Closed conformal Killing-Yano initial data

TL;DR

This paper derives conditions on initial data for Einstein's equations that guarantee the development admits a special 2-form related to hidden symmetries, extending known results to higher dimensions.

Contribution

It introduces the closed conformal Killing-Yano initial data (cCYKID) equations, providing necessary and sufficient conditions for higher-dimensional spacetimes to admit such symmetries.

Findings

Produced a 5-parameter family of propagation identities for the CKY equation.

Identified conditions ensuring the development admits a closed conformal Killing-Yano 2-form.

Handled the special 4-dimensional case separately.

Abstract

Through an exhaustive search, we produce a 5-parameter family of propagation identities for the closed conformal Killing-Yano equation on 2-forms, which hold on an Einstein cosmological vacuum spacetime in any dimension $n>4$. It is well-known that spacetimes admitting a non-degenerate 2-form of this type are exhausted by the Kerr-NUT-(A)dS family of exact higher dimensional black hole solutions. As a consequence, we identify a set of necessary and sufficient conditions ensuring that the cosmological vacuum development of an initial data set for Einstein's field equations admits a closed conformal Killing-Yano 2-form. We refer to these conditions as \emph{closed conformal Killing-Yano initial data} (cCYKID) equations. The 4-dimensional case is special and is treated separately, where we can also handle the conformal Killing-Yano equation without the closed condition.