Spherically symmetric brane in a bulk of f(R) and Gauss–Bonnet gravity

Abstract

Effective gravitational field equations on a four-dimensional brane embedded in a five-dimensional bulk have been considered. Using the Einstein–Hilbert action along with the Gauss–Bonnet correction term, we have derived static spherically symmetric vacuum solution to the effective field equations, first order in the Gauss–Bonnet coupling parameter. The solution so obtained, has one part corresponding to general relativity with an additional correction term, proportional to the Gauss–Bonnet coupling parameter. The correction term modifies the spacetime structure, in particular, the location of the event horizon. Proceeding further, we have derived effective field equations for gravity with Gauss–Bonnet correction term and a static spherically symmetric solution has been obtained. In this case the Gauss–Bonnet term modifies both the event and cosmological horizon of the spacetime. There exists another way of obtaining the brane metric—expanding the bulk gravitational field equations in the ratio of bulk to brane curvature scale and assuming a separable bulk metric ansatz. It turns out that static, spherically symmetric solutions obtained from this perturbative method can be matched exactly, with the solutions derived earlier. This will hold for Einstein–Hilbert plus Gauss–Bonnet as well as for f(R) with the Gauss–Bonnet correction. Implications of these results are discussed.