On the absence of conformally flat slicings of the Kerr spacetime

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, phi) 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 phi.