Slice Stretching Effects for Maximal Slicing of a Schwarzschild Black Hole

TL;DR

This paper investigates the effects of slice stretching in maximal slicing of Schwarzschild black holes, proposing boundary conditions and techniques to mitigate these effects in numerical simulations.

Contribution

It introduces boundary conditions and lapse functions that delay or prevent slice stretching effects in black hole simulations.

Findings

Favorable boundary conditions delay slice stretching effects.

Averages of odd and even lapse functions are effective in numerical simulations.

Analytic methods to avoid slice stretching are discussed.

Abstract

Slice stretching effects such as slice sucking and slice wrapping arise when foliating the extended Schwarzschild spacetime with maximal slices. For arbitrary spatial coordinates these effects can be quantified in the context of boundary conditions where the lapse arises as a linear combination of odd and even lapse. Favorable boundary conditions are then derived which make the overall slice stretching occur late in numerical simulations. Allowing the lapse to become negative, this requirement leads to lapse functions which approach at late times the odd lapse corresponding to the static Schwarzschild metric. Demanding in addition that a numerically favorable lapse remains non-negative, as result the average of odd and even lapse is obtained. At late times the lapse with zero gradient at the puncture arising for the puncture evolution is precisely of this form. Finally, analytic arguments are given on how slice stretching effects can be avoided. Here the excision technique and the working mechanism of the shift function are studied in detail.