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

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 ${\mathcal{C}}^2=C_{\alpha \beta \gamma \delta }{C}^{\alpha \beta \gamma \delta }$. 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 ${\mathcal{C}}^2$ 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).

12 pages, 4 figures. A few references added, some details added to better illustrate the result. Version accepted in Physical Review D

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• Qualifying ringdown and shadow of black holes under general parametrized metrics with photon orbits
• Beyond Schwarzschild-de Sitter spacetimes: I. A new exhaustive class of metrics inspired by Buchdahl for pure $R^2$ gravity in a compact form
• On effective models of regular black holes inspired by higher-derivative and nonlocal gravity
• The static charged black holes with Weyl corrections
• Emerging Newtonian potential in pure $R^2$ gravity on a de Sitter background
• Black Hole Solutions in $R^2$ Gravity