Kerr black holes with Proca hair
Carlos Herdeiro, Eugen Radu, Helgi Runarsson

TL;DR
This paper demonstrates the existence of rotating black holes with Proca hair, challenging previous no-hair theorems, by constructing explicit solutions and analyzing their properties in Einstein-Proca theory.
Contribution
It proves that stationary black holes can have Proca hair when matter does not inherit spacetime symmetries, and explicitly constructs such solutions.
Findings
Existence of Kerr black holes with Proca hair confirmed.
Proca hair solutions branch from Kerr black holes with clouds.
Some solutions exhibit counter-rotation with respect to the horizon.
Abstract
Bekenstein proved that in Einstein's gravity minimally coupled to one (or many) real, Abelian, Proca field, stationary black holes (BHs) cannot have Proca hair. Dropping Bekenstein's assumption that matter inherits spacetime symmetries, we show this model admits asymptotically flat, stationary, axi-symmetric, regular on and outside an event horizon BHs with Proca hair, for an even number of real (or an arbitrary number of complex) Proca fields. To establish it, we start by showing that a test, complex Proca field can form bound states, with real frequency, around Kerr BHs: stationary Proca clouds. These states exist at the threshold of superradiance. It was conjectured in arXiv:1403.2757, that the existence of such clouds at the linear level implies the existence of a new family of BH solutions at the non-linear level. We confirm this expectation and explicitly construct examples of…
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