On the existence of certain axisymmetric interior metrics

TL;DR

This paper investigates the existence of specific axisymmetric metrics inspired by noncommutative geometry, focusing on how noncommutativity affects singularities in Kerr solutions through Einstein equations.

Contribution

It derives Einstein equations for noncommutative geometry inspired Kerr metrics and proves theorems on their existence.

Findings

Proved theorems on the existence of certain noncommutative Kerr metrics.

Derived Einstein equations for axisymmetric, noncommutative-inspired metrics.

Analyzed the impact of noncommutativity on ring-like singularities.

Abstract

One of the effects of noncommutative coordinate operators is that the delta-function connected to the quantum mechanical amplitude between states sharp to the position operator gets smeared by a Gaussian distribution. Although this is not the full account of effects of noncommutativity, this effect is in particular important, as it removes the point singularities of Schwarzschild and Reissner-Nordstr\"{o}m solutions. In this context, it seems to be of some importance to probe also into ring-like singularities which appear in the Kerr case. In particular, starting with an anisotropic energy-momentum tensor and a general axisymmetric ansatz of the metric together with an arbitrary mass distribution (e.g. Gaussian) we derive the full set of Einstein equations that the Noncommutative Geometry inspired Kerr solution should satisfy. Using these equations we prove two theorems regarding the existence of certain Kerr metrics inspired by Noncommutative Geometry.