -- though the leading-order mutual information per unit area between the two CFTs taken as wholes may be nonzero. Some planar AdS-dS-wormholes also fail to have plane-symmetric surfaces that would compute the total entropy of either CFT. We suggest this to remain true of less-symmetric surfaces so that the HRT entropy is ill-defined and some modified prescription is required. It may be possible to simply extend HRT or the closely-related maximin construction by a limiting procedure, though complex extremal surfaces could also play an important role., Comment: 27+10 pages. v2: minor modifications to address referee comments. v3: fixed typos

We propose that holographic entanglement entropy can be calculated at arbitrary orders in the bulk Planck constant using the concept of a "quantum extremal surface": a surface which extremizes the generalized entropy, i.e. the sum of area and bulk entanglement entropy. At leading order in bulk quantum corrections, our proposal agrees with the formula of Faulkner, Lewkowycz, and Maldacena, which was derived only at this order; beyond leading order corrections, the two conjectures diverge. Quantum extremal surfaces lie outside the causal domain of influence of the boundary region as well as its complement, and in some spacetimes there are barriers preventing them from entering certain regions. We comment on the implications for bulk reconstruction., Comment: 28 pages, 6 figures. Version 3: typos fixed, minor changes to references

The Second Law of black hole thermodynamics is shown to hold for arbitrarily complicated theories of higher curvature gravity, so long as we allow only linearized perturbations to stationary black holes. Some ambiguities in Wald's Noether charge method are resolved. The increasing quantity turns out to be the same as the holographic entanglement entropy calculated by Dong. It is suggested that only the linearization of the higher-curvature Second Law is important, when consistently truncating a UV-complete quantum gravity theory., Comment: 10 pages. Honorable mention 2015 Gravity Research Foundation Essay; Appendix A, footnotes, some citations, and minor textual changes added later. v2 corrected typo in Eq. (14) for the linearized Dong entropy. To appear in IJMPD

The averaged null energy conditions (ANEC) states that, along a complete null curve, the negative energy fluctuations of a quantum field must be balanced by positive energy fluctuations. We use the AdS/CFT correspondence to prove the ANEC for a class of strongly coupled conformal field theories in flat spacetime. A violation of the ANEC in the field theory would lead to acausal propagation of signals in the bulk., 13 pages, 1 figure. v4: corrected an error in the proof, results remain unchanged, v3: added additional references along with minor editorial changes, v2: added additional references and acknowledgments

We propose bulk duals for certain coarse-grained entropies of boundary regions. The `one-point entropy' is defined in the conformal field theory by maximizing the entropy in a domain of dependence while fixing the one-point functions. We conjecture that this is dual to the area of the edge of the region causally accessible to the domain of dependence (i.e. the `causal holographic information' of Hubeny and Rangamani). The `future one-point entropy' is defined by generalizing this conjecture to future domains of dependence and their corresponding bulk regions. We show that the future one-point entropy obeys a nontrivial second law. If our conjecture is true, this answers the question "What is the field theory dual of Hawking's area theorem?", 43 pages, 9 figures. v3: minor changes suggested by referee v2: added a few additional references

An extension of Penrose's singularity theorem is proved for spacetimes where black holes are allowed to form from non-singular initial data. With standard assumptions about the spacetime, and assuming the existence of a trapped surface which lies outside of black hole horizons and is not completely surrounded by horizons, we show that the spacetime region outside (or on) the horizons must contain singularities. If the trapped surface is surrounded by horizons, we show that the horizons divide spacetime into causally disconnected pieces. Unlike the original Penrose's theorem, our theorems provide some information about the location of singularities. We illustrate how they can be used to rule out some cosmological scenarios., Comment: References added

We present a generic condition for Lorentzian manifolds to have a barrier that limits the reach of boundary-anchored extremal surfaces of arbitrary dimension. We show that any surface with nonpositive extrinsic curvature is a barrier, in the sense that extremal surfaces cannot be continuously deformed past it. Furthermore, the outermost barrier surface has nonnegative extrinsic curvature. Under certain conditions, we show that the existence of trapped surfaces implies a barrier, and conversely. In the context of AdS/CFT, these barriers imply that it is impossible to reconstruct the entire bulk using extremal surfaces. We comment on the implications for the firewall controversy., 24 pages, 7 figures. Version 3: fixed typos and made a small correction to the proof of theorem 2.2

