arXiv subjects

Robert M. Wald

Publications and source records attributed to Robert M. Wald.

At least 19 recordsLinked to original sources

arXiv2026

Alternative Boundary Conditions in Asymptotically Anti-de Sitter Spacetimes

Anti-de Sitter (AdS) spacetimes are not globally hyperbolic, so boundary conditions are generally needed for well-defined dynamics. Ishibashi and Wald (IW) determined boundary conditions yielding well-posed dynamics with conserved positive energy for scalar, electromagnetic, and linearized gravitational fields, using nonlocal potentials and individual spherical harmonic modes. In this paper, we determine which IW conditions are local in the primary fields and which are AdS-invariant. In four spacetime dimensions, the standard AdS-invariant boundary conditions set the rescaled magnetic field to zero for electromagnetism and the rescaled magnetic Weyl tensor to zero for Einstein gravity on the conformal boundary. We generalize these standard boundary conditions by setting arbitrary linear combinations of the electric and magnetic fields, or the corresponding Weyl tensors, to zero, and we conjecture that these exhaust the local, AdS-invariant, conservative boundary conditions in AdS$_{1,3}$. Boundary conditions with non-AdS-invariance and in different spacetime dimensions are also studied. The AdS-invariant conditions in AdS$_{1,3}$ extend to Yang-Mills theory and nonlinear gravity in a general Fefferman-Graham setting. We construct phase spaces for these general boundary conditions. The covariant phase spaces for general conditions are naturally associated with Lagrangians containing Pontryagin terms and, in gravity, a Gauss-Bonnet term. In gravity, except for the standard boundary condition, no nontrivial asymptotic gauge symmetries exist and all conserved charges such as ADM mass vanish identically, as in a closed universe. Spacetimes with Killing fields including AdS then exhibit linearization instability in the sense of Fischer and Marsden and Moncrief. For nonstandard boundary conditions, the modified symplectic structure yields new expressions for the entropy formula of black holes.

hep-th↗

arXiv2025

Local Description of Decoherence of Quantum Superpositions by Black Holes and Other Bodies

It was previously shown that if an experimenter, Alice, puts a massive or charged body in a quantum spatial superposition, then the presence of a black hole (or more generally any Killing horizon) will eventually decohere the superposition [arXiv:2205.06279, arXiv:2301.00026, arXiv:2311.11461]. This decoherence was identified as resulting from the radiation of soft photons/gravitons through the horizon, thus suggesting that the global structure of the spacetime is essential for describing the decoherence. In this paper, we show that the decoherence can alternatively be described in terms of the local two-point function of the quantum field within Alice's lab, without any direct reference to the horizon. From this point of view, the decoherence of Alice's superposition in the presence of a black hole arises from the extremely low frequency Hawking quanta present in Alice's lab. We explicitly calculate the decoherence occurring in Schwarzschild spacetime in the Unruh vacuum from the local viewpoint. We then use this viewpoint to elucidate (i) the differences in decoherence effects that would occur in Schwarzschild spacetime in the Boulware and Hartle-Hawking vacua; (ii) the difference in decoherence effects that would occur in Minkowski spacetime filled with a thermal bath as compared with Schwarzschild spacetime; (iii) the lack of decoherence in the spacetime of a static star even though the vacuum state outside the star is similar in many respects to the Boulware vacuum around a black hole; and (iv) the requirements on the degrees of freedom of a material body needed to produce a decoherence effect that mimics that of a black hole.

hep-th↗

arXiv2025

How to Minimize the Decoherence Caused by Black Holes

We consider an experimentalist, Alice, who creates a quantum superposition of a charged or massive body outside of a black hole (or, more generally, in the presence of a Killing horizon). It was previously shown that emission of soft photons/gravitons into the black hole will result in the decoherence of the components of the superposition if it is held open for a sufficiently long span of time. However, at any finite time, $t_c$, during the process, it is not obvious how much decoherence has irrevocably occurred. Equivalently, it is not obvious how much information an observer inside the black hole can extract about Alice's superposition prior to time $t_c$. In this paper, we solve for the optimal experimental protocol to be followed by Alice for $t > t_c$ so as to minimize the decoherence of the components of her superposition. More precisely, given the entangling radiation that has passed through the horizon prior to the cross-section $\mathcal C$ corresponding to the time $t = t_c$ in Alice's lab, we determine the "optimal purification" of this radiation beyond $\mathcal C$ such that the global quantum state of the radiation through the horizon has maximal overlap (quantum fidelity) with the Hartle-Hawking or Unruh vacuum. Due to the intricate low frequency entanglement structure of the quantum field theory vacuum state, we find this optimal purification to be nontrivial. In particular, even if Alice has already "closed" her superposition by bringing the components back together, we find that she can decrease the net decoherence of the components of her superposition somewhat by re-opening it and performing further manipulations.

