A Critical Assessment of Black Hole Solutions With a Linear Term in Their Redshift Function

TL;DR

This paper investigates the linear term in the redshift function of black holes across various gravity theories, analyzing its geometric and astrophysical implications, and exploring how thermodynamics can distinguish between theories and affect black hole remnants.

Contribution

It provides a detailed analysis of the linear term in black hole solutions across multiple gravity theories, linking it to physical parameters and thermodynamics to differentiate models.

Findings

The parameter β relates to curvature invariants and cosmological constants.

Black hole thermodynamics can break metric degeneracy between theories.

Remnant masses can differ across theories, affecting dark matter viability.

Abstract

Different theories of gravity can admit the same black hole solution, but the parameters usually have different physical interpretations. In this work we study in depth the linear term $\beta r$ in the redshift function of black holes, which arises in conformal gravity, de Rham-Gabadadze-Tolley (dRGT) massive gravity, $f(R)$ gravity (as approximate solution) and general relativity. Geometrically we quantify the parameter $\beta$ in terms of the curvature invariants. Astrophysically we found that $\beta$ can be expressed in terms of the cosmological constant, the photon orbit radius and the innermost stable circular orbit (ISCO) radius. The metric degeneracy can be broken once black hole thermodynamics is taken into account. Notably, we show that under Hawking evaporation, different physical theories with the same black hole solution (at the level of the metric) can lead to black hole remnants with different values of their physical masses with direct consequences on their viability as dark matter candidates. In particular, the mass of the graviton in massive gravity can be expressed in terms of the cosmological constant and of the formation epoch of the remnant. Furthermore the upper bound of remnant mass can be estimated to be around $0.5 \times 10^{27}$ kg.