arXiv subjects

Donald Marolf

Publications and source records attributed to Donald Marolf.

At least 37 records · Page 2Linked to original sources

arXiv2024

Null states and time evolution in a toy model of black hole dynamics

Spacetime wormholes can provide non-perturbative contributions to the gravitational path integral that make the actual number of states $e^S$ in a gravitational system much smaller than the number of states $e^{S_{\mathrm{p}}}$ predicted by perturbative semiclassical effective field theory. The effects on the physics of the system are naturally profound in contexts in which the perturbative description actively involves $N = O(e^S)$ of the possible $e^{S_{\mathrm{p}}}$ perturbative states; e.g., in late stages of black hole evaporation. Such contexts are typically associated with the existence of non-trivial quantum extremal surfaces. However, by forcing a simple topological gravity model to evolve in time, we find that such effects can also have large impact for $N\ll e^S$ (in which case no quantum extremal surfaces can arise). In particular, even for small $N$, the insertion of generic operators into the path integral can cause the non-perturbative time evolution to differ dramatically from perturbative expectations. On the other hand, this discrepancy is small for the special case where the inserted operators are non-trivial only in a subspace of dimension $D \ll e^S$. We thus study this latter case in detail. We also discuss potential implications for more realistic gravitational systems.

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arXiv2024

Constrained HRT Surfaces and their Entropic Interpretation

Consider two boundary subregions $A$ and $B$ that lie in a common boundary Cauchy surface, and consider also the associated HRT surface $γ_B$ for $B$. In that context, the constrained HRT surface $γ_{A:B}$ can be defined as the codimension-2 bulk surface anchored to $A$ that is obtained by a maximin construction restricted to Cauchy slices containing $γ_B$. As a result, $γ_{A:B}$ is the union of two pieces, $γ^B_{A:B}$ and $γ^{\bar B}_{A:B}$ lying respectively in the entanglement wedges of $B$ and its complement $\bar B$. Unlike the area $\mathcal{A}\left(γ_A\right)$ of the HRT surface $γ_A$, at least in the semiclassical limit, the area $\mathcal{A}\left(γ_{A:B}\right)$ of $γ_{A:B}$ commutes with the area $\mathcal{A}\left(γ_B\right)$ of $γ_B$. To study the entropic interpretation of $\mathcal{A}\left(γ_{A:B}\right)$, we analyze the Rényi entropies of subregion $A$ in a fixed-area state of subregion $B$. We use the gravitational path integral to show that the $n\approx1$ Rényi entropies are then computed by minimizing $\mathcal{A}\left(γ_A\right)$ over spacetimes defined by a boost angle conjugate to $\mathcal{A}\left(γ_B\right)$. In the case where the pieces $γ^B_{A:B}$ and $γ^{\bar B}_{A:B}$ intersect at a constant boost angle, a geometric argument shows that the $n\approx1$ Rényi entropy is then given by $\frac{\mathcal{A}(γ_{A:B})}{4G}$. We discuss how the $n\approx1$ Rényi entropy differs from the von Neumann entropy due to a lack of commutativity of the $n\to1$ and $G\to0$ limits. We also discuss how the behaviour changes as a function of the width of the fixed-area state. Our results are relevant to some of the issues associated with attempts to use standard random tensor networks to describe time dependent geometries.

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arXiv2024

Link-area commutators in AdS${}_3$ area-networks

Random tensor networks (RTNs) have proved to be fruitful tools for modelling the AdS/CFT correspondence. Due to their flat entanglement spectra, when discussing a given boundary region $R$ and its complement $\bar R$, standard RTNs are most analogous to fixed-area states of the bulk quantum gravity theory, in which quantum fluctuations have been suppressed for the area of the corresponding HRT surface. However, such RTNs have flat entanglement spectra for all choices of $R, \bar R,$ while quantum fluctuations of multiple HRT-areas can be suppressed only when the corresponding HRT-area operators mutually commute. We probe the severity of such obstructions in pure AdS$_3$ Einstein-Hilbert gravity by constructing networks whose links are codimension-2 extremal-surfaces and by explicitly computing semiclassical commutators of the associated link-areas. Since $d=3,$ codimension-2 extremal-surfaces are geodesics, and codimension-2 `areas' are lengths. We find a simple 4-link network defined by an HRT surface and a Chen-Dong-Lewkowycz-Qi constrained HRT surface for which all link-areas commute. However, the algebra generated by the link-areas of more general networks tends to be non-Abelian. One such non-Abelian example is associated with entanglement-wedge cross sections and may be of more general interest.

