Perturbative evolution of nonlinear initial data for binary black holes: Zerilli vs. Teukolsky

TL;DR

This paper compares the effectiveness of Zerilli and Teukolsky formalisms in evolving nonlinear initial data for binary black holes, finding Teukolsky's approach more stable and accurate in the close limit regime.

Contribution

It demonstrates that Weyl scalar $$ evolved via the Teukolsky equation outperforms metric perturbations via Zerilli in nonlinear regimes, informing perturbative and numerical relativity methods.

Findings

Teukolsky's $$ remains accurate up to larger separations.

Zerilli's metric perturbations break down prematurely.

Results apply to a broader class of initial data.

Abstract

We consider the problem of evolving nonlinear initial data in the close limit regime. Metric and curvature perturbations of nonrotating black holes are equivalent to first perturbative order, but Moncrief waveform in the former case and Weyl scalar $\psi_4$ in the later differ when nonlinearities are present. For exact Misner initial data (two equal mass black holes initially at rest), metric perturbations evolved via the Zerilli equation suffer of a premature break down (at proper separation of the holes $L/M\approx2.2$) while the exact Weyl scalar $\psi_4$ evolved via the Teukolsky equation keeps a very good agreement with full numerical results up to $L/M\approx3.5$. We argue that this inequivalent behavior holds for a wider class of conformally flat initial data than those studied here. We then discuss the relevance of these results for second order perturbative computations and for perturbations to take over full numerical evolutions of Einstein equations.