Towards the entanglement entropy of two quantum black holes

Abstract

Starting from a Wheeler–DeWitt type equation for an uncharged black hole \(Q=0\), and by choosing the order parameter \((s=2)\) and running the gravitational degrees of freedom it is possible to reduce the Wheeler–DeWitt equation to the canonical form of a quantum harmonic oscillator. In this direction, a natural frequency of oscillation is identified for the black hole. The entanglement entropy of a pair of interacting quantum black holes of mass M is obtained and analyzed. Here we consider as a starting model a pair of identical oscillators of frequency \(\omega \) coupled by a quadratic potential and with interaction constant given by \(\omega _{c}\). Given the relation between the oscillation frequency \(\omega \) of an isolated quantum black hole and its mass, the entanglement entropy of this system is obtained by analogy with a pair of quantum oscillators coupled by a quadratic interaction potential of frequency \(\omega _{c}\). The analysis of the entanglement entropy is performed by introducing the reduced variables \({\widetilde{\omega }}\) and \({\widetilde{A}}\). An interesting result arises when we consider \({\widetilde{A}}\gg 1\) and the interaction parameter is set \({\widetilde{\omega }}=1\). In this case, the entanglement entropy can be replaced by its asymptotic expansion, where the dominant term is of logarithmic character in the reduced area. Another case of analysis emerges when the reduced area \({\widetilde{A}}=1\) and \({\widetilde{\omega }}\) varies. In this case, the entanglement entropy depends uniquely on the interaction parameter between the two black holes.