13.4 f (R) theories in extra dimensions

Although f (R) theories have been introduced mainly in four dimensions, the same models may appear in the context of braneworld [502501] in which our universe is described by a brane embedded in extra dimensions (see [404] for a review). This scenario implies a careful use of f (R) theories, because a boundary (brane) appears. Before looking at the real working scenario in braneworld, it is necessary to focus on the mathematical description of f (R) models through a sensible definition of boundary conditions for the metric elements on the surface of the brane.

Some works appeared regarding the possibility of introducing f (R) theories in the context of braneworld scenarios [499409651397Jump To The Next Citation Point]. In doing so one requires a surface term [222482694849286], 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

∫ √ --- ∫ ∘ --- S = dnx − gf(R ) + 2 dn−1x |γ |F K, (13.38 ) Ω ∂Ω
where F ≡ ∂f ∕∂R, γ is the determinant of the induced metric on the n − 1 dimensional boundary, and K 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 y = 0. However its derivatives with respect to y can be discontinuous at y = 0. The Ricci tensor Rμν in Eq. (2.4View Equation) is made of the metric up to the second derivatives ′′ g with respect to y. This means that ′′ g 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) [8786536]. In general this also leads to the discontinuity of the Ricci scalar R across the brane. Since the discontinuity of R 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

ds2 = dy2 + γμνdxμdx ν. (13.39 )
In terms of the extrinsic curvature tensor K μν = − ∂yγμν∕2 for a brane, the l.h.s. of the equations of motion tensor [which is analogous to the l.h.s. of Eq. (2.4View Equation) in 4 dimensions] is defined by
1- ΣAB ≡ FRAB − 2fgAB − ∇A ∇BF (R ) + gAB □F (R ). (13.40 )
This has a delta function behavior for the μ-ν components, leading to [207Jump To The Next Citation Point]
D μν ≡ [F (K μν − K γμν) + γμνF,R∂yR ]+− = T μν, (13.41 )
where Tμν is the matter stress-energy tensor on the brane. Hence R is continuous, whereas its first derivative is not, in general. This imposes an extra condition on the metric crossing the brane at y = 0, compared to General Relativity in which the condition for the continuity of R is not present. However, it is not easy to find a solution for which the metric derivative is discontinuous but R 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 R is continuous. Such a matter action can be found by analyzing the action in the Einstein frame and introducing a scalar field ψ coupled to the scalaron ϕ on the brane as follows [207]
∫ [ ] SM = dn−1x√ −-γ-exp[(n − 1)C (ϕ )] − 1-exp[− 2C (ϕ)]γμν∇ μψ ∇ νψ − V (ψ) . (13.42 ) 2
The presence of the coupling C (ϕ) with the field ϕ modifies the Israel junction conditions. Indeed, if C = 0, then R must be continuous, but if C ⁄= 0, R can have a delta function profile. This method may help for finding a solution for the bulk that satisfies boundary conditions on the brane.
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