Charged Black Holes In Quadratic Theories

TL;DR

This paper investigates how quadratic gravitational theories alter the solutions and thermodynamics of charged black holes, finding that while some properties mirror Reissner-Nordström black holes, their entropy differs.

Contribution

It demonstrates that Reissner-Nordström black holes are not solutions in general quadratic theories and provides approximate solutions and thermodynamic analysis within these frameworks.

Findings

Reissner-Nordström solutions are not valid in most quadratic theories.

Approximate charged black hole solutions are derived for small coupling.

Black hole thermodynamics remains similar to RN but with different entropy.

Abstract

We point out that in general the Reissner-Nordstr\"om (RN) charged black holes of general relativity are not solutions of the four dimensional quadratic gravitational theories. They are, e.g., exact solutions of the $R+R^2$ quadratic theory but not of a theory where a $R_{ab}R^{ab}$ term is present in the gravitational Lagrangian. In the case where such a non linear curvature term is present with sufficiently small coupling, we obtain an approximate solution for a charged black hole of charge $Q$ and mass $M$. For $Q\ll M$ the validity of this solution extends down to the horizon. This allows us to explore the thermodynamic properties of the quadratic charged black hole and we find that, to our approximation, its thermodynamics is identical to that of a RN black hole. However our black hole's entropy is not equal to the one fourth of the horizon area. Finally we extend our analysis to the rotating charged black hole and qualitatively similar results are obtained.