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 $R_0$ (with $0<R_0\le M$) that serves as the free parameter for the family. The slices also have an asymptotically flat end at spatial infinity. In the limit $R_0=0$ the spatial slices lose their trumpet geometry and become flat -- in this limit, our coordinates reduce to Painlevé-Gullstrand coordinates.

arXiv comment7 pages, 3 figures

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