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Erik Verlinde

Publications and source records attributed to Erik Verlinde.

At least 19 recordsLinked to original sources
arXiv2026

Quantum Fluctuations of the Black Hole Horizon

Classical black holes have sharply defined event horizons, but quantum mechanically the horizon acquires a quantum uncertainty, called the `quantum width' by Marolf. We propose a definition of the quantum width by a physical experiment involving the last moment a signal emitted from an ingoing light ray can escape to infinity. Calculations of this observable for spherically symmetric black holes in perturbative quantum gravity reveal that the quantum width depends on the resolution of the probe, and is often much larger than the Planck scale. For example, for Schwarzschild black holes in four dimensions in a particular regime of parameters, a piece of the horizon of size $\sigma_\perp$ has quantum width roughly $\sqrt{l_P r_s^2/\sigma_\perp}$.

hep-th↗
arXiv2025

Quantum Symmetry and Geometry in Double-Scaled SYK

The emergence of the quantum $R$-matrix in the double-scaled SYK model points to an underlying quantum group structure. In this work, we identify the quantum group $\mathcal{U}_q(\mathfrak{su}(1,1))$ as a subalgebra of the chord algebra. Specifically, we construct the generators of $\mathcal{U}_q(\mathfrak{s} \mathfrak{u}(1,1))$ from combinations of operators within the chord algebra and show that the one-particle chord Hilbert space decomposes into the positive discrete series representations of $\mathcal{U}_q(\mathfrak{s} \mathfrak{u}(1,1))$. Using the coproduct structure of the quantum group, we build the multi-particle Hilbert space and establish its equivalence with previous results defined by the chord rules. In particular, we show that the quantum $R$-matrix acts as a swapping operator that reverses the ordering of open chords in each fusion channel while incorporating the corresponding $q$-weighted penalty factors. This action enables an explicit derivation of the chord Yang-Baxter relation. We further explore a realization of the quantum group generators on the quantum disk, and present a novel factorization formula for the bulk gravitational wavefunction in the presence of matter. We further discuss the relation between the $\mathcal{U}_q(\mathfrak{s} \mathfrak{u}(1,1))$ structure uncovered here and the $\mathcal{U}_q(\mathfrak{s} \mathfrak{l}(2, \mathbb{R}))$ algebra previously studied from the boundary perspective. Finally, we study the gravitational wavefunction with matter in the Schwarzian regime.

hep-th↗
arXiv2024

An Operator Algebraic Approach To Black Hole Information

We present an operator algebraic perspective on the black hole information problem. For a black hole after Page time that is entangled with the early radiation we formulate a version of the information puzzle that is well-posed in the $G\to 0$ limit. We then give a description of the information recovery protocol in terms of von Neumann algebras using elements of the Jones index theory of type II$_1$ subfactors. The subsequent evaporation and recovery steps are represented by Jones's basic construction, and an operation called the canonical shift. A central element in our description is the Jones projection, which leads to an entanglement swap and implements an operator algebraic version of a quantum teleportation protocol. These aspects are further elaborated on in a microscopic model based on type I algebras. Finally, we argue that in the emergent type III algebra the canonical shift may be interpreted as a spacetime translation and, hence, that at the microscopic level "translation = teleportation".

hep-th↗
arXiv2022

A Conversation on ER = EPR

Ten years ago this week, the two authors had an email conversation about black holes, ER bridges, and EPR entanglement. This brief note contains a verbatim translation of these emails. While the ideas expressed in this email dialogue linking [1] and [2] are more mainstream now than they were back then, there are still many unresolved puzzles, some of which are discussed in this old correspondence.

hep-th↗
arXiv2022

Quantum chaos in 2D gravity

We present a quantitative and fully non-perturbative description of the ergodic phase of quantum chaos in the setting of two-dimensional gravity. To this end we describe the doubly non-perturbative completion of semiclassical 2D gravity in terms of its associated universe field theory. The guiding principle of our analysis is a flavor-matrix theory (fMT) description of the ergodic phase of holographic gravity, which exhibits $\mathrm{U}(n|n)$ causal symmetry breaking and restoration. JT gravity and its 2D-gravity cousins alone do not realize an action principle with causal symmetry, however we demonstrate that their {\it universe field theory}, the Kodaira-Spencer (KS) theory of gravity, does. After directly deriving the fMT from brane-antibrane correlators in KS theory, we show that causal symmetry breaking and restoration can be understood geometrically in terms of different (topological) D-brane vacua. We interpret our results in terms of an open-closed string duality between holomorphic Chern-Simons theory and its closed-string equivalent, the KS theory of gravity. Emphasis will be put on relating these geometric principles to the characteristic spectral correlations of the quantum ergodic phase.

hep-th↗
arXiv2022

Modular Fluctuations from Shockwave Geometries

Modular fluctuations have previously been shown to obey an area law $\langle ΔK^2 \rangle = \langle K \rangle = {A}/{4 G_N}$. Furthermore, modular fluctuations generate fluctuations in the spacetime geometry of empty causal diamonds. Here we demonstrate the physical origin of these fluctuations, showing that the modular area law, in $d-$dimensionsal Minkowski space, can be reproduced from shockwaves arising from vacuum fluctuations. The size of the vacuum fluctuations is fixed by commutation relations in light-ray operators, of the same form postulated by 't Hooft in the context of black hole horizons.

