No-Go Theorems for Hairy Black Holes in Scalar- or Vector-Tensor-Gauss-Bonnet Theory

TL;DR

This paper proves a no-go theorem showing that static spherically symmetric black holes cannot have vector hair in a specific scalar- and vector-tensor Gauss-Bonnet theory, as the vector field must vanish at and outside the horizon.

Contribution

It establishes a no-go theorem for vector hair in Einstein-Λ-Vector-Tensor-Gauss-Bonnet theory, extending understanding of black hole solutions in modified gravity.

Findings

Vector field vanishes at the event horizon.

No static spherically symmetric black holes with vector hair.

Vanishing of the vector field outside the horizon when degenerated.

Abstract

In this paper, we show a no-go theorem for static spherically symmetric black holes with vector hair in Einstein-$\Lambda$-Vector-Tensor-Gauss-Bonnet theory where a complex vector field non-minimally couples with Gauss-Bonnet invariant. For this purpose, we expand metric functions and radial functions of a vector field around the event horizon, and substitute the expansions into equations of motion. Demanding that the equations of motion are satisfied in each order, we show that the complex vector field vanishes on the event horizon. Moreover, when the event horizon is degenerated, it is also implied that the complex vector field vanishes on and outside the horizon.