Geometric Invariant Measuring the Deviation from Kerr Data

TL;DR

This paper introduces a geometric invariant that measures how much a given vacuum Einstein initial data deviates from Kerr black hole data, using approximate Killing spinors to characterize non-Kerr behavior.

Contribution

The authors construct a new geometric invariant for vacuum initial data that vanishes precisely for Kerr slices, aiding in the analysis of black hole uniqueness and stability.

Findings

Invariant vanishes for Kerr data

Provides a quantitative measure of non-Kerr behavior

Introduces approximate Killing spinors as a key tool

Abstract

A geometrical invariant for regular asymptotically Euclidean data for the vacuum Einstein field equations is constructed. This invariant vanishes if and only if the data correspond to a slice of the Kerr black hole spacetime --thus, it provides a measure of the non-Kerr-like behavior of generic data. In order to proceed with the construction of the geometric invariant, we introduce the notion of approximate Killing spinors.