Black Holes in Higher Dimensional Gravity Theory with Quadratic in Curvature Corrections

TL;DR

This paper investigates how quadratic curvature corrections, motivated by quantum effects, modify higher-dimensional black hole solutions, revealing the emergence of secondary hair and deriving a master equation for solutions.

Contribution

It demonstrates that quantum-inspired quadratic curvature terms alter higher-dimensional black hole solutions, introducing secondary hair and providing a master equation for their analysis.

Findings

Schwarzschild solution remains valid in 4D with Weyl correction.

Higher dimensions exhibit modifications to Tangherlini black holes due to quantum corrections.

A reduction to a single third-order ODE enables analysis of solutions beyond perturbation.

Abstract

Static spherically symmetric black holes are discussed in the framework of higher dimensional gravity with quadratic in curvature terms. Such terms naturally arise as a result of quantum corrections induced by quantum fields propagating in the gravitational background. We focus our attention on the correction of the form ${\cal C}^2=C_{\alpha\beta\gamma\delta} C^{\alpha\beta\gamma\delta}$. The Gauss-Bonnet equation in four-dimensional (4D) spacetime enables one to reduce this term in the action to the terms quadratic in the Ricci tensor and scalar curvature. As a result the Schwarzschild solution which is Ricci flat will be also a solution of the theory with the Weyl scalar ${\cal C}^2$ correction. An important new feature of the spaces with dimension $D > 4$ is that in the presence of the Weyl curvature-squared term a solution necessary differs from the corresponding `classical' vacuum Tangherlini metric. This difference is related to the presence of {\em secondary} or {\em induced} hair. We explore how the Tangherlini solution is modified by `quantum corrections', assuming that the gravitational radius $r_0$ is much larger than the scale of the quantum corrections. We also demonstrated that finding a general solution beyond the perturbation method can be reduced to solving a single third order ODE (master equation).