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 nonzero for the Kerr solution. Thus, the Kerr metric admits no conformally flat slices. An analysis of higher orders in the expansions of the Cotton-York tensor for solutions such that the obstruction vanishes suggests that the only stationary solution admitting conformally flat slices are the Schwarzschild family of solutions.

Revised version to appear in Class. Quantum Grav. with 13 pages. Section 2 regarding multipolar expansions of stationary spacetimes largely expanded. A Maple script demonstrating the calculations in the axially symmetric case is available upon request from the author