Rotating black holes with primary hair in five-dimensional generalized Proca theory

TL;DR

This paper introduces a new class of exact rotating black hole solutions with primary hair in five-dimensional generalized Proca theories, extending known solutions like Myers-Perry with additional vector field features.

Contribution

It provides the first analytic rotating black hole solutions with primary hair in five-dimensional generalized Proca theories using a Kerr-Schild ansatz.

Findings

Solutions include a cosmological constant and two angular momenta.

The solutions exhibit primary hair as an arbitrary function of an angular coordinate.

They generalize the Myers-Perry black holes with additional Proca field contributions.

Abstract

This work presents a new class of exact analytic rotating black hole solutions within five-dimensional generalized Proca theories. Through a Kerr-Schild ansatz where the Proca field is set along a null geodesic congruence, the non-linear field equations reduce to a consistent set of three master equations. This geometric configuration ensures that the vector field remains light-like on-shell, effectively restricting the theory's functional couplings to discrete constants and allowing for a fully analytic treatment. The resulting solutions, incorporating a cosmological constant and two independent angular momenta, exhibit primary hair given by an arbitrary function of the non-Killing angular coordinate. We identify several solution branches defined by specific algebraic relations between the Proca coupling constants, providing a significant generalization of the Myers-Perry family. Notably, the metric retains a Kerr-Schild form identical to the Myers-Perry representation, with an additional contribution constructed from the tensor product of the Proca one-form with itself.