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arXiv2023

Stability of saddles and choices of contour in the Euclidean path integral for linearized gravity: Dependence on the DeWitt Parameter

Due to the conformal factor problem, the definition of the Euclidean gravitational path integral requires a non-trivial choice of contour. The present work examines a generalization of a recently proposed rule-of-thumb \cite{Marolf:2022ntb} for selecting this contour at quadratic order about a saddle. The original proposal depended on the choice of an indefinite-signature metric on the space of perturbations, which was taken to be a DeWitt metric with parameter $α=-1$. This choice was made to match previous results, but was otherwise admittedly {\it ad hoc}. To begin to investigate the physics associated with the choice of such a metric, we now explore contours defined using analogous prescriptions for $α\neq -1$. We study such contours for Euclidean gravity linearized about AdS-Schwarzschild black holes in reflecting cavities with thermal (canonical ensemble) boundary conditions, and we compare path-integral stability of the associated saddles with thermodynamic stability of the classical spacetimes. While the contour generally depends on the choice of DeWitt parameter $α$, the precise agreement between these two notions of stability found at $α=-1$ continues to hold over the finite interval $(-2,-2/d)$, where $d$ is the dimension of the bulk spacetime. This agreement manifestly fails for $α> -2/d$ when the DeWitt metric becomes positive definite. However, we also find dramatic failures for $α< -2$ that correlate with breakdowns of the de Donder-like gauge condition defined by $α$, and at which the relevant fluctuation operator fails to be diagonalizable. This provides criteria that may be useful in predicting metrics on the space of perturbations that give physically-useful contours in more general settings. Along the way, we also identify an interesting error in \cite{Marolf:2022ntb}, though we show this error to be harmless.

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arXiv2023

A trace inequality for Euclidean gravitational path integrals (and a new positive action conjecture)

The AdS/CFT correspondence states that certain conformal field theories are equivalent to string theories in a higher-dimensional anti-de Sitter space. One aspect of the correspondence is an equivalence of density matrices or, if one ignores normalizations, of positive operators. On the CFT side of the correspondence, any two positive operators $A,B$ will satisfy the trace inequality $\operatorname{Tr}(AB) \leq \operatorname{Tr}(A) \operatorname{Tr}(B)$. This relation holds on any Hilbert space ${\cal H}$ and is deeply associated with the fact that the algebra $B({\cal H})$ of bounded operators on ${\cal H}$ is a type I von Neumann factor. Holographic bulk theories must thus satisfy a corresponding condition, which we investigate below. In particular, we argue that the Euclidean gravitational path integral respects this inequality at all orders in the semi-classical expansion and with arbitrary higher-derivative corrections. The argument relies on a conjectured property of the classical gravitational action, which in particular implies a positive action conjecture for quantum gravity wavefunctions. We prove this conjecture for Jackiw-Teitelboim gravity and we also motivate it for more general theories.

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arXiv2022

The action of HRT-areas as operators in semiclassical gravity

We study the action of Hubeny-Rangamani-Takayanagi (HRT) area operators on the covariant phase space of classical solutions. It has been previously proposed that this action generates a transformation which, roughly speaking, boosts the entanglement wedge on one side of the HRT surface relative to the entanglement wedge on the other side. We give a sharp argument for a precise result of this form in a general theory of Einstein-Hilbert gravity minimally coupled to matter, taking appropriate care with asymptotically Anti-de Sitter (AdS) boundary conditions. The result agrees with direct computations of commutators involving HRT areas in pure 2+1 dimensional Einstein-Hilbert gravity on spacetimes asymptotic to planar AdS. We also clarify the sense in which this transformation is singular in the deep UV when the HRT-surface is anchored to an asymptotically AdS boundary.

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arXiv2022

Gravitational thermodynamics without the conformal factor problem: Partition functions and Euclidean saddles from Lorentzian Path Integrals

Thermal partition functions for gravitational systems have traditionally been studied using Euclidean path integrals. But in Euclidean signature the gravitational action suffers from the conformal factor problem, which renders the action unbounded below. This makes it difficult to take the Euclidean formulation as fundamental. However, despite their familiar association with periodic imaginary time, thermal gravitational partition functions can also be described by real-time path integrals over contours defined by real Lorentzian metrics. The one caveat is that we should allow certain codimension-2 singularities analogous to the familiar Euclidean conical singularities. With this understanding, we show that the usual Euclidean-signature black holes (or their complex rotating analogues) define saddle points for the real-time path integrals that compute our partition functions. Furthermore, when the black holes have positive specific heat, we provide evidence that a codimension-2 subcontour of our real Lorentz-signature contour of integration can be deformed so as to show that these black holes saddles contribute with non-zero weight to the semiclassical limit, and that the same is then true of the remaining two integrals.