hep-th↗

arXiv2024

Black Holes Inside and Out 2024: visions for the future of black hole physics

The gravitational physics landscape is evolving rapidly, driven by our ability to study strong-field regions, in particular black holes. Black Holes Inside and Out gathered world experts to discuss the status of the field and prospects ahead. We hope that the ideas and perspectives are a source of inspiration. Structure: Black Hole Evaporation - 50 Years by William Unruh The Stability Problem for Extremal Black Holes by Mihalis Dafermos The Entropy of Black Holes by Robert M. Wald The Non-linear Regime of Gravity by Luis Lehner Black Holes Galore in D > 4 by Roberto Emparan Same as Ever: Looking for (In)variants in the Black Holes Landscape by Carlos A. R. Herdeiro Black Holes, Cauchy Horizons, and Mass Inflation by Matt Visser The Backreaction Problem for Black Holes in Semiclassical Gravity by Adrian del Rio Black Holes Beyond General Relativity by Enrico Barausse and Jutta Kunz Black Holes as Laboratories: Searching for Ultralight Fields by Richard Brito Primordial Black Holes from Inflation by Misao Sasaki Tests of General Relativity with Future Detectors by Emanuele Berti Black Holes as Laboratories: Tests of General Relativity by Ruth Gregory and Samaya Nissanke Simulating Black Hole Imposters by Frans Pretorius Black Hole Spectroscopy: Status Report by Gregorio Carullo VLBI as a Precision Strong Gravity Instrument by Paul Tiede Testing the nature of compact objects and the black hole paradigm by Mariafelicia De Laurentis and Paolo Pani Some Thoughts about Black Holes in Asymptotic Safety by Alessia Platania Black Hole Evaporation in Loop Quantum Gravity by Abhay Ashtekar How the Black Hole Puzzles are Resolved in String Theory by Samir D. Mathur Quantum Black Holes: From Regularization to Information Paradoxes by Niayesh Afshordi and Stefano Liberati

gr-qc↗

arXiv2024

The Entropy of Dynamical Black Holes

We propose a new formula for the entropy of a dynamical black hole$-$valid to leading order for perturbations off of a stationary black hole background$-$in an arbitrary classical diffeomorphism covariant Lagrangian theory of gravity in $n$ dimensions. In stationary eras, this formula agrees with the usual Noether charge formula, but in nonstationary eras, we obtain a nontrivial correction term. In general relativity, our formula for the entropy of a dynamical black hole is $1/4$ of the horizon area plus a term involving the integral of the expansion of the null generators of the horizon, which we show is $1/4$ of the area of the apparent horizon to leading order. Our formula for entropy in a general theory of gravity obeys a "local physical process version" of the first law of black hole thermodynamics. For first order perturbations sourced by external matter that satisfies the null energy condition, our entropy obeys the second law of black hole thermodynamics. For vacuum perturbations, the second law is obeyed at leading order if and only if the "modified canonical energy flux" is positive (as is the case in general relativity but presumably would not hold in general theories). We obtain a general relationship between our formula for the entropy of a dynamical black hole and a formula proposed independently by Dong and by Wall. We then consider the generalized second law in semiclassical gravity for first order perturbations of a stationary black hole. We show that the validity of the quantum null energy condition (QNEC) on a Killing horizon is equivalent to the generalized second law using our notion of black hole entropy but using a modified notion of von Neumann entropy for matter. On the other hand, the generalized second law for the Dong-Wall entropy is equivalent to an integrated version of QNEC, using the unmodified von Neumann entropy for the entropy of matter.

