A set of invariant quality factors measuring the deviation from the Kerr metric

TL;DR

This paper introduces scalar invariant quality factors that measure how closely a stationary space-time resembles the Kerr solution, useful for stability analysis and gravitational collapse studies.

Contribution

It presents new scalar invariant quality factors that quantify the deviation from Kerr geometry without prior knowledge of the Kerr solution.

Findings

Quality factors range from 0 to 1, with 1 indicating exact Kerr geometry.

Different sets of quality factors are constructed for various space-time conditions.

These factors are useful for stability analysis and characterizing gravitational collapse.

Abstract

A number of scalar invariant characterizations of the Kerr solution are presented. These characterizations come in the form of {\em quality factors} defined in stationary space-times. A quality factor is a scalar quantity varying in the interval $[0,1]$ with the value 1 being attained if and only if the space-time is locally isometric to the Kerr solution. No knowledge of the Kerr solution is required to compute these quality factors. A number of different possibilities arise depending on whether the space-time is Ricci-flat and asymptotically flat, just Ricci-flat, or Ricci non-flat. In each situation a number of quality factors are constructed and analysed. The relevance of these quality factors is clear in any situation where one seeks a rigorous formulation of the statement that a space-time is "close" to the Kerr solution, such as: its non-linear stability problem, the asymptotic settlement of a radiating isolated system undergoing gravitational collapse, or in the formulation of some uniqueness results.