Darboux diagonalization of the spatial 3-metric in Kerr spacetime

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.

18 pages

References in corpus (3)

• Vaidya spacetimes, black-bounces, and traversable wormholes
• Thin-shell traversable wormhole crafted from a regular black hole with asymptotically Minkowski core
• On the space-time curvature experienced by quasiparticle excitations in the Painleve-Gullstrand effective geometry

Cited by in corpus (7)

• Killing tensor and Carter constant for Painleve-Gullstrand form of Lense-Thirring spacetime
• Killing horizons and surface gravities for a well-behaved three-function generalisation of the Kerr spacetime
• Constant-$r$ geodesics in the Painleve-Gullstrand form of Lense-Thirring spacetime
• Physically motivated ansatz for the Kerr spacetime
• Painleve-Gullstrand coordinates versus Kerr spacetime geometry
• The eye of the storm: A regular Kerr black hole
• Two discs and a missing triangle: the maximally extended Kerr black hole revisited