Abstract
The island rule for the entanglement entropy is applied to an eternal Reissner–Nordström black hole. The key ingredient is that the black hole is assumed to be in thermal equilibrium with a heat bath of an arbitrary temperature and so the generalized entropy is treated as being off-shell. Taking the on-shell condition to the off-shell generalized entropy, we find the generalized entropy and then obtain the entanglement entropy following the island rule. For the non-extremal black hole, the entanglement entropy grows linearly in time and can be saturated after the Page time as expected. The entanglement entropy also has a well-defined Schwarzschild limit. In the extremal black hole, the island prescription provides a logarithmically growing entanglement entropy in time and a constant entanglement entropy after the Page time. In the extremal black hole, the boundary of the island hits the curvature singularity where the semi-classical approximations appear invalid. To avoid encountering the curvature singularity, we apply this procedure to the Hayward black hole regular at the origin. Consequently, the presence of the island in extremal black holes can provide a finite entanglement entropy, which might imply non-trivial vacuum configurations of extremal black holes.
Highlights
S = min ext Sgen = min ext A(∂ I ) 4G N + Smatter ( R ∪ I) (1)where Sgen is the generalized entropy and A(∂ I ) is the area of the boundary of island I
We suggest that a black hole with an island is assumed to be in thermal equilibrium with a heat bath of an arbitrary temperature in such a way that at the interim stage the off-shell generalized entropy Sgen(κ) can be constructed
We investigated the entanglement entropy of the Reissner–Nordström black hole and the Hayward black
Summary
Where Sgen is the generalized entropy and A(∂ I ) is the area of the boundary of island I. Starting from a black hole in thermal equilibrium with a heat bath of an arbitrary temperature β, one can obtain the area law of the black hole entropy by taking as β → βH where βH is the inverse temperature of the black hole [32,33] In this respect, we suggest that a black hole with an island is assumed to be in thermal equilibrium with a heat bath of an arbitrary temperature in such a way that at the interim stage the off-shell generalized entropy Sgen(κ) can be constructed. Where κ = 2π T = 2πβ−1 are off-shell quantities and κH is the surface gravity at the horizon This formula will be used in calculating the entanglement entropy for non-extremal and extremal black holes.
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