The entropy of finite gravitating regions

Abstract

We develop a formalism for calculating the entanglement entropy of an arbitrary spatial region of a gravitating spacetime at a moment of time symmetry. The crucial ingredient is a path integral over embeddings of the region into the overall spacetime, interpretable as a sum over the edge modes associated with the region. We find that the entanglement entropy of a gravitating region equals the minimal surface area among all regions that enclose it. This suggests a notion of "terrestrial holography" where regions of space can encode larger ones, in contrast to the standard form of holography, in which degrees of freedom on the celestial sphere at the boundary of the universe encode the interior.

Figures (8)

• Figure 1: The full Hilbert space $\mathcal{H}_\Sigma$ is defined on the complete lattice. When restricting to a subset $a$ of the nodes on the lattice, described by a reduced Hilbert space $\mathcal{H}_a$, one must also include additional degrees of freedom to account for the endpoints of Wilson lines not present in $\mathcal{H}_\Sigma$. These new degrees of freedom are the edge modes in $\mathcal{H}_{e.m.}$.
• Figure 2: $a$ is a manifold with boundary, independent of $\Sigma$. By specifying an embedding $\phi$, we can think of $a$ as a subset of $\Sigma$ in a gauge invariant way. Throughout this paper, we refer to $\phi_e(a)$ as $A$.
• Figure 3: Two examples of diffeomorphisms which can change the embedding map. Left: The deformation moves the new region (blue) strictly inside the original region $A$ (blue and black). The residual gauge symmetry is diffeomorphisms with support on $A$ (blue and black): such a diffeomorphism can push the boundary back to the original configuration. Right: The deformation moves the new region (blue and red) strictly outside the original one $A$ (blue). The residual gauge symmetry is diffeomorphisms with support on $A$ (blue): such a diffeomorphism can never pull the boundary back to the original configuration.
• Figure 4: ( a) Penrose diagram of AdS$_2$ showing the domains of dependence of a half-infinite interval and its complement. Herer, the blue interval is $A$. ( b) The edge modes allow us to split the left and right wedges into two disconnected spacetimes. Here, the blue interval is $a$. They are disconnected because the extended Hilbert space factorizes, and thus the two subregions are unentangled. To regain the original AdS$_2$ spacetime, we must glue these wedges at the corner (the black dot) using the entangling product. ( c) Euclidean state preparation of $a$ and its edge modes. The solid blue line is the "ket" and the dashed blue line is the "bra". The direction of the Euclidean time evolution is shown with the arrow. This is not a Cauchy slice for AdS$_2$: it is a density matrix for the left Rindler wedge.
• Figure 5: ( a) Penrose diagram of AdS$_2$ showing the domains of dependence of a finite interval and its complement. Here, the blue interval is $A$. ( b) The edge modes allow us to split the interval and its complement into disconnected spacetimes (we only show the finite region here). Here, the blue interval is $a$. They are disconnected because the extended Hilbert space factorizes, and thus the two subregions are unentangled. To regain the original AdS$_2$ spacetime, we must glue $a$ and its complement at their corners (the black dots) using the entangling product. ( c) Euclidean state preparation of the density matrix of $a$ and its edge modes. The solid blue line is the "ket" and the dashed blue line is the "bra". The direction of the Euclidean time evolution is shown with the arrow. One might worry whether this evolution is well-defined, in other words, whether is a sensible Hamiltonian on just this wedge and its edge modes. The evolution is in fact defined, because the Hamiltonian charges generating diffeomorphisms (in particular, time evolution) are integrable Ciambelli_2022, so the dynamics is unitary.