Darboux diagonalization of the spatial 3-metric in Kerr spacetime

TL;DR

This paper investigates the possibility of diagonalizing the spatial 3-metric in Kerr spacetime using Darboux's method, revealing fundamental incompatibilities with other desirable metric properties.

Contribution

It demonstrates a no-go theorem showing that Darboux diagonalization of Kerr's spatial slices cannot coexist with unit-lapse conditions and axial symmetry.

Findings

Darboux diagonalization is incompatible with unit-lapse form in Kerr spacetime.

The spatial 3-slices of Kerr cannot be made conformally flat while maintaining certain conditions.

A no-go theorem analogous to the conformal flatness restriction is established.

Abstract

The astrophysical importance of the Kerr spacetime cannot be overstated. Of the currently known exact solutions to the Einstein field equations, the Kerr spacetime stands out in terms of its direct applicability to describing astronomical black hole candidates. In counterpoint, purely mathematically, there is an old classical result of differential geometry, due to Darboux, that all 3-manifolds can have their metrics recast into diagonal form. In the case of the Kerr spacetime the Boyer-Lindquist coordinates provide an explicit example of a diagonal spatial 3-metric. Unfortunately, as we demonstrate herein, Darboux diagonalization of the spatial 3-slices of the Kerr spacetime is incompatible with simultaneously putting the Kerr metric into unit-lapse form while retaining manifest axial symmetry. This no-go theorem is somewhat reminiscent of the no-go theorem to the effect that the spatial 3-slices of the Kerr spacetime cannot be chosen to be conformally flat.