Perturbative Metric of Charged Black Holes in Quadratic Gravity

We consider perturbative solutions to the classical field equations coming from a quadratic gravitational lagrangian in four dimensions. We study the charged, spherically symmetric black hole and explicitly give corrections up to third order (in the coupling constant $\beta$ multiplying the $R_{\mu \nu }{R}^{\mu \nu }$ term) to the Reissner–Nordstr"om hole metric. We consider the thermodynamics of such black holes, in particular, we compute explicitly its temperature and entropy–area relation which deviates from the usual $S=A\mathrm{/}4$ expression.