Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:gr-qc/9504049 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Entanglement entropy in curved spacetimes with event horizons

Rainer Müller, C. O. Loustó

Published Oct 15, 1995• 65 citations
arXivHTMLPDF

Abstract

We consider the computation of the entanglement entropy in curved backgrounds with event horizons. We use a Hamiltonian approach to the problem and perform numerical computations on a spherical lattice of spacing a. We study the cosmological case and make explicit computations for the Friedmann-Robertson-Walker universe. Our results for a massless, minimally coupled scalar field can be summarized by ${\mathit{S}}_{\mathrm{ent}}$=0.30${\mathit{r}}_{\mathit{H}}^{2}$/${\mathit{a}}^{2}$, which resembles the flat space formula, although here the horizon radius ${\mathit{r}}_{\mathit{H}}$ is time dependent.

DOI
10.1103/physrevd.52.4512

Related papers

Thermodynamics of entanglement in Schwarzschild spacetime

Shinji Mukohyama A, Masafumi Seriu, Hideo Kodama

Extending the analysis in our previous paper, we construct the entanglement thermodynamics for a massless scalar field on Schwarzschild spacetime. Contrary to the flat case, the entanglement energy ${E}_{\mathrm{ent}}$ turns out to be proportional to the area radius of the boundary if it is near the horizon. This peculiar behavior of ${E}_{\mathrm{ent}}$ can be understood by the redshift effect caused by the curved background. Combined with the behavior of the entanglement entropy, this result yields, quite surprisingly, the entanglement thermodynamics of the same structure as black hole thermodynamics. On the basis of these results, we discuss the relevance of the concept of entanglement as the microscopic origin of black hole thermodynamics.

arXiv:gr-qc/9712018•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jul 31, 1998•26 citations

Entanglement entropy of a scalar field across a spherical boundary in the Einstein universe

Katja Ried

A scalar field in the ground state, when partially hidden from observation by a spherical boundary, acquires entanglement entropy $S$ proportional to the area of the surface. This area law is well established in flat space, where it follows almost directly from dimensional arguments. We study its validity in an Einstein universe, whose curvature provides an additional physical parameter on which the entropy could, in principle, depend. The surprisingly simple result is that the entanglement entropy still scales linearly with the area. This is supported by other observations to the effect that the entanglement entropy arises mostly from degrees of freedom near the boundary, making it insensitive to the large-scale geometry of the background space.

arXiv:1309.7380•Sep 27, 2013•1 citation

Entanglement entropy of linearized gravitons in a sphere

Valentín Benedetti, Horacio Casini

We compute the entanglement entropy of a massless spin 2 field in a sphere in flat Minkowski space. We describe the theory with a linearized metric perturbation field ${h}_{\ensuremath{\mu}\ensuremath{\nu}}$ and decompose the field into independent modes using tensor spherical harmonics. We fix the gauge such that (a) the two dynamical modes for each angular momentum decouple and have the dynamics of scalar spherical modes, and (b) the gauge-fixed field degrees of freedom inside the sphere represent gauge invariant operators of the theory localized in the same region. In this way the entanglement entropy turns out to be equivalent to the one of a pair of free massless scalars where the contributions of the $l=0$ and $l=1$ modes have been subtracted. The result for the coefficient of the universal logarithmic term is $\ensuremath{-}61/45$ and coincides with that computed using a regularization based on the mutual information.

arXiv:1908.01800•Physical review. D/Physical review. D.•Feb 7, 2020•19 citations

Entanglement entropy in quantum field theory and entanglement entropic force between black holes

Noburo Shiba, ノブロウ シバ, 暢郎 芝

First, we study the entanglement entropy S AB of a d + 1 dimensional massless free scalar field on two disjoint compact spatial regions A and B whose sizes are R 1 and R 2 , respectively, and the distance between them is r.The state of the massless free scalar field is the vacuum state.We consider the mutual information S A;B ≡ S A + S B -S AB where S A and S B are the entanglement entropy on the inside region of A and B, respectively.We develop the computational method based on that of Bombelli et al and obtain the result thatWhen A and B are spheres in d = 2 and 3, we obtain explicit resultsBy using the method used in the Minkowski spacetime case with some modifications, we study the entanglement entropy, S C , of the massless free scalar field on the outside region C of two black holes A and B whose radii are R 1 and R 2 and how it depends on the distance, r( R 1 , R 2 ) , between two black holes.If we can consider the entanglement entropy as thermodynamic entropy, we can see the entropic force (we call this entanglement entropic force) acting on the two black holes from the r dependence of S C .We consider the property of the entropic force.

•Sep 25, 2013•0 citations

Entanglement Entropy for D3-, M2- and M5-brane backgrounds

Edward Quijada, Henrique Boschi-Filho

In the context of the gauge/gravity duality applications, we study and compute the entanglement entropy of gauge theories corresponding to string/M theories on D3-, M2- and M5-brane backgrounds. This is achieved using the Ryu-Takayanagi formula. We obtain the entanglement entropy for the general cases of the D3-, M2- and M5-brane backgrounds and also for the near horizon AdS limit, as well as the non-conformal flat space limit in each case.

