Abstract
We derive a novel four-dimensional black hole with planar horizon that asymptotes to the linear dilaton background. The usual growth of its entanglement entropy before Page’s time is established. After that, emergent islands modify to a large extent the entropy, which becomes finite and is saturated by its Bekenstein-Hawking value in accordance with the finiteness of the von Neumann entropy of eternal black holes. We demonstrate that viewed from the string frame, our solution is the two-dimensional Witten black hole with two additional free bosons. We generalize our findings by considering a general class of linear dilaton black hole solutions at a generic point along the σ-model renormalization group (RG) equations. For those, we observe that the entanglement entropy is “running” i.e. it is changing along the RG flow with respect to the two-dimensional worldsheet length scale. At any fixed moment before Page’s time the aforementioned entropy increases towards the infrared (IR) domain, whereas the presence of islands leads the running entropy to decrease towards the IR at later times. Finally, we present a four-dimensional charged black hole that asymptotes to the linear dilaton background as well. We compute the associated entanglement entropy for the extremal case and we find that an island is needed in order for it to follow the Page curve.
Highlights
Manifestation” of an inherent quantum clock by way of inner entanglement and depletion of gravitons in the N-portrait language
We generalize our findings by considering a general class of linear dilaton black hole solutions at a generic point along the σ-model renormalization group (RG) equations
This is further substantiated by the nontrivial fact that in all the examples studied in the literature so far, the entanglement entropy turns out to follow the Page curve when the aforementioned prescription is employed. Even though these considerations initially focused on two-dimensional systems, it is clear that the “island rule” extends well beyond two-dimensional spacetimes, e.g. [11, 17, 21, 23]. It saves the day for the higher dimensional Schwarzschild black hole solution, by rendering its entanglement entropy unitary; this was studied in [17] for pure Einsteinian gravity in D ≥ 4 dimensions, while [21] generalized these findings when higher derivative curvature invariants are included in the gravitational action
Summary
The equations of motion for the theory are obtained by varying the. Let us focus on static, isotropic vacuum solutions of the form ds2 = −B(r)dt2 + A(r)dr2 + r2 dx2 + dy , σ ≡ σ(r) = α log(kr) ,. Meaning that the four-dimensional theory described by (2.1) admits the following black hole geometry ds2 = −r2. In analogy with the Schwarzschild black hole, let us work in terms of the following tortoise coordinate r∗ = rh dr r. R∗ ∼ rh log(r/rh) for r rh and r∗ → −∞ for r = rh where the event horizon lies. V = rhe rh , in terms of which the metric is written as ds2 = −dudv + r2(u, v) dx2 + dy , σ = −2 log kr(u, v) ,. The Penrose diagram of the solution is shown in figure 1
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