hep-th↗

arXiv2023

The Operator Product Expansion in Quantum Field Theory

Operator product expansions (OPEs) in quantum field theory (QFT) provide an asymptotic relation between products of local fields defined at points $x_1, \dots, x_n$ and local fields at point $y$ in the limit $x_1, \dots, x_n \to y$. They thereby capture in a precise way the singular behavior of products of quantum fields at a point as well as their ``finite trends.'' In this article, we shall review the fundamental properties of OPEs and their role in the formulation of interacting QFT in curved spacetime, the ``flow relations'' in coupling parameters satisfied by the OPE coefficients, the role of OPEs in conformal field theories, and the manner in which general theorems -- specifically, the PCT theorem -- can be formulated using OPEs in a curved spacetime setting.

hep-th↗

arXiv2023

Killing Horizons Decohere Quantum Superpositions

We recently showed that if a massive (or charged) body is put in a quantum spatial superposition, the mere presence of a black hole in its vicinity will eventually decohere the superposition. In this paper we show that, more generally, decoherence of stationary superpositions will occur in any spacetime with a Killing horizon. This occurs because, in effect, the long-range field of the body is registered on the Killing horizon which, we show, necessitates a flux of "soft horizon gravitons/photons" through the horizon. The Killing horizon thereby harvests "which path" information of quantum superpositions and will decohere any quantum superposition in a finite time. It is particularly instructive to analyze the case of a uniformly accelerating body in a quantum superposition in flat spacetime. As we show, from the Rindler perspective the superposition is decohered by "soft gravitons/photons" that propagate through the Rindler horizon with negligible (Rindler) energy. We show that this decoherence effect is distinct from--and larger than--the decoherence resulting from the presence of Unruh radiation. We further show that from the inertial perspective, the decoherence is due to the radiation of high frequency (inertial) gravitons/photons to null infinity. (The notion of gravitons/photons that propagate through the Rindler horizon is the same notion as that of gravitons/photons that propagate to null infinity.) We also analyze the decoherence of a spatial superposition due to the presence of a cosmological horizon in de Sitter spacetime. We provide estimates of the decoherence time for such quantum superpositions in both the Rindler and cosmological cases. Although we explicitly treat the case of spacetime dimension $d=4$, our analysis applies to any dimension $d \geq 4$.

hep-th↗

arXiv2023

Gravitationally Mediated Entanglement: Newtonian Field vs. Gravitons

We argue that if the Newtonian gravitational field of a body can mediate entanglement with another body, then it should also be possible for the body producing the Newtonian field to entangle directly with on-shell gravitons. Our arguments are made by revisiting a gedankenexperiment previously analyzed by Belenchia et al., which showed that a quantum superposition of a massive body requires both quantized gravitational radiation and local vacuum fluctuations of the spacetime metric in order to avoid contradictions with complementarity and causality. We provide a precise and rigorous description of the entanglement and decoherence effects occurring in this gedankenexperiment, thereby significantly improving upon the back-of-the-envelope estimates given in the analysis of Belenchia et al. and also showing that their conclusions are valid in much more general circumstances. As a by-product of our analysis, we show that under the protocols of the gedankenexperiment, there is no clear distinction between entanglement mediated by the Newtonian gravitational field of a body and entanglement mediated by on-shell gravitons emitted by the body. This suggests that Newtonian entanglement implies the existence of graviton entanglement and supports the view that the experimental discovery of Newtonian entanglement may be viewed as implying the existence of the graviton.

quant-ph↗

arXiv2022

Cross-Section Continuity of Definitions of Angular Momentum

We introduce a notion of "cross-section continuity" as a criterion for the viability of definitions of angular momentum, $J$, at null infinity: If a sequence of cross-sections, ${\mathcal C}_n$, of null infinity converges uniformly to a cross-section ${\mathcal C}$, then the angular momentum, $J_n$, on ${\mathcal C}_n$ should converge to the angular momentum, $J$, on ${\mathcal C}$. The Dray-Streubel (DS) definition of angular momentum automatically satisfies this criterion by virtue of the existence of a well defined flux associated with this definition. However, we show that the one-parameter modification of the DS definition proposed by Compere and Nichols (CN) -- which encompasses numerous other alternative definitions -- does not satisfy cross-section continuity. On the other hand, we prove that the Chen-Wang-Yau (CWY) definition does satisfy the cross-section continuity criterion.

