Towards a Gordon form of the Kerr spacetime

TL;DR

This paper explores the possibility of expressing the Kerr spacetime metric in Gordon form, providing approximate solutions in slow-rotation and near-null limits to aid in developing analogue models.

Contribution

It presents the first Gordon form approximations for Kerr spacetime in slow-rotation and near-null limits, advancing toward an exact Gordon form.

Findings

Gordon form for slow-rotation Kerr approximation derived

Gordon form near-null limit for Kerr spacetime obtained

Progress suggests potential for an exact Gordon form of Kerr spacetime

Abstract

It is not currently known how to put the Kerr spacetime metric into the so-called Gordon form, although the closely related Kerr-Schild form of the Kerr metric is well known. A Gordon form for the Kerr geometry, if it could be found, would be particularly useful in developing analogue models for the Kerr spacetime, since the Gordon form is explicitly given in terms of the 4-velocity and "refractive index" of an effective medium. In the current article we report progress toward this goal. First we present the Gordon form for an approximation to Kerr spacetime in the slow-rotation limit, obtained by suitably modifying the well-known Lense-Thirring form of the slow-rotation metric. Second we present the Gordon form for the Kerr spacetime in the near-null limit, (the 4-velocity of the medium being close to null). That these two perturbative approximations to the Kerr spacetime in Gordon form exist gives us some confidence that ultimately one might be able to write the exact Kerr spacetime in this form.