hep-th↗
arXiv2022

No Page Curves for the de Sitter Horizon

We investigate the fine-grained entropy of the de Sitter cosmological horizon. Starting from three-dimensional pure de Sitter space, we consider a partial reduction approach, which supplies an auxiliary system acting as a heat bath both at future infinity and inside the static patch. This allows us to study the time-dependent entropy of radiation collected for both observers in the out-of-equilibrium Unruh-de Sitter state, analogous to black hole evaporation for a cosmological horizon. Central to our analysis in the static patch is the identification of a weakly gravitating region close to the past cosmological horizon; this is suggestive of a relation between observables at future infinity and inside the static patch. We find that in principle, while the meta-observer at future infinity naturally observes a pure state, the static patch observer requires the use of the island formula to reproduce a unitary Page curve. However, in practice, catastrophic backreaction occurs at the Page time, and neither observer will see unitary evaporation.

hep-th↗
arXiv2022

A universe field theory for JT gravity

We present a field theory description for the non-perturbative splitting and joining of baby universes in Euclidean Jackiw-Teitelboim (JT) gravity. We show how the gravitational path integral, defined as a sum over topologies, can be reproduced from the perturbative expansion of a Kodaira-Spencer (KS) field theory for the complex structure deformations of the spectral curve. We use that the Schwinger-Dyson equations for the KS theory can be mapped to the topological recursion relations. We refer to this dual description of JT gravity as a `universe field theory'. By introducing non-compact D-branes in the target space geometry, we can probe non-perturbative aspects of JT gravity. The relevant operators are obtained through a modification of the JT path integral with Neumann boundary conditions. The KS/JT identification suggests that the ensemble average for JT gravity can be understood in terms of a more standard open/closed duality in topological string theory.

hep-th↗
arXiv2021

From the BTZ black hole to JT gravity: geometrizing the island

We study the evaporation of two-dimensional black holes in JT gravity from a three-dimensional point of view. A partial dimensional reduction of AdS$_3$ in Poincaré coordinates leads to an extremal 2D black hole in JT gravity coupled to a 'bath': the holographic dual of the remainder of the 3D spacetime. Partially reducing the BTZ black hole gives us the finite temperature version. We compute the entropy of the radiation using geodesics in the three-dimensional spacetime. We then focus on the finite temperature case and describe the dynamics by introducing time-dependence into the parameter controlling the reduction. The energy of the black hole decreases linearly as we slowly move the dividing line between black hole and bath. Through a re-scaling of the BTZ parameters we map this to the more canonical picture of exponential evaporation. Finally, studying the entropy of the radiation over time leads to a geometric representation of the Page curve. The appearance of the island region is explained in a natural and intuitive fashion.

hep-th↗
arXiv2021

Spacetime Fluctuations in AdS/CFT

We compute fluctuations in the modular energy of the vacuum associated with a Rindler-wedge in AdS spacetime in the context of AdS/CFT. We discuss the possible effect of these energy fluctuations on the spacetime geometry, and on the traversal time of a light beam propagating from the boundary to the bulk and back.

hep-th↗
arXiv2021

The Weak Lensing Radial Acceleration Relation: Constraining Modified Gravity and Cold Dark Matter theories with KiDS-1000

We present measurements of the radial gravitational acceleration around isolated galaxies, comparing the expected gravitational acceleration given the baryonic matter with the observed gravitational acceleration, using weak lensing measurements from the fourth data release of the Kilo-Degree Survey. These measurements extend the radial acceleration relation (RAR) by 2 decades into the low-acceleration regime beyond the outskirts of the observable galaxy. We compare our RAR measurements to the predictions of two modified gravity (MG) theories: MOND and Verlinde's emergent gravity. We find that the measured RAR agrees well with the MG predictions. In addition, we find a difference of at least $6σ$ between the RARs of early- and late-type galaxies (split by Sérsic index and $u-r$ colour) with the same stellar mass. Current MG theories involve a gravity modification that is independent of other galaxy properties, which would be unable to explain this behaviour. The difference might be explained if only the early-type galaxies have significant ($M_{gas} \approx M_*$) circumgalactic gaseous haloes. The observed behaviour is also expected in $Λ$CDM models where the galaxy-to-halo mass relation depends on the galaxy formation history. We find that MICE, a $Λ$CDM simulation with hybrid halo occupation distribution modelling and abundance matching, reproduces the observed RAR but significantly differs from BAHAMAS, a hydrodynamical cosmological galaxy formation simulation. Our results are sensitive to the amount of circumgalactic gas; current observational constraints indicate that the resulting corrections are likely moderate. Measurements of the lensing RAR with future cosmological surveys will be able to further distinguish between MG and $Λ$CDM models if systematic uncertainties in the baryonic mass distribution around galaxies are reduced.