gr-qc↗

arXiv2022

Black Holes Decohere Quantum Superpositions

We show that if a massive body is put in a quantum superposition of spatially separated states, the mere presence of a black hole in the vicinity of the body will eventually destroy the coherence of the superposition. This occurs because, in effect, the gravitational field of the body radiates soft gravitons into the black hole, allowing the black hole to acquire "which path" information about the superposition. A similar effect occurs for quantum superpositions of electrically charged bodies. We provide estimates of the decoherence time for such quantum superpositions. We believe that the fact that a black hole will eventually decohere any quantum superposition may be of fundamental significance for our understanding of the nature of black holes in a quantum theory of gravity.

hep-th↗

arXiv2022

Local and Covariant Flow Relations for OPE Coefficients in Lorentzian Spacetimes

For Euclidean quantum field theories, Holland and Hollands have shown operator product expansion (OPE) coefficients satisfy "flow equations": For interaction parameter $λ$, the partial derivative of any OPE coefficient with respect to $λ$ is given by an integral over Euclidean space of a sum of products of other OPE coefficients. In this paper, we generalize these results for flat Euclidean space to curved Lorentzian spacetimes in the context of the solvable "toy model" of massive Klein-Gordon scalar field theory, with $m^2$ viewed as the "self-interaction parameter". Even in Minkowski spacetime, a serious difficulty arises from the fact that all integrals must be taken over a compact spacetime region to ensure convergence but any integration cutoff necessarily breaks Lorentz covariance. We show how covariant flow relations can be obtained by adding compensating "counterterms" in a manner similar to that of the Epstein-Glaser renormalization scheme. We also show how to eliminate dependence on the "infrared-cutoff scale" $L$, thereby yielding flow relations compatible with almost homogeneous scaling of the fields. In curved spacetime, the spacetime integration will cause the OPE coefficients to depend non-locally on the spacetime metric, in violation of the requirement that quantum fields should depend locally and covariantly on the metric. We show how this potentially serious difficulty can be overcome by replacing the metric with a suitable local polynomial approximation about the OPE expansion point. We thereby obtain local and covariant flow relations for the OPE coefficients of Klein-Gordon theory in curved Lorentzian spacetimes. In an appendix, we develop an algorithm for constructing local and covariant flow relations beyond our "toy model" based on the associativity properties of OPE coefficients, and we apply our method to $λϕ^4$-theory.

gr-qc↗

arXiv2022

Infrared Finite Scattering Theory in Quantum Field Theory and Quantum Gravity

Infrared (IR) divergences arise in scattering theory with massless fields and are manifestations of the memory effect. There is nothing singular about states with memory, but they do not lie in the standard Fock space. IR divergences are artifacts of trying to represent states with memory in the standard Fock space. For collider physics, one can impose an IR cutoff and calculate inclusive quantities. But, this approach cannot treat memory as a quantum observable and is highly unsatisfactory if one views the S-matrix as fundamental in QFT and quantum gravity, since the S-matrix is undefined. For a well-defined S-matrix, it is necessary to define in/out Hilbert spaces with memory. Such a construction was given by Faddeev and Kulish (FK) for QED. Their construction "dresses" momentum states of the charged particles by pairing them with memory states of the electromagnetic field to produce states of vanishing large gauge charges at spatial infinity. However, in massless QED, due to collinear divergences, the "dressing" has an infinite energy flux so these states are unphysical. In Yang-Mills theory the "soft particles" used for dressing also contribute to the current flux, invalidating the FK procedure. In quantum gravity, the analogous FK construction would attempt to produce a Hilbert space of eigenstates of supertranslation charges at spatial infinity. However, we prove that there are no eigenstates of supertranslation charges except the vacuum. Thus, the FK construction fails in quantum gravity. We investigate some alternatives to FK constructions but find that these also do not work. We believe that to treat scattering at a fundamental level in quantum gravity - as well as in massless QED and YM theory - it is necessary to take an algebraic viewpoint rather than shoehorn the in/out states into some fixed Hilbert space. We outline the framework of such an IR finite scattering theory.

hep-th↗

arXiv2021

Blázquez-Salcedo-Knoll-Radu Wormholes Are Not Solutions to the Einstein-Dirac-Maxwell Equations

