Some works appeared regarding the possibility of introducing f (R) theories in the context of braneworld scenarios [499, 40, 96, 513, 97]. In doing so one requires a surface term [222, 482, 69, 48, 49, 286], which is known as the Hawking–Luttrell term [295] (analogous to the York–Gibbons–Hawking one for General Relativity). The action we consider is given by

where , is the determinant of the induced metric on the dimensional boundary, and is the trace of the extrinsic curvature tensor.In this case particular attention should be paid to boundary conditions on the brane, that is, the Israel junction conditions [323]. In order to have a well-defined geometry in five dimensions, we require that the metric is continuous across the brane located at . However its derivatives with respect to can be discontinuous at . The Ricci tensor in Eq. (2.4) is made of the metric up to the second derivatives with respect to . This means that have a delta-function dependence proportional to the energy-momentum tensor at a distributional source (i.e., with a Dirac’s delta function centered on the brane) [87, 86, 536]. In general this also leads to the discontinuity of the Ricci scalar across the brane. Since the discontinuity of can lead to inconsistencies in f (R) gravity, one should add this extra-constraint as a junction condition. In other words, one needs to impose that, although the metric derivative is discontinuous, the Ricci scalar should still remain continuous on the brane.

This is tantamount to imposing that the extra scalar degree of freedom introduced is continuous on the brane. We use Gaussian normal coordinates with the metric

In terms of the extrinsic curvature tensor for a brane, the l.h.s. of the equations of motion tensor [which is analogous to the l.h.s. of Eq. (2.4) in 4 dimensions] is defined by This has a delta function behavior for the - components, leading to [207] where is the matter stress-energy tensor on the brane. Hence is continuous, whereas its first derivative is not, in general. This imposes an extra condition on the metric crossing the brane at , compared to General Relativity in which the condition for the continuity of is not present. However, it is not easy to find a solution for which the metric derivative is discontinuous but is not. Therefore some authors considered matter on the brane which is not universally coupled with the induced metric. This approach leads to the relaxation of the condition that is continuous. Such a matter action can be found by analyzing the action in the Einstein frame and introducing a scalar field coupled to the scalaron on the brane as follows [207] The presence of the coupling with the field modifies the Israel junction conditions. Indeed, if , then must be continuous, but if , can have a delta function profile. This method may help for finding a solution for the bulk that satisfies boundary conditions on the brane.http://www.livingreviews.org/lrr-2010-3 |
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