Analysis of a Bianchi-like equation satisfied by the Mars-Simon tensor

TL;DR

This paper studies a Bianchi-like equation satisfied by the Mars-Simon tensor, establishing conditions for solution existence, uniqueness, and asymptotic behavior, especially in the context of positive cosmological constant.

Contribution

It analyzes the Mars-Simon tensor's Bianchi-like equation, proving well-posedness and Fuchsian type properties at scri for the asymptotic Cauchy problem.

Findings

Constraints are preserved under a generalized Buchdahl condition.

Existence and uniqueness of solutions are established for the Fuchsian type equation.

The asymptotic Cauchy problem is well-posed at spacelike scri with positive cosmological constant.

Abstract

The Mars-Simon tensor (MST), which e.g. plays a crucial role to provide gauge invariant characterizations of the Kerr-NUT-(A)(dS) family, satisfies a Bianchi-like equation. In this paper we analyze this equation in close analogy to the Bianchi equation, in particular it will be shown that the constraints are preserved supposing that a generalized Buchdahl condition holds. This permits the systematic construction of solutions to this equation in terms of a well-posed Cauchy problem. A particular emphasis lies on the asymptotic Cauchy problem, where data are prescribed on a spacelike scri (i.e. for $\Lambda >0$). In contrast to the Bianchi equation, the MST equation is of Fuchsian type at scri, for which existence and uniqueness results are derived.