Degeneracy measures for the algebraic classification of numerical spacetimes

TL;DR

This paper introduces a new measure of algebraic degeneracy for the Weyl tensor, clarifying the Petrov classification of numerical spacetimes like black hole mergers and showing that previous measures may misinterpret the spacetime type.

Contribution

It proposes a geometrically motivated degeneracy measure that improves the interpretation of Petrov types in numerical relativity simulations.

Findings

The new measure indicates black hole mergers settle directly to Petrov Type D.

Previous measures suggested a Type II 'hangup' which is an artifact.

The study clarifies the algebraic classification of dynamical spacetimes.

Abstract

We study the issue of algebraic classification of the Weyl curvature tensor, with a particular focus on numerical relativity simulations. The spacetimes of interest in this context, binary black hole mergers, and the ringdowns that follow them, present subtleties in that they are generically, strictly speaking, Type I, but in many regions approximately, in some sense, Type D. To provide meaning to any claims of "approximate" Petrov class, one must define a measure of degeneracy on the space of null rays at a point. We will investigate such a measure, used recently to argue that certain binary black hole merger simulations ring down to the Kerr geometry, after hanging up for some time in Petrov Type II. In particular, we argue that this hangup in Petrov Type II is an artefact of the particular measure being used, and that a geometrically better-motivated measure shows a black hole merger produced by our group settling directly to Petrov Type D.