On the absence of conformally flat slicings of the Kerr spacetime

Abstract

This work investigates the possibility of achieving a conformally flat slicing of the Kerr spacetime. We consider a hypersurface of the form $t=F(r,\theta ,a)$, where $(t,r,\theta ,\varphi )$ are the Boyer-Lindquist coordinates, solve for a vanishing Cotton-York tensor of the induced metric order by order in the spin parameter $a$, and show that the procedure fails at the fifth order. We also prove that no coordinate change can induce a spatially flat recasting of the Kerr(-de Sitter) metric, beyond linear order in $a$, adopting a more general ansatz depending on $\varphi$.

12 pages No-go result on spatially flat slicings upgraded from Kerr spacetimes to Kerr-de Sitter spacetimes