Dynamical perturbations of black-hole punctures: effects of slicing conditions

TL;DR

This paper analyzes how different slicing conditions in numerical relativity affect the behavior of black-hole punctures, revealing that the mean curvature determines whether the lapse decays exponentially or oscillates harmonically.

Contribution

It provides an analytical framework linking the mean curvature at the puncture to the qualitative behavior of the lapse under different slicing conditions.

Findings

1+log slicing leads to exponential decay of the lapse.

Shock-avoiding slices cause harmonic oscillation of the lapse.

Mean curvature at the puncture is key to the behavior.

Abstract

While numerous numerical relativity simulations adopt a 1+log slicing condition, shock-avoiding slicing conditions form a viable and sometimes advantageous alternative. Despite both conditions satisfying similar equations, recent numerical experiments point to a qualitative difference in the behavior of the lapse in the vicinity of the black-hole puncture: for 1+log slicing, the lapse appears to decay approximately exponentially, while for shock-avoiding slices it performs approximately harmonic oscillation. Motivated by this observation, we consider dynamical coordinate transformations of the Schwarzschild spacetime to describe small perturbations of static trumpet geometries analytically. We find that the character of the resulting equations depends on the (unperturbed) mean curvature at the black-hole puncture: for 1+log slicing it is positive, predicting exponential decay in the lapse, while for shock-avoiding slices it vanishes, leading to harmonic oscillation. In addition to identifying the value of the mean curvature as the origin of these qualitative differences, our analysis provides insight into the dynamical behavior of black-hole punctures for different slicing conditions.