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arXiv2022

The Spacetime Geometry of Fixed-Area States in Gravitational Systems

The concept of fixed-area states has proven useful for recent studies of quantum gravity, especially in connection with gravitational holography. We explore the Lorentz-signature spacetime geometry intrinsic to such fixed-area states in this paper. This contrasts with previous treatments which focused instead on Euclidean-signature saddles for path integrals that prepare such states. We analyze general features of fixed-area state geometries and construct explicit examples. The spacetime metrics are real at real times and have no conical singularities. With enough symmetry the classical metrics are in fact smooth, though more generally their curvatures feature power-law divergences along null congruences launched orthogonally from the fixed-area surface. While we argue that such divergences are not problematic at the classical level, quantum fields in fixed-area states feature stronger divergences. At the quantum level we thus expect fixed-area states to be well-defined only when the fixed-area surface is appropriately smeared.

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arXiv2022

Stability of the microcanonical ensemble in Euclidean Quantum Gravity

This work resolves a longstanding tension between the physically-expected stability of the microcanonical ensemble for gravitating systems and the fact that the known negative mode of the asymptotically flat Schwarzschild black hole decays too rapidly at infinity to affect the ADM energy boundary term at infinity. The key to our study is that we fix an appropriate {\it off-shell} notion of energy, which we obtain by constructing the microcanonical partition function as an integral transform of the canonical partition function. After applying the rule-of-thumb for Wick rotations from our recent companion paper to deal with the conformal mode problem of Euclidean gravity, we find a positive definite action for linear perturbations about any Euclidean Schwarzchild (-AdS) black hole. Most of our work is done in a cavity with reflecting boundary conditions, but the cavity wall can be removed by taking an appropriate limit.

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arXiv2022

The Canonical Ensemble Reloaded: The Complex-Stability of Euclidean quantum gravity for Black Holes in a Box

We revisit the stability of black hole saddles for the Euclidean path integral describing the canonical partition function $Z(β)$ for gravity inside a spherical reflecting cavity. The boundary condition at the cavity wall couples the transverse-traceless (TT) and pure-trace modes that are traditionally used to describe fluctuations about Euclidean Schwarzschild black holes in infinite-volume asymptotically flat and asymototically AdS spacetimes. This coupling obstructs the familiar Gibbons-Hawking-Perry treatment of the conformal factor problem, as Wick rotation of the pure-trace modes would require that the TT modes be rotated as well. The coupling also leads to complex eigenvalues for the Łoperator. We nevertheless find that the Łoperator can be diagonalized in the space of coupled modes. This observation allows the eigenmodes to define a natural generalization of the pure-trace Wick-rotation recipe used in infinite volume, with the result that a mode with eigenvalue $λ$ is stable when ${\rm Re}\,λ> 0$. In any cavity, and with any cosmological constant $Λ\le 0$, we show this recipe to reproduce the expectation from black hole thermodynamics that large Euclidean black holes define stable saddles while the saddles defined by small Euclidean black holes are unstable.

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arXiv2022

Baby Universes and Worldline Field Theories

The quantum gravity path integral involves a sum over topologies that invites comparisons to worldsheet string theory and to Feynman diagrams of quantum field theory. However, the latter are naturally associated with the non-abelian algebra of quantum fields, while the former has been argued to define an abelian algebra of superselected observables associated with partition-function-like quantities at an asymptotic boundary. We resolve this apparent tension by pointing out a variety of discrete choices that must be made in constructing a Hilbert space from such path integrals, and arguing that the natural choices for quantum gravity differ from those used to construct QFTs. We focus on one-dimensional models of quantum gravity in order to make direct comparisons with worldline QFT.

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arXiv2021

Microcanonical Path Integrals and the Holography of small Black Hole Interiors

We use a microcanonical path integral closely related to that introduced by Brown and York in 1992 to add new entries to the AdS/CFT dictionary concerning the interiors of small black holes. Stationary points of such path integrals are also stationary points of more standard canonical-type path integrals with fixed boundary metric, but the condition for dominance is now maximizing Hubeny-Rangamani-Takayanagi entropy at fixed energy. As a result, such path integrals can bring to the fore saddles that fail to dominate in more familiar contexts. We use this feature to argue that the standard Kruskal-like two-sided extension of small AdS black holes with energy $E_0$ is dual to a microcanonical version of the thermofield double state for AdS black holes that maximize the microcanonical bulk entropy at this energy. We also comment on entanglement in such states and on quantum effects that become large when the energy-width of the microcanonical ensemble is sufficiently small.

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arXiv2021

Real-time gravitational replicas: Low dimensional examples

We continue the study of real-time replica wormholes initiated in arXiv:2012.00828. Previously, we had discussed the general principles and had outlined a variational principle for obtaining stationary points of the real-time gravitational path integral. In the current work we present several explicit examples in low-dimensional gravitational theories where the dynamics is amenable to analytic computation. We demonstrate the computation of Rényi entropies in the cases of JT gravity and for holographic two-dimensional CFTs (using the dual gravitational dynamics). In particular, we explain how to obtain the large central charge result for subregions comprising of disjoint intervals directly from the real-time path integral.

