An effective density matrix for vacua in asymptotically flat gravity

Abstract

We explicitly construct the density matrix associated to the vacuum state of a large spherically symmetric causal diamond in four-dimensional asymptotically flat gravity. We achieve this using the soft effective action, which characterizes the low-energy gravitational degrees of freedom that arise in the long-distance limit of the Einstein-Hilbert action and consists of both the soft graviton mode and the Goldstone mode arising from the spontaneous breaking of supertranslation symmetry. Once we have the density matrix, we straightforwardly compute the mean and variance of the modular Hamiltonian and in particular show that the variance obeys an area law.