# Braneworld gravity within non-conservative gravitational theory

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## Abstract

We investigate the braneworld gravity starting from the non-conservative gravitational field equations in a five-dimensional bulk. The approach is based on the Gauss–Codazzi formalism along with the study of the braneworld consistency conditions. The effective gravitational equations on the brane are obtained and the constraint leading to a brane energy-momentum conservation is analyzed.

## 1 Introduction

Despite the ubiquity of dissipative processes in the real world, it is intriguing to notice their absence in the standard formulations of the principle of least action. In the traditional classical mechanics, dissipative phenomena are handled by means of the Rayleigh dissipation function which comes into play through an extra term in the Euler-Lagrange equations, where one does not abandon however the underlying variational formalism, so that the Lagrangian of the system is kept untouched [1]. A first effort attempting to construct, within the classical mechanics context, a full formalism to describe dissipative systems from the perspective of a principle of least action dates back to the Herglotz’s work [2]. In his approach he argue that it would be possible to describe a physical system endowed with dissipation by assuming an action-dependent Lagrangian. For instance, when considering a linear dependence on the action he has shown the appearance of a typical velocity-proportional frictional term in the corresponding equations of motion derived from such Lagrangian. Almost ninety years separated the pioneer Herglotz’s contribution from a covariant extension of his formalism, which was just recently accomplished by Lazo et al. [3]. From this covariant formulation the authors constructed a new theory of gravity consisting of a set of modified field equations along with a non-conservation for the energy-momentum tensor. They make a discussion on the possible consequences of this “geometric” dissipation effects on the cosmological scenario, pointing out that these new degrees of freedom can account for the dark energy content in the universe. Besides, they add an study on the gravitational waves propagation within this theory.

In a very recent paper, a more complete investigation of cosmological aspects in this non-conservative gravity is performed [4]. At the background level, the authors show an equivalence between this non-conservative cosmology with the bulk viscous model in the Eckart’s formalism [5, 6]. Whereas at the perturbative level, they verified that the linear perturbations indicate a possible way out to alleviate the problems faced by the viscous cosmology.

Braneworld models have attracted the attention of the scientific community due its possible application to the hierarchy problem [7]. Soon after the appearance of such a possibility, the gravitational aspects of these models started to be under investigation. In particular, a systematic study performed by means of the Gauss–Codazzi formalism [8, 9] made possible a broad range of applications of braneworld scenarios in gravitation and cosmology. From among the several interesting prospects resulting from this investigation, in the context of braneworld gravity, is the impossibility of covariant conservation of the brane stress tensor when matter is present in the bulk [10]. The main purpose of this paper is to investigate the physical consequences of such a geometric induced non-conservation of the energy-momentum on the construction of braneworld models.

After a short introduction highlighting some of the main aspects of non-conservative gravity in Sect. 2, we apply in Sect. 3 the Gauss–Codazzi formalism assuming that the bulk gravity is governed by its precepts. It is shown that the non-conservative aspect of the bulk gravity can counterbalance the bulk matter effect leading to a covariant conservation of the brane stress-tensor. It is also shown a complete gravitational effective field equation, along with a corrected four-dimensional gravitational ‘constant’, which now acquires a dependence upon the coordinates. In Sect. 4 we approach non-conservative braneworld models with the aid of the well known braneworld sum rules, a complete formalism resulting in a one-parameter family of consistency conditions. It is shown that, in this specific context, it is possible to derive an extension of the Randall–Sundrum model without using a negative brane tension. In the final Section we conclude emphasizing the possible applications in cosmology.

## 2 A toolkit on non-conservative gravity

*S*. Such a condition leads to a generalized version of the Euler-Lagrangian equation

^{1}This non-conservative theory of gravity presents the following set of field equations

## 3 Applying the Gauss–Codazzi formalism

*r*is the index of the fifth dimension. Besides, one denotes \(g_{\mu \nu }=q_{\mu \nu }+n_\mu n_\nu \), where \(n_\mu \) is a unitary vector orthogonal everywhere on the brane, provided it is orientable. In terms of (the variation of) \(n_\mu \) the extrinsic curvature reads \(K_{\mu \nu }=q_\mu ^\alpha q_\nu ^\beta \nabla _\alpha n_\beta \). It is clear from these choices that from 0 to 4 in the indexes we are restricted to the brane, leaving the last index value to the extra dimension. Notice that the physical content of Eq. (5) may be simply stayed as follows: the brane curvature is given by the projection of the bulk curvature, also having into account the way the brane is embedded in the bulk.

*Q*. Expressing, then, the relevant quantities by means of the Heaviside distribution, its derivatives and products must fulfill the rules of the distributional calculus, from which the Israel–Darmois junctions conditions arise. It is to be noticed, however, that \(\mathcal{K}_{\mu \nu }\) does not have second derivatives in the metric and therefore both junction conditions are nothing else but the usual ones. Thus, attributing a energy-momentum tensor of the form

*v*(in the case of a homogeneous and isotropic brane), usually called the brane tension, from the stress-tensor on the brane, \(\tau _{\beta \delta }\). Taking advantage of Eq. (10) along with (9) one arrives at the effective gravitational equation on the brane given by

We shall finalize this section pointing out that Eq. (18) must be implemented for any (braneworld) model builder who want to ensure conservation of the brane stress tensor in the context studied here. It shall imprint a severe constraint on the model in question. In the next section we dedicate more attention to this question, not by investigating a particular model, but instead appreciating the consequences of (18) which are to be shared by any model constructed in such a scope.

## 4 Braneworld sum rules

In trying to find out consistency conditions for braneworlds whose orbifold character is present, i.e., whose internal space is indeed compact, it was conceived an important formalism giving the necessary rules to be fulfilled by the plethora of models conceived since the publication of [7]. This formalism was presented in Ref. [12], generalized in Ref. [13], and studied under several different aspects [14, 15, 16]. We shall depict here the main relevant aspects for our purposes. When thinking of possible using the braneworld sum rules in the non-conservative gravity context a word of warning is in order. It seems possible, though nontrivial, to find out the