Recently, Blázquez-Salcedo, Knoll, and Radu (BSKR) have given a class of static, spherically symmetric, traversable wormhole spacetimes with Dirac and Maxwell fields. The BSKR wormholes are obtained by joining a classical solution to the Einstein-Dirac-Maxwell (EDM) equations on the "up" side of the wormhole ($r \geq 0$) to a corresponding solution on the "down" side of the wormhole ($r \leq 0$). However, it can be seen that the BSKR metric fails to be $C^3$ on the wormhole throat at $r=0$. We prove that if the matching were done in such a way that the resulting spacetime metric, Dirac field, and Maxwell field composed a solution to the EDM equations in a neighborhood of $r=0$, then all of the fields would be smooth at $r=0$ in a suitable gauge. Thus, the BSKR wormholes cannot be solutions to the EDM equations. The failure of the BSKR wormholes to solve the EDM equations arises both from the failure of the Maxwell field to satisfy the required matching conditions (which implies the presence of an additional shell of charged matter at $r=0$) and, more significantly, from the failure of the Dirac field to satisfy required matching conditions (which implies the presence of a spurious source term for the Dirac field at $r=0$.

gr-qc↗

arXiv2021

Phase effects from strong gravitational lensing of gravitational waves

Assessing the probability that two or more gravitational waves (GWs) are lensed images of the same source requires an understanding of the image properties, including their relative phase shifts in strong lensing (SL). For non-precessing, circular binaries dominated by quadrupole radiation these phase shifts are degenerate with either a shift in the coalescence phase or a detector and inclination dependent shift in the orientation angle. This degeneracy is broken by the presence of higher harmonic modes with $|m|\ne 2$ in the former and $|m| \ne l$ in the latter. Precession or eccentricity will also break this degeneracy. This implies that lensed GWs will not necessarily be consistent with (unlensed) predictions from general relativity (GR). Therefore, unlike EM lensing, GW SL can lead to images with an observable modified phase evolution. However, for a wide parameter space, the lensed waveform is similar enough to an unlensed waveform that detection pipelines will still find it. For present detectors, templates with a shifted detector-dependent orientation angle have SNR differences of less than $1\%$ for mass ratios up to 0.1, and less than $5\%$ for precession parameters up to 0.5 and eccentricities up to 0.4 at 20Hz. The mismatch is lower than $10\%$ with the alternative detector-independent coalescence phase shift. Nonetheless, for a loud enough source, even with one image it may be possible to directly identify it as a SL image from its non-GR waveform. In more extreme cases, lensing may lead to considerable distortions, and the lensed GWs may be undetected with current searches. Nevertheless, an exact template with a phase shift in Fourier space can always be constructed to fit any lensed GW. We conclude that an optimal search strategy would incorporate phase information in all stages, with an exact treatment in the final assessment of the probability of multiple lensed events.

gr-qc↗

arXiv2020

Quantum Instability of the Cauchy Horizon in Reissner-Nordström-deSitter Spacetime

In classical General Relativity, the values of fields on spacetime are uniquely determined by their values at an initial time within the domain of dependence of this initial data surface. However, it may occur that the spacetime under consideration extends beyond this domain of dependence, and fields, therefore, are not entirely determined by their initial data. This occurs, for example, in the well-known (maximally) extended Reissner-Nordström or Reissner-Nordström-deSitter (RNdS) spacetimes. The boundary of the region determined by the initial data is called the "Cauchy horizon." It is located inside the black hole in these spacetimes. The strong cosmic censorship conjecture asserts that the Cauchy horizon does not, in fact, exist in practice because the slightest perturbation (of the metric itself or the matter fields) will become singular there in a sufficiently catastrophic way that solutions cannot be extended beyond the Cauchy horizon. Thus, if strong cosmic censorship holds, the Cauchy horizon will be converted into a "final singularity," and determinism will hold. Recently, however, it has been found that, classically this is not the case in RNdS spacetimes in a certain range of mass, charge, and cosmological constant. In this paper, we consider a quantum scalar field in RNdS spacetime and show that quantum theory comes to the rescue of strong cosmic censorship. We find that for any state that is nonsingular (i.e., Hadamard) within the domain of dependence, the expected stress-tensor blows up with affine parameter, $V$, along a radial null geodesic transverse to the Cauchy horizon as $T_{VV} \sim C/V^2$ with $C$ independent of the state and $C \neq 0$ generically in RNdS spacetimes. This divergence is stronger than in the classical theory and should be sufficient to convert the Cauchy horizon into a strong curvature singularity.

