A simple family of analytical trumpet slices of the Schwarzschild spacetime

Abstract

We describe a simple family of analytical coordinate systems for the Schwarzschild spacetime. The coordinates penetrate the horizon smoothly and are spatially isotropic. Spatial slices of constant coordinate time t feature a trumpet geometry with an asymptotically cylindrical end inside the horizon at a prescribed areal radius R0 (with 0 < R0 ⩽ M) that serves as the free parameter for the family. The slices also have an asymptotically flat end at spatial infinity. In the limit R0 = 0 the spatial slices lose their trumpet geometry and become flat—in this limit, our coordinates reduce to Painlevé–Gullstrand coordinates.