astro-ph.GA↗
arXiv2021

Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds

Motivated by the vast success of deep convolutional networks, there is a great interest in generalizing convolutions to non-Euclidean manifolds. A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. The underlying reason for this ambiguity is that general manifolds do not come with a canonical choice of reference frames (gauge). Kernels and features therefore have to be expressed relative to arbitrary coordinates. We argue that the particular choice of coordinatization should not affect a network's inference -- it should be coordinate independent. A simultaneous demand for coordinate independence and weight sharing is shown to result in a requirement on the network to be equivariant under local gauge transformations (changes of local reference frames). The ambiguity of reference frames depends thereby on the G-structure of the manifold, such that the necessary level of gauge equivariance is prescribed by the corresponding structure group G. Coordinate independent convolutions are proven to be equivariant w.r.t. those isometries that are symmetries of the G-structure. The resulting theory is formulated in a coordinate free fashion in terms of fiber bundles. To exemplify the design of coordinate independent convolutions, we implement a convolutional network on the Möbius strip. The generality of our differential geometric formulation of convolutional networks is demonstrated by an extensive literature review which explains a large number of Euclidean CNNs, spherical CNNs and CNNs on general surfaces as specific instances of coordinate independent convolutions.

cs.LG↗
arXiv2019

Probing typical black hole microstates

We investigate the possibility that the geometry dual to a typical AdS black hole microstate corresponds to the extended AdS-Schwarzschild geometry, including a region spacelike to the exterior. We argue that this region can be described by the mirror operators, a set of state-dependent operators in the dual CFT. We probe the geometry of a typical state by considering state-dependent deformations of the CFT Hamiltonian, which have an interpretation as a one-sided analogue of the Gao-Jafferis-Wall traversable wormhole protocol for typical states. We argue that the validity of the conjectured bulk geometry requires that out-of-time-order correlators of simple CFT operators on typical pure states must exhibit the same chaotic effects as thermal correlators at scrambling time. This condition is related to the question of whether the product of operators separated by scrambling time obey the Eigenstate Thermalization Hypothesis. We investigate some of these statements in the SYK model and discuss similarities with state-dependent perturbations of pure states in the SYK model previously considered by Kourkoulou and Maldacena. Finally, we discuss how the mirror operators can be used to implement an analogue of the Hayden-Preskill protocol.

hep-th↗
arXiv2018

Monstrous Product CFTs in the Grand Canonical Ensemble

We study symmetric products of the chiral 'Monster' conformal field theory with c=24 in the grand canonical ensemble by introducing a complex parameter ρ, whose imaginary part represents the chemical potential μconjugate to the number of copies of the CFT. The grand canonical partition function is given by the DMVV product formula in terms of the multiplicities of the seed CFT. It possesses an O(2,2;\ZZ) symmetry that enhances the familiar SL(2,\ZZ) modular invariance of the canonical ensemble and mixes the modular parameter τwith the parameter ρ. By exploiting this enhanced modular symmetry and the pole structure of the DMVV product formula we are able to extend the region of validity of Cardy's formula, and explain why it matches the semi-classical Bekenstein-Hawking formula for black holes all the way down to the AdS-scale. We prove that for large c the spectrum contains a universal perturbative sector whose degeneracies obey Hagedorn growth. The transition from Cardy to Hagedorn growth is found to be due to the formation of a Bose-Einstein condensate of ground state CFTs at low temperatures. The grand canonical partition function has an interesting phase structure, which in addition to the standard Hawking-Page transition between low and high temperature, exhibits a wall-crossing transition that exchanges the roles of τand ρ.

hep-th↗
arXiv2018

On the interior geometry of a typical black hole microstate

We argue that the region behind the horizon of a one-sided black hole can be probed by an analogue of the double-trace deformation protocol of Gao-Jafferis-Wall. This is achieved via a deformation of the CFT Hamiltonian by a term of the form ${\cal O} \widetilde{\cal O}$, where $\widetilde{\cal O}$ denote the state-dependent "mirror operators". We argue that this deformation creates negative energy shockwaves in the bulk, which allow particles inside the horizon to escape and to get directly detected in the CFT. This provides evidence for the smoothness of the horizon of black holes dual to typical states. We argue that the mirror operators allow us to perform an analogue of the Hayden-Preskill decoding protocol. Our claims rely on a technical conjecture about the chaotic behavior of out-of-time-order correlators on typical pure states at scrambling time.

hep-th↗
arXiv2013

Behind the Horizon in AdS/CFT

We extend the recent proposal of Papadodimas and Raju of a CFT construction of operators inside the black hole interior to arbitrary non-maximally mixed states. Our construction builds on the general prescription given in earlier work, based on ideas from quantum error correction. We indicate how the CFT state dependence of the interior modes can be removed by introducing an external system, such as an observer, that is entangled with the CFT.

hep-th↗
arXiv2013

Passing through the Firewall

We propose that black hole information is encoded in non-local correlations between microscopic interior and exterior degrees of freedom. We give a simple qubit representation of this proposal, and show herein that for every black hole state, the apparent firewall can be removed via a universal, state independent unitary transformation. A central element in our discussion is the distinction between virtual qubits, which are in a specified vacuum state, and real qubits, that carry the free quantum information of the black hole. We outline how our proposal may be realized in AdS/CFT

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