Black hole entropy from entanglement: A review

Abstract

We review aspects of the thermodynamics of black holes and in particular take into account the fact that the quantum entanglement between the degrees of freedom of a scalar field, traced inside the event horizon, can be the origin of black hole entropy. The main reason behind such a plausibility is that the well-known Bekenstein-Hawking entropy-area proportionality -the so-called 'area law' of black hole physics -holds for entanglement entropy as well, provided the scalar field is in its ground state, or in other minimum uncertainty states, such as a generic coherent state or squeezed state. However, when the field is either in an excited state or in a state which is a superposition of ground and excited states, a power-law correction to the area law is shown to exist. Such a correction term falls off with increasing area, so that eventually the area law is recovered for large enough horizon area. On ascertaining the location of the microscopic degrees of freedom that lead to the entanglement entropy of black holes, it is found that although the degrees of freedom close to the horizon contribute most to the total entropy, the contributions from those that are far from the horizon are more significant for excited/superposed states than for the ground state. Thus, the deviations from the area law for excited/superposed states may, in a way, be attributed to the far-away degrees of freedom. Finally, taking the scalar field (which is traced over) to be massive, we explore the changes on the area law due to the mass. Although most of our computations are done in flat space-time with a hypothetical spherical region, considered to be the analogue of the horizon, we show that our results hold as well in curved space-times representing static asymptotically flat spherical black holes with single horizon.

Entanglement Entropy of Black Holes

“…The functions ψ − (x) and ψ + (x) are the boundary values defined on the part of the hypersurface H which is respectively inside (H − ) and outside (H + ) the horizon Σ. As was shown in [16] the wave function (53) corresponds to the Hartle-Hawking vacuum state [126].…”

“…According to refs. [55], [54], [52], [53], if one starts with a mixed state of the vacuum and an excited state, the entanglement entropy receives some power law corrections,…”

“…The progress made in the recent years and the intensity of the on-going research indicate that the entanglement entropy is a very promising direction which in the coming years may lead to a breakthrough in our understanding of the black holes and Quantum Gravity. A number of very nice reviews appeared in the recent years that address the role of the entanglement entropy for black holes [20], [90], [146], [53]; review the calculation of the entanglement entropy in quantum field theory in flat spacetime [81], [37] and the role of the conformal symmetry [29]; focus on the holographic aspects of the entanglement entropy [184], [12]. In the present review I build on these works and focus on the study of the entanglement entropy as applied to black holes.…”

Entropic corrections to Einstein equations

“…The entanglement entropy of the ground state obeys the Hawking area law. Only the excited state contributes to the correction, and more excitations produce more deviation from the area law [37] (also see [38] for a review on the origin of black hole entropy through entanglement). The power-law corrected entropy is written as [39,40]…”

Power-Law Entropy-Corrected HDE and NADE in Brans-Dicke Cosmology

“…The ground state obeys the usual Bekenstein-Hawking entropyarea law. The corrections to entropy arise only from the excited state, and more excitations produce more deviation from the entropy-area law [20] (also see [21] for a review on power-law entropy corrections).…”