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

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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 C²=Cαᵦᵧδ Cαβγδ. 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 C² 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 secondary or induced hair. We explore how the Tangherlini solution is modified by `quantum corrections', assuming that the gravitational radius r₀ 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).