Schwarzschild-de Sitter Spacetimes, McVittie Coordinates, and Trumpet Geometries

TL;DR

This paper derives analytical trumpet geometries in Schwarzschild-de Sitter spacetimes, generalizing static maximal trumpet slices to cosmological settings, aiding numerical simulations of black holes in expanding universes.

Contribution

It introduces new analytical expressions for trumpet geometries in Schwarzschild-de Sitter spacetimes using generalized slicings and coordinates analogous to McVittie coordinates.

Findings

Analytical expressions for trumpet geometries in Schwarzschild-de Sitter spacetimes.

Coordinate systems that interpolate between black hole horizons and cosmological asymptotics.

Clarification of the role of time-dependence and boundary conditions in cosmological trumpet slices.

Abstract

Trumpet geometries play an important role in numerical simulations of black hole spacetimes, which are usually performed under the assumption of asymptotic flatness. Our Universe is not asymptotically flat, however, which has motivated numerical studies of black holes in asymptotically de Sitter spacetimes. We derive analytical expressions for trumpet geometries in Schwarzschild-de Sitter spacetimes by first generalizing the static maximal trumpet slicing of the Schwarzschild spacetime to static constant mean curvature trumpet slicings of Schwarzschild-de Sitter spacetimes. We then switch to a comoving isotropic radial coordinate which results in a coordinate system analogous to McVittie coordinates. At large distances from the black hole the resulting metric asymptotes to a Friedmann-Lemaitre-Robertson-Walker metric with an exponentially-expanding scale factor. While McVittie coordinates have another asymptotically de Sitter end as the radial coordinate goes to zero, so that they generalize the notion of a "wormhole" geometry, our new coordinates approach a horizon-penetrating trumpet geometry in the same limit. Our analytical expressions clarify the role of time-dependence, boundary conditions and coordinate conditions for trumpet slices in a cosmological context, and provide a useful test for black hole simulations in asymptotically de Sitter spacetimes.