Physically motivated ansatz for the Kerr spacetime

Abstract

Despite some 60 years of work on the subject of the Kerr rotating black hole there is as yet no widely accepted physically based and pedagogically viable ansatz suitable for deriving the Kerr solution without significant computational effort. (Typically involving computer-aided symbolic algebra.) Perhaps the closest one gets in this regard is the Newman-Janis trick; a trick which requires several physically unmotivated choices in order to work. Herein we shall try to make some progress on this issue by using a non-ortho-normal tetrad based on oblate spheroidal coordinates to absorb as much of the messy angular dependence as possible, leaving one to deal with a relatively simple angle-independent tetrad-component metric. That is, we shall write $g_{ab}=g_{AB}{e}^A{}_a{e}^B{}_b$ seeking to keep both the tetrad-component metric $g_{AB}$ and the non-ortho-normal co-tetrad $e^A{}_a$ relatively simple but non-trivial. We shall see that it is possible to put all the mass dependence into $g_{AB}$, while the non-ortho-normal co-tetrad $e^A{}_a$ can be chosen to be a mass-independent representation of flat Minkowski space in oblate spheroidal coordinates: $(g_{\mathrm{Minkowski}}{)}_{ab}={\eta }_{AB}{e}^A{}_a{e}^B{}_b$. This procedure separates out, to the greatest extent possible, the mass dependence from the rotational dependence, and makes the Kerr solution perhaps a little less mysterious.

arXiv commentV1:19 pages; no figures. V2: one minor typo fixed; two new references

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Cited by in corpus (4)

• Killing horizons and surface gravities for a well-behaved three-function generalisation of the Kerr spacetime
• Fully extremal black holes: a black hole graveyard?
• Deriving Schwarzschild and Kerr from the Einstein Equivalence Principle
• Kerr metric from two commuting complex structures