gr-qc↗

arXiv2020

Black Hole Memory

The memory effect at null infinity, $\mathcal{I}^+$, can be defined in terms of the permanent relative displacement of test particles (at leading order in $1/r$) resulting from the passage of a burst of gravitational radiation. In $D=4$ spacetime dimensions, the memory effect can be characterized by the supertranslation relating the "good cuts" of $\mathcal{I}^+$ in the stationary eras at early and late retarded times. It also can be characterized in terms of charges and fluxes associated with supertranslations. Black hole event horizons are in many ways analogous to $\mathcal{I}^+$. We consider here analogous definitions of memory for a black hole, assuming that the black hole is approximately stationary at early and late advanced times, so that its event horizon is described by a Killing horizon (assumed nonextremal) at early and late times. We give prescriptions for defining preferred foliations of nonextremal Killing horizons. We give a definition of the memory tensor for a black hole in terms of the "permanent relative displacement" of the null geodesic generators of the event horizon between the early and late time stationary eras. We show that preferred foliations of the event horizon in the early and late time eras are related by a Chandrasekaran-Flanagan-Prabhu (CFP) supertranslation. However, we find that the memory tensor for a black hole horizon does not appear to be related to the CFP symmetries or their charges and fluxes in a manner similar to that occurring at $\mathcal{I}^+$.

gr-qc↗

arXiv2019

Spin Self-Force

We consider the motion of charged and spinning bodies on the symmetry axis of a non-extremal Kerr-Newman black hole. If one treats the body as a test point particle of mass, $m$, charge $q$, and spin $S$, then by dropping the body into the black hole from sufficiently near the horizon, the first order area increase, $δA$, of the black hole can be made arbitrarily small, i.e., the process can be done in a ``reversible'' manner. At second order, there may be effects on the energy delivered to the black hole---quadratic in $q$ and $S$---resulting from (i) the finite size of the body and (ii) self-force corrections to the energy. Sorce and Wald have calculated these effects for a charged, non-spinning body on the symmetry axis of an uncharged Kerr black hole. We consider the generalization of this process for a charged and spinning body on the symmetry axis of a Kerr-Newman black hole, where the self-force effects have not been calculated. A spinning body (with negligible extent along the spin axis) must have a mass quadrupole moment $\gtrsim S^2/m$, so at quadratic order in the spin, we must take into account quadrupole effects on the motion. After taking into account all such finite size effects, we find that the condition $δ^2 A \geq 0$ yields a nontrivial lower bound on the self-force energy, $E_{SF}$, at the horizon. In particular, for an uncharged, spinning body on the axis of a Kerr black hole of mass $M$, the net contribution of spin self-force to the energy of the body at the horizon is $E_{SF} \geq S^2/8M^3$, corresponding to an overall repulsive spin self-force. A lower bound for the self-force energy, $E_{SF}$, for a body with both charge and spin at the horizon of a Kerr-Newman black hole is given. This lower bound will be the correct formula for $E_{SF}$ if the dropping process can be done reversibly to second order.

gr-qc↗

arXiv2019

The Particle and Energy Cost of Entanglement of Hawking Radiation with the Final Vacuum State

A semiclassical analysis shows that in the process of black hole formation and evaporation, an initial pure state will evolve to a mixed state, i.e., information will be lost. One way of avoiding this conclusion without invoking drastic modifications of the local laws of physics in a low curvature regime would be for the information to be restored at the very end of the evaporation process. It is normally envisioned that this would require a final burst of particles entangled with the early time Hawking radiation. This would imply the emission of an extremely large number of particles from an object of Planck size and mass and would appear to be blatantly ruled out by energy considerations. However, Hotta, Schutzhold, and Unruh have analyzed a $(1+1)$-dimensional moving mirror analog of the Hawking process and have found that, in this model, information is restored via entanglement of the early time Hawking radiation with vacuum fluctuations in the spacetime region to the future of the event where the mirror returns to inertial motion. We analyze their model here and give a precise formulation of this entanglement by introducing the notion of "Milne particles." We then analyze the inertial particle and energy cost of such an entanglement of Hawking radiation with vacuum fluctuations. We show that that, in fact, the entanglement of early time Hawking radiation with vacuum fluctuations requires the emission of at least as many late time inertial particles as Hawking particles. Although the energy cost can be made small in the $(1+1)$-dimensional mirror system, this should not be the case for the $(3+1)$-dimensional evaporating black hole system. Thus, vacuum entanglement has the same difficulties as the more usual burst scenarios for attempting to avoid information loss.

gr-qc↗