Asymptotic expansions of the Cotton–York tensor on slices of stationary spacetimes

Abstract

We discuss expansions for the Cotton–York tensor near infinity for arbitrary slices of stationary spacetimes. From these expansions, it follows directly that a necessary condition for the existence of conformally flat slices in stationary solutions is the vanishing of a certain quantity of quadrupolar nature (obstruction). The obstruction is non-zero for the Kerr solution. Thus, the Kerr metric admits no conformally flat slices. An analysis of the next order terms in the expansions in the case of solutions such that the obstruction vanishes, suggests that the only stationary solutions admitting conformally flat slices are the Schwarzschild family of solutions.