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 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.

Extreme Bowen–York initial data

“…It is not a priori obvious that in the limit we do not obtain extreme Kerr data. However, this follows from the theorem proved in [28], because our data are conformally flat and there are no conformally flat slices in Kerr (including the extreme limit). Moreover (using the theorem proved in [16]) we also conclude that for the Bowen-York data the strict inequality √ |J | < m holds (cf equation (2)), where m is the total mass of the data (for extreme Kerr we have √ |J | = m).…”

Conformally flat slices of asymptotically flat spacetimes

“…Therefore several distinct precise formulations of the concept have arisen. A key development in these definitions has been the use of conformal compactification [32], which was used [14,15] in the demonstration that there is no conformally flat slice of the Kerr spacetime. We instead work with a more pedestrian definition, which is motivated and stated in the following.…”

“…Simplifying choices may however have unfortunate physical consequences on the data being constructed. It is known, for example, that the Kerr spacetime admits no spatial slice which is conformally flat [14,15]. Therefore the use of this restriction, even in the construction of a single spinning black hole, must result in data which corresponds not to Kerr, but to some physical deformation thereof.…”

On the characterization of non-degenerate foliations of pseudo-Riemannian manifolds with conformally flat leaves

“…Let (V, g) be a pseudo-Riemannian manifold with its standard Levi-Civita connection ∇ and let P, be any pair fulfilling (4). Use them to define the quantity L a bc by means of (7). The bi-conformal connection is by definition the only linear connection such that its difference with the Levi-Civita connection results in the tensor L a bc .…”

“…This existence problem has been addressed in very particular cases and with very specific motivations. 2,7 The results presented in this paper enable us to address the existence problem in full generality. This paper is structured as follows: in Sec.…”