Well-behaved harmonic time slices of a charged, rotating, boosted black hole

Harmonic time slicings are used in some hyperbolic formulations of Einstein's equations and are therefore of considerable interest to the field of numerical relativity. We construct an analytic coordinate representation of the Kerr-Newman geometry that is harmonic in both its spatial and temporal coordinates. The metric is independent of time and the spacelike, $t=\mathrm{}\mathrm{const}$ slices extend from spatial infinity smoothly through the event horizon at ${r=r}_{+}$ and end at the Cauchy horizon at ${r=r}_{-}.$ When the spatial harmonic coordinate condition is imposed, there is also a spatial coordinate singularity at $r=M$, but this fully harmonic metric can be trivially boosted to yield an analytic solution for a harmonically sliced translating, spinning black hole. We also examine the behavior of evolutions which obey the harmonic slicing condition but start from initial data that is not in the time-independent harmonic slicing foliation. We find that with a suitable choice of the spatial gauge, the evolving three-geometry is “attracted” to the time-independent three-geometry we present in this paper.