arXiv:1711.08505•Nov 22, 2017•2 citations

Entanglement Entropy of Yukawa-Coupled Fields Across a Rindler Horizon

Akshay Kulkarni, Rahul Nigam

We compute the entanglement entropy across a Rindler horizon in scalar field theory with Yukawa interaction. Starting from a microscopic scalar-mediator theory in flat spacetime, we integrate out the massive mediator to obtain a quadratic but nonlocal effective kernel that determines the ground-state wavefunctional. The reduced density matrix for a single Rindler wedge is constructed explicitly by tracing over the complementary wedge, allowing the entanglement entropy to be evaluated directly from the kernel without replica or geometric methods. Exploiting translational invariance parallel to the horizon, the problem decomposes into independent transverse momentum sectors that reduce effectively to one-dimensional nonlocal systems and can be diagonalized analytically in the weak-coupling regime. The interaction-induced entropy obeys an area law, with leading corrections controlled by the Yukawa screening mass and logarithmically sensitive to the transverse ultraviolet cutoff, reflecting the localization of correlations near the horizon. Although the modular Hamiltonian depends on the Rindler acceleration, the entanglement spectrum and entropy are independent of this choice, demonstrating the observer-independent nature of vacuum entanglement. Our framework provides a direct and microscopically transparent approach to computing interaction-induced corrections to horizon entanglement using nonlocal effective kernels.

arXiv:2602.11038•Feb 11, 2026•0 citations
We consider the computation of the entanglement entropy in curved backgrounds with event horizons. We use a Hamiltonian approach to the problem and perform numerical computations on a spherical lattice of spacing a. We study the cosmological case and make explicit computations for the Friedmann-Robertson-Walker universe. Our results for a massless, minimally coupled scalar field can be summarized by Sent{\mathit{S}}_{\mathrm{ent}}Sent​=0.30rH2{\mathit{r}}_{\mathit{H}}^{2}rH2​/a2{\mathit{a}}^{2}a2, which resembles the flat space formula, although here the horizon radius rH{\mathit{r}}_{\mathit{H}}rH​ is time dependent.
Extending the analysis in our previous paper, we construct the entanglement thermodynamics for a massless scalar field on Schwarzschild spacetime. Contrary to the flat case, the entanglement energy Eent{E}_{\mathrm{ent}}Eent​ turns out to be proportional to the area radius of the boundary if it is near the horizon. This peculiar behavior of Eent{E}_{\mathrm{ent}}Eent​ can be understood by the redshift effect caused by the curved background. Combined with the behavior of the entanglement entropy, this result yields, quite surprisingly, the entanglement thermodynamics of the same structure as black hole thermodynamics. On the basis of these results, we discuss the relevance of the concept of entanglement as the microscopic origin of black hole thermodynamics.
A scalar field in the ground state, when partially hidden from observation by a spherical boundary, acquires entanglement entropy SSS proportional to the area of the surface. This area law is well established in flat space, where it follows almost directly from dimensional arguments. We study its validity in an Einstein universe, whose curvature provides an additional physical parameter on which the entropy could, in principle, depend. The surprisingly simple result is that the entanglement entropy still scales linearly with the area. This is supported by other observations to the effect that the entanglement entropy arises mostly from degrees of freedom near the boundary, making it insensitive to the large-scale geometry of the background space.
We compute the entanglement entropy of a massless spin 2 field in a sphere in flat Minkowski space. We describe the theory with a linearized metric perturbation field ${h}_{\ensuremath{\mu}\ensuremath{\nu}}$ and decompose the field into independent modes using tensor spherical harmonics. We fix the gauge such that (a) the two dynamical modes for each angular momentum decouple and have the dynamics of scalar spherical modes, and (b) the gauge-fixed field degrees of freedom inside the sphere represent gauge invariant operators of the theory localized in the same region. In this way the entanglement entropy turns out to be equivalent to the one of a pair of free massless scalars where the contributions of the l=0l=0l=0 and l=1l=1l=1 modes have been subtracted. The result for the coefficient of the universal logarithmic term is $\ensuremath{-}61/45$ and coincides with that computed using a regularization based on the mutual information.
First, we study the entanglement entropy S AB of a d + 1 dimensional massless free scalar field on two disjoint compact spatial regions A and B whose sizes are R 1 and R 2 , respectively, and the distance between them is r.The state of the massless free scalar field is the vacuum state.We consider the mutual information S A;B ≡ S A + S B -S AB where S A and S B are the entanglement entropy on the inside region of A and B, respectively.We develop the computational method based on that of Bombelli et al and obtain the result thatWhen A and B are spheres in d = 2 and 3, we obtain explicit resultsBy using the method used in the Minkowski spacetime case with some modifications, we study the entanglement entropy, S C , of the massless free scalar field on the outside region C of two black holes A and B whose radii are R 1 and R 2 and how it depends on the distance, r( R 1 , R 2 ) , between two black holes.If we can consider the entanglement entropy as thermodynamic entropy, we can see the entropic force (we call this entanglement entropic force) acting on the two black holes from the r dependence of S C .We consider the property of the entropic force.
We compute the entanglement entropy across a Rindler horizon in scalar field theory with Yukawa interaction. Starting from a microscopic scalar-mediator theory in flat spacetime, we integrate out the massive mediator to obtain a quadratic but nonlocal effective kernel that determines the ground-state wavefunctional. The reduced density matrix for a single Rindler wedge is constructed explicitly by tracing over the complementary wedge, allowing the entanglement entropy to be evaluated directly from the kernel without replica or geometric methods. Exploiting translational invariance parallel to the horizon, the problem decomposes into independent transverse momentum sectors that reduce effectively to one-dimensional nonlocal systems and can be diagonalized analytically in the weak-coupling regime. The interaction-induced entropy obeys an area law, with leading corrections controlled by the Yukawa screening mass and logarithmically sensitive to the transverse ultraviolet cutoff, reflecting the localization of correlations near the horizon. Although the modular Hamiltonian depends on the Rindler acceleration, the entanglement spectrum and entropy are independent of this choice, demonstrating the observer-independent nature of vacuum entanglement. Our framework provides a direct and microscopically transparent approach to computing interaction-induced corrections to horizon entanglement using nonlocal effective kernels.