Quantum field theory in curved spacetime may be defined either through a manifestly unitary canonical approach or via the manifestly covariant path integral formalism. For gauge theories, these two approaches have produced conflicting results, leading to the question of whether the canonical approach is covariant, and whether the path integral approach is unitary. We show the unitarity of the covariant U(1) Maxwell theory, defined via the Wick rotation of a Euclidean path integral. We begin by gauge-fixing the path integral, taking care with zero modes, large gauge transformations, and nontrivial bundles. We find an extra geometric factor in the partition function that has been overlooked in previous work, coming from the zero mode of the gauge symmetry, which affects the entropy and stress-energy tensor. With this extra factor, the covariant calculation agrees with the canonical result for ultrastatic manifolds, and in D = 2. Finally, we argue that if there exists a unitary (but not necessarily covariant) canonical formulation, then the covariant formulation must also be unitary, even if the two approaches disagree., 10 pages

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., 19 pages, no figues, v2 references and minor comments added, v3 Ref. 4 & typos in Eq. 14, 27, 40 corrected

It has been argued i) that Lorentz-signature solutions with wormholes connecting n asymptotically AdS regions describe bulk quantum states dual to n entangled but non-interacting CFTs and ii) that such bulk wormhole states should be identified with similar entangled but non-interacting bulk systems, each describing quantum geometries with only a single asymptotic region. But if the wormhole is to behave semiclassically, we show that conjecture (ii) cannot hold. Instead, the theory of asymptotically AdS bulk quantum gravity must admit superselection sectors with respect to the CFT observables that are labeled by the type of wormhole connections allowed between black holes. Moreover, these superselection sectors are indistinguishable in the dual CFT. Finally, we describe restrictions on the possible superselection sectors associated with the spin-statistics relation and the expectation that black holes lying in distinct asymptotically AdS regions may be approximated by well-separated black holes in a single asymptotically AdS region., 16 pages, 1 figure, v2 references and minor comments added, v3 includes "Note added in Proof"

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., Comment: 26 pg, 10 fig. v2 Existence proof extended to certain black hole spacetimes. (Assuming existence, the other results (i-iv) already applied to black hole spacetimes.) New section about stable vs. unstable maximin surfaces (a critical technical issue), 3 new figures, more commentary and a Discussion. Some proofs corrected to make them more valid, e.g. 4(c). v3 updated refs & footnotes v4 fixed 4(a)

A discrete model of Lorentzian quantum gravity is proposed. The theory is completely background free, containing no reference to absolute space, time, or simultaneity. The states at one slice of time are networks in which each vertex is labelled with two arrows, which point along an adjacent edge, or to the vertex itself. The dynamics is specified by a set of unitary replacement rules, which causally propagate the local degrees of freedom. The inner product between any two states is given by a sum over histories. Assuming it converges (or can be Abel resummed), this inner product is proven to be hermitian and fully gauge-degenerate under spacetime diffeomorphisms. At least for states with a finite past, the inner product is also positive. This allows a Hilbert space of physical states to be constructed., Comment: 38 pages, 9 figures, v3 added to exposition and references, v4 expanded prospects section

Quantum fluctuations of matter fields contribute to the thermal entropy of black holes. For free minimally-coupled scalar and spinor fields, this contribution is precisely the entanglement entropy. For gauge fields, Kabat found an extra negative divergent "contact term" with no known statistical interpretation. We compare this contact term to a similar term that arises for nonminimally-coupled scalar fields. Although both divergences may be interpreted as terms in the Wald entropy, we point out that the contact term for gauge fields comes from a gauge-dependent ambiguity in Wald's formula. Revisiting Kabat's derivation of the contact term, we show that it is sensitive to the treatment of infrared modes. To explore these infrared issues, we consider two-dimensional compact manifolds, such as Euclidean de Sitter space, and show that the contact term arises from an incorrect treatment of zero modes. In a manifestly gauge-invariant reduced phase space quantization, the gauge field contribution to the entropy is positive, finite, and equal to the entanglement entropy., Comment: 27 pages

