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Abstract: We study the entanglement entropy, $S C$, of a massless free scalar field onthe 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, $r\gg R 1,R 2$, between two blackholes. If we can consider the entanglement entropy as thermodynamic entropy, wecan see the entropic force acting on the two black holes from the $r$dependence of $S C$. We develop the computational method based on that ofBombelli et al to obtain the $r$ dependence of $S C$ of scalar fields whoseLagrangian is quadratic with respect to the scalar fields. First we study $S C$in $d+1$ dimensional Minkowski spacetime. In this case the state of themassless free scalar field is the Minkowski vacuum state and we replace twoblack holes by two imaginary spheres, and we take the trace over the degrees offreedom residing in the imaginary spheres. We obtain the leading term of $S C$with respect to $1-r$. The result is $S C=S A+S B+\tfrac{1}{r^{2d-2}}GR 1,R 2$, where $S A$ and $S B$ are the entanglement entropy on the insideregion of $A$ and $B$, and $GR 1,R 2 \leq 0$. We do not calculate$GR 1,R 2$ in detail, but we show how to calculate it. In the black hole casewe use the method used in the Minkowski spacetime case with some modifications.We show that $S C$ can be expected to be the same form as that in the Minkowskispacetime case. But in the black hole case, $S A$ and $S B$ depend on $r$, sowe do not fully obtain the $r$ dependence of $S C$. Finally we assume that theentanglement entropy can be regarded as thermodynamic entropy, and consider theentropic force acting on two black holes. We argue how to separate theentanglement entropic force from other force and how to cancel $S A$ and $S B$whose $r$ dependence are not obtained. Then we obtain the physical predictionwhich can be tested experimentally in principle.



Author: Noburo Shiba

Source: https://arxiv.org/







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