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

TL;DR

This paper investigates the behavior of the Cotton-York tensor near infinity in stationary spacetimes, revealing conditions under which conformally flat slices exist, and concludes that only Schwarzschild solutions admit such slices.

Contribution

It provides asymptotic expansions of the Cotton-York tensor and identifies a quadrupolar obstruction preventing conformally flat slices in Kerr spacetime.

Findings

Kerr spacetime has a nonzero obstruction, so it admits no conformally flat slices.

Schwarzschild solutions are the only stationary solutions with conformally flat slices.

Higher order analysis suggests Schwarzschild is unique in this regard.

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.