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arXiv2021

Time-independence of gravitational Rényi entropies and unitarity in quantum gravity

The Hubeny-Rangamani-Takayanagi surface $γ_{HRT}$ computing the entropy $S(D)$ of a domain of dependence $D$ on an asymptotically AdS boundary is known to be causally inaccessible from $D$. We generalize this gravitational result to higher replica numbers $n >1$ by considering the replica-invariant surfaces (aka `splitting surfaces') $\boldsymbol{\mathbfγ}$ of real-time replica-wormhole saddle-points computing Rényi entropies $S_n(D)$ and showing that there is a sense in which $D$ must again be causally inaccessible from $\boldsymbol{\mathbfγ}$ when the saddle preserves both replica and conjugation symmetry. This property turns out to imply the $S_n(D)$ to be independent of any choice of any Cauchy surface $Σ_D$ for $D$, and also that the $S_n(D)$ are independent of the choice of boundary sources within $D$. This is a key hallmark of unitary evolution in any dual field theory. Furthermore, from the bulk point of view it adds to the evidence that time evolution of asymptotic observables in quantum gravity is implemented by a unitary operator in each baby universe superselection sector. Though we focus here on pure Einstein-Hilbert gravity and its Kaluza-Klein reductions, we expect the argument to extend to any two-derivative theory who satisfies the null convergence condition. We consider both classical saddles and the effect of back-reaction from quantum corrections.

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arXiv2021

Real-time gravitational replicas: Formalism and a variational principle

This work is the first step in a two-part investigation of real-time replica wormholes. Here we study the associated real-time gravitational path integral and construct the variational principle that will define its saddle-points. We also describe the general structure of the resulting real-time replica wormhole saddles, setting the stage for construction of explicit examples. These saddles necessarily involve complex metrics, and thus are accessed by deforming the original real contour of integration. However, the construction of these saddles need not rely on analytic continuation, and our formulation can be used even in the presence of non-analytic boundary-sources. Furthermore, at least for replica- and CPT-symmetric saddles we show that the metrics may be taken to be real in regions spacelike separated from a so-called `splitting surface'. This feature is an important hallmark of unitarity in a field theory dual.

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arXiv2021

The Page curve and baby universes

Black hole thermodynamics suggests that, in order to describe the physics of distant observers, one may model a black hole as a standard quantum system with density of states set by the Bekenstein-Hawking entropy $S_\mathrm{BH}$. This idea has long been considered to be in strong tension with Hawking's prediction that radiation from black holes is nearly thermal, and with low-energy gravity more generally. But the past two years have shown that low-energy gravity does offer a self-consistent description of black hole evaporation consistent with the above idea, and which in particular reproduces the famous Page curve. We provide a brief overview of this new paradigm, focusing on Lorentz-signature asymptotically-flat spacetimes, and emphasising operationally-defined observables that probe the entropy of Hawking radiation.

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arXiv2021

Traversability of Multi-Boundary Wormholes

We generalize the Gao-Jafferis-Wall construction of traversable two-sided wormholes to multi-boundary wormholes. In our construction, we take the background spacetime to be multi-boundary black holes in AdS$_3$. We work in the hot limit where the dual CFT state in certain regions locally resembles the thermofield double state. Furthermore, in these regions, the hot limit makes the causal shadow exponentially small. Based on these two features of the hot limit, and with the three-boundary wormhole as our main example, we show that traversability between any two asymptotic regions in a multi-boundary wormhole can be triggered using a double-trace deformation. In particular, the two boundary regions need not have the same temperature and angular momentum. We discuss the non-trivial angular dependence of traversability in our construction, as well as the effect of the causal shadow region.

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arXiv2021

Multi-mouth Traversable Wormholes

We describe the construction of traversable wormholes with multiple mouths in four spacetime dimensions and discuss associated quantum entanglement. Our wormholes may be traversed between any pair of mouths. In particular, in the three-mouth case they have fundamental group $F_2$ (the free group on two generators). By contrast, connecting three regions $A,B,C$ in pairs ($AB$, $BC$, and $AC$) using three separate wormholes would give fundamental group $F_3$. Our solutions are asymptotically flat up to the presence of possible magnetic fluxes or cosmic strings that extend to infinity. The construction begins with a two-mouth traversable wormhole supported by backreaction from quantum fields. Inserting a sufficiently small black hole into its throat preserves traversability between the original two mouths. This black hole is taken to be the mouth of another wormhole connecting the original throat to a new distant region of spacetime. Making the new wormhole traversable in a manner similar to the original two-mouth wormhole provides the desired causal connections. From a dual field theory point of view, when AdS asymptotics are added to our construction, multiparty entanglement may play an important role in the traversability of the resulting wormhole.

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