Rotating Killing horizons in generic $F(R)$ gravity theories

TL;DR

This paper explores properties of rotating Killing horizons in generic $F(R)$ gravity, demonstrating the constancy of surface gravity and ruling out certain hairy solutions under specific conditions.

Contribution

It extends the understanding of horizon properties in $F(R)$ gravity, showing surface gravity constancy and constraints on hairy solutions in these theories.

Findings

Surface gravity is constant on the horizon.

No hairy solutions for massive vector fields under positive scalar potential.

Effective gravitational coupling varies with polar coordinate on the horizon.

Abstract

We discuss various properties of rotating Killing horizons in generic $F(R)$ theories of gravity in dimension four for spacetimes endowed with two commuting Killing vector fields. Assuming there is no curvature singularity anywhere on or outside the horizon, we construct a suitable $(3+1)$-foliation. We show that similar to Einstein's gravity, we must have $T_{ab}k^ak^b=0$ on the Killing horizon, where $k^a$ is a null geodesic tangent to the horizon. For axisymmetric spacetimes, the effective gravitational coupling $\sim\,F'^{-1}(R)$ should usually depend upon the polar coordinate and hence need not necessarily be a constant on the Killing horizon. We prove that the surface gravity of such a Killing horizon must be a constant, irrespective of whether $F'(R)$ is a constant there or not. We next apply these results to investigate some further basic features. In particular, we show that any hairy solution for the real massive vector field in such theories is clearly ruled out, as long as the potential of the scalar field generated in the corresponding Einstein's frame is a positive definite quantity.