Quadratic gravity as hair tonic for black holes

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Abstract
The field equations of a semiclassical theory of gravitation in 4+n dimensions with a Lagrangian consisting of the Hilbert-Einstein term (linear in curvature) and the Gauss-Bonnet combination (quadratic in curvature) are written in a four-dimensional form involving the 4-metric tensor and its derivatives, as well as the radius of the compactified internal space and its derivatives. It is found that in regions of spacetime where the curvature is small, so that the quadratic terms can be neglected, this theory reduces to the four-dimensional Einstein theory if the internal space possesses a flat geometry and its radius is constant. The influence of the quadratic terms on the Schwarzschild geometry is then evaluated using a simple approximated form of the field equations. It is found that the internal radius, regular at the event horizon, acts as scalar hair, and its effect on the resulting geometry is investigated.