  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.
