标题： 两黑洞的纠缠熵和纠缠熵力 显示英文标题

标题： Entanglement Entropy of Two Black Holes and Entanglement Entropic Force

摘要： 我们研究了质量为零的自由标量场在两个黑洞 $A$ 和 $B$ 的外部区域 $C$ 的纠缠熵 $S_C$，以及它如何依赖于两个黑洞之间的距离 $r(\gg R_1,R_2)$，这两个黑洞的半径分别为 $R_1$ 和 $R_2$。 如果我们把纠缠熵看作热力学熵，就可以从$r$对$S_C$的依赖关系中看到作用在两个黑洞上的熵力。我们基于 Bombelli 等人的计算方法，得到了标量场的$r$对$S_C$的依赖关系，这些标量场的拉格朗日量相对于标量场是二次的。我们首先研究了$S_C$在$d+1$维闵可夫斯基时空中的情况。在这种情况下，无质量自由标量场的状态是闵可夫斯基真空态，我们将两个黑洞替换为两个虚数球体，并对存在于虚数球体中的自由度进行求迹。 我们得到$S_C$关于$1/r$的主项。 结果是$S_C=S_A+S_B+\tfrac{1}{r^{2d-2}} G(R_1,R_2)$，其中$S_A$和$S_B$是在$A$和$B$内部区域的纠缠熵，以及$G(R_1,R_2) \leq 0$。 我们不详细计算$G(R_1,R_2)$，但我们将展示如何计算它。 在黑洞情况下，我们使用在闵可夫斯基时空情况下使用的方法，并做一些修改。 我们证明$S_C$可以预期与闵可夫斯基时空情况下的形式相同。 但在黑洞情况下，$S_A$和$S_B$依赖于$r$，因此我们不能完全获得$r$对$S_C$的依赖性。 最后我们假设纠缠熵可以视为热力学熵，并考虑作用在两个黑洞上的熵力。 我们论述如何将纠缠熵力与其他力分离，以及如何消除$S_A$和$S_B$，它们的$r$依赖关系尚未获得。 然后我们得到了原则上可以进行实验检验的物理预测。 显示英文摘要

摘要： We study the entanglement entropy, $S_C$, of a 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, $r(\gg R_1,R_2)$, between two black holes. If we can consider the entanglement entropy as thermodynamic entropy, we can 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 of Bombelli et al to obtain the $r$ dependence of $S_C$ of scalar fields whose Lagrangian 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 the massless free scalar field is the Minkowski vacuum state and we replace two black holes by two imaginary spheres, and we take the trace over the degrees of freedom residing in the imaginary spheres. We obtain the leading term of $S_C$ with respect to $1/r$. The result is $S_C=S_A+S_B+\tfrac{1}{r^{2d-2}} G(R_1,R_2)$, where $S_A$ and $S_B$ are the entanglement entropy on the inside region of $A$ and $B$, and $G(R_1,R_2) \leq 0$. We do not calculate $G(R_1,R_2)$ in detail, but we show how to calculate it. In the black hole case we 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 Minkowski spacetime case. But in the black hole case, $S_A$ and $S_B$ depend on $r$, so we do not fully obtain the $r$ dependence of $S_C$. Finally we assume that the entanglement entropy can be regarded as thermodynamic entropy, and consider the entropic force acting on two black holes. We argue how to separate the entanglement 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 prediction which can be tested experimentally in principle.