Improving the "No-Hair" Theorem for the Proca Field

TL;DR

This paper refines the proof of the no-hair theorem for Proca fields around black holes by employing more natural horizon arguments and Einstein equations, strengthening the theoretical foundation of the theorem.

Contribution

It improves Bekenstein's original proof by justifying the horizon measure and clarifying the role of Einstein equations in establishing the no-hair theorem for Proca fields.

Findings

Horizon contribution to the integral identity is more involved.

The vanishing of the Proca field is confirmed using Einstein equations.

The proof is now based on more natural and justified arguments.

Abstract

This paper reconsider the problem of a Proca field in the exterior of a static black hole. The original Bekenstein's demonstration on the vanishing of this field, based on an integral identity, is improved by using more natural arguments at the event horizon. In particular, the use of the so-called standard integration measure in the horizon is fully justified. Accordingly, the horizon contribution to the Bekenstein integral identity is more involved and its vanishing can be only established using the related Einstein equations. With the new reasoning the ``no-hair'' theorem for the Proca field now rest on better founded grounds.