Braneworlds scenarios in a gravity model with higher order spatial three-curvature terms

Abstract

In this work we study a Hořava-like 5-dimensional model in the context of braneworld theory. The equations of motion of such model are obtained and, within the realm of warped geometry, we show that the model is consistent if and only if λ takes its relativistic value 1. Furthermore, we show that the elimination of problematic terms involving the warp factor second order derivatives are eliminated by imposing detailed balance condition in the bulk. Afterwards, Israel’s junction conditions are computed, allowing the attainment of an effective Lagrangian in the visible brane. In particular, we show that the resultant effective Lagrangian in the brane corresponds to a (3+1)-dimensional Hořava-like model with an emergent positive cosmological constant but without detailed balance condition. Now, restoration of detailed balance condition, at this time imposed over the brane, plays an interesting role by fitting accordingly the sign of the arbitrary constant β, insuring a positive brane tension and a real energy for the graviton within its dispersion relation. Also, the brane consistency equations are obtained and, as a result, the model admits positive brane tensions in the compactification scheme if, and only if, β is negative and the detailed balance condition is imposed.

Appendix: Evaluation of \(\tilde{\varLambda}\)

From (30a), (30b), it is straightforward to obtain

$$ \tilde{\varLambda}=\frac{1}{2} \bigl(\alpha\,{}^{\hbox{\tiny {(4)}}} \tilde{R}^2+\beta\,{}^{(4)}\tilde{R} ^{ab}{}^{(4)} \tilde{R}_{ab} \bigr)+\frac{\tilde {q}^{ab}V_{ab}}{\tilde{N}}, $$

(84)

where the first term in the right hand side of the above equation can be found in (33). The more intricate term is $\tilde {q}^{ab}V_{ab}$. By defining $v_{ab}\equiv\tilde{N}f_{ab}$ ($v\equiv \tilde{g}^{ab}v_{ab}$) and by keeping in mind that v _4i =0, it can be expressed as [see (20)]

(85)

We have benefited from the identity

$$ {}^{(4)}\tilde{\varGamma}^4_{4a}={}^{\hbox{\tiny {(4)}}} \tilde{\varGamma}^a_{44}=0. $$

(86)

Note that

$$ {}^{(4)}\tilde{\varGamma}^4_{ij}=- \frac{\tilde {g}_{ij}}{2}\frac{\partial_y w}{w}, $$

(87)

while

$$ {}^{(4)}\tilde{\varGamma}^i_{4j}= \frac{\delta ^i_j}{2}\frac{\partial_y w}{w}. $$

(88)

Purely spatial indices lead to

$$ {}^{(4)}\tilde{\varGamma}^i_{jk}= \frac{q^{il}}{2} (2\partial_{(j}q_{k)l}-\partial _lq_{jk} )={}^{(3)}\varGamma^i_{jk}. $$

(89)

Although v _i4=0, this is not true for $\tilde{D}_{a}v_{4i}$. The important relations one needs at this point are

(90)

(91)

(92)

together with $\tilde{D}_{4}v_{i4}=0$. With those relations in mind one is able to get

(93)

(94)

(95)

and

(96)

By collecting the results (90)–(96) into (85), dividing it by $\tilde{N}=\sqrt{w}N$, writing $v_{ab}=\sqrt {w}Nf_{ab}$ and then inserting it into (84) one finally gets the result quoted in (78).