A Fanfare of Trumpet Black Holes

The trumpet slicing of a black hole spacetime has a number of interesting properties: the slicing is horizon penetrating but never enters an ``alternative universe'' and the central singularity is automatically excised without the need for any boundary conditions. These properties have been exploited in moving-puncture numerical relativity codes. Analytic solutions for trumpet coordinates in the Schwarzschild geometry are well known. We present the analytic solution for the extremal trumpet slicing of the Reissner-Nordstrom geometry, and use an unexpected structure present in this solution to derive new insights into the Schwarzschild trumpet coordinates. A new method for obtaining the limiting surface of a trumpet slicing is exploited to obtain the limiting surface for the Kerr geometry order by order in $a^2$. These results are then combined to construct the leading-order extremal trumpet solution for the Kerr geometry.