Maximal Slicing for Puncture Evolutions of Schwarzschild and Reissner-Nordstr\"om Black Holes

TL;DR

This paper analytically constructs maximal slices of Schwarzschild and Reissner-Nordstr"om black holes, confirming numerical observations and deriving the late-time lapse value at the horizon, aiding black hole simulations.

Contribution

It provides explicit analytical maximal slices with zero gradient lapse at the puncture, validating numerical results and extending to Reissner-Nordstr"om black holes.

Findings

Analytical maximal slices match numerical simulations.

Lapse at the horizon approaches 0.3248 at late times.

Generalization to Reissner-Nordstr"om black holes.

Abstract

We prove by explicit construction that there exists a maximal slicing of the Schwarzschild spacetime such that the lapse has zero gradient at the puncture. This boundary condition has been observed to hold in numerical evolutions, but in the past it was not clear whether the numerically obtained maximal slices exist analytically. We show that our analytical result agrees with numerical simulation. Given the analytical form for the lapse, we can derive that at late times the value of the lapse at the event horizon approaches the value ${3/16}\sqrt{3} \approx 0.3248$, justifying the numerical estimate of 0.3 that has been used for black hole excision in numerical simulations. We present our results for the non-extremal Reissner-Nordstr\"om metric, generalizing previous constructions of maximal slices.