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arXiv:gr-qc/9712018 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Thermodynamics of entanglement in Schwarzschild spacetime

Shinji Mukohyama A, Masafumi Seriu, Hideo Kodama

Published Jul 31, 1998• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.58.064001

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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.
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.
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.
It has been known for decades that black holes behave as thermodynamic objects and, as such, have an entropy, which was acquired in exact form as the Bekenstein-Hawking entropy. However, the origin of this entropy, and in particular its proportionality to the area of the horizon, remained unexplained. Motivated by this mystery, several authors examined the possibility of entanglement entropy as a source of black hole entropy, by considering a scalar ?eld in the ground state and\ntracing out the degrees of freedom inside an imaginary sphere, analogous to the interior of the black hole. The resulting entanglement entropy between the two regions turned out to scale with the area of their mutual boundary. In the simplest model the scalar ?eld is represented by a system of coupled harmonic oscillators. After an introduction to the phenomenon of quantum entanglement and the associated entanglement entropy, we will review the problem of two coupled oscillators in the ground state and derive the entanglement entropy using an approach in Fock space, as opposed to the standard method involving integrals in positon space. This will also\nallow us to look at some excited states of the system. Subsequently we will try to extend this approach to a system of N coupled harmonic oscillators and derive the entanglement entropy between an inner and outer region of oscillators.
Entanglement entropy is often speculated as a strong candidate for the origin of black-hole entropy. To judge whether or not this speculation is true, it is effective to investigate the whole structure of thermodynamics obtained from the entanglement entropy, rather than just to examine the apparent structure of the entropy alone or to compare it with that of black-hole entropy. This is because entropy acquires a physical significance only when it is related to the energy and the temperature of a system. From this point of view, we construct a ``thermodynamics of entanglement'' by introducing an entanglement energy and compare it with black-hole thermodynamics. We consider two possible definitions of entanglement energy. Then we construct two different kinds of thermodynamics by combining each of these different definitions of entanglement energy with the entanglement entropy. We find that both of these two kinds of thermodynamics show significant differences from black-hole thermodynamics if no gravitational effects are taken into account. These differences are, in particular, highlighted in the context of the third law of thermodynamics. Finally we see how the inclusion of gravity alters the thermodynamics of the entanglement. We give a suggestive argument that the thermodynamics of the entanglement behaves like black-hole thermodynamics if the gravitational effects are included properly. Thus the entanglement entropy passes a nontrivial check to be the origin of black-hole entropy.