The generalized second law is proven for semiclassical quantum fields falling across a causal horizon, minimally coupled to general relativity. The proof is much more general than previous proofs in that it permits the quantum fields to be rapidly changing with time, and shows that entropy increases when comparing any slice of the horizon to any earlier slice. The proof requires the existence of an algebra of observables restricted to the horizon, satisfying certain axioms (Determinism, Ultralocality, Local Lorentz Invariance, and Stability). These axioms are explicitly verified in the case of free fields of various spins, as well as 1+1 conformal field theories. The validity of the axioms for other interacting theories is discussed., 44 pages, 1 fig. v3: clarified Sec. 2; signs, factors/notation corrected in Eq. 75-80, 105-107; reflects published version. v4: clearer axioms in Sec. 2.3, fixed compensating factor of 2 errors in Eq. 54,74 etc., and other errors. Results unaffected. v5: fixed typos. v6: replaced faulty 1+1 CFT argument, added note on recent progress

The anomalous conformal transformation law of the generalized entropy is found for dilaton gravity coupled to a 1+1 conformal matter sector with central charges $c = \tilde{c}$. (When $c \ne \tilde{c}$ the generalized entropy is not invariant under local Lorentz boosts.) It is shown that a certain second null derivative of the entropy, $S_\text{gen}" + (6/c)(S_\text{out}')^2$, is primary, and therefore retains its sign under a general conformal transformation. Consequently all conformal vacua have increasing entropy on causal horizons. Alternative definitions of the horizon, including apparent or dynamical horizons, can have decreasing entropy in any dimension $D \ge 2$. This indicates that the generalized second law should be defined using the causal horizon., Comment: 19 pages, 2 figures, v2 significant additions clarifying sections 3, 5, and 6, v3 fixed typo in Eq. (35)

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. An introductory section 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.) A proof of the coarse-grained ordinary second law is given., 46 pages, 8 figures. v2: discussion of global hyperbolicity revised (4.1, 5.2), more comments on AdS. v3: major revisions including change of title. v4: similar to published version, but with corrections to plan of paper (1) and definition of global hyperbolicity (3.2). v5: fixed proof of Thm. 1, changed wording of Thm. 3 & proof of Thm. 4, revised Sec. 5.2, new footnotes

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 boost and null translation symmetry. Possible extensions to more general horizons and matter-gravity couplings are discussed., 16 pages, 2 figures. v2 edited abstract

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 compensate this entropy decrease., Comment: 15 pages, 1 figure, v2 Title change & minor revisions to match published version, v3 fixed accidental deletion of author names

A null line is a complete achronal null geodesic. It is proven that for any quantum fields minimally coupled to semiclassical Einstein gravity, the averaged null energy condition (ANEC) on null lines is a consequence of the generalized second law of thermodynamics for causal horizons. Auxiliary assumptions include CPT and the existence of a suitable renormalization scheme for the generalized entropy. Although the ANEC can be violated on general geodesics in curved spacetimes, as long as the ANEC holds on null lines there exist theorems showing that semiclassical gravity should satisfy positivity of energy, topological censorship, and should not admit closed timelike curves. It is pointed out that these theorems fail once the linearized graviton field is quantized, because then the renormalized shear squared term in the Raychaudhuri equation can be negative. A "shear-inclusive" generalization of the ANEC is proposed to remedy this, and is proven under an additional assumption about perturbations to horizons in classical general relativity., 19 pages, 1 figure. Added 2 paragraphs to end of section 2

Ten attempts to prove the Generalized Second Law of Thermodyanmics (GSL) are described and critiqued. Each proof provides valuable insights which should be useful for constructing future, more complete proofs. Rather than merely summarizing previous research, this review offers new perspectives, and strategies for overcoming limitations of the existing proofs. A long introductory section addresses some choices that must be made in any formulation the GSL: Should one use the Gibbs or the Boltzmann entropy? Should one use the global or the apparent horizon? Is it necessary to assume any entropy bounds? If the area has quantum fluctuations, should the GSL apply to the average area? The definition and implications of the classical, hydrodynamic, semiclassical and full quantum gravity regimes are also discussed. A lack of agreement regarding how to define the "quasi-stationary" regime is addressed by distinguishing it from the "quasi-steady" regime., 60 pages, 2 figures, 1 table. v2: corrected typos and added a footnote to match the published version

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 {\it perpetuum mobile}, but argue that these can be neglected if the black hole mass is sufficiently large., 5 pages, 1 figure; v2. Added reference, and a remark concerning the assumption that the entropy radiated in fields responsible for Lorentz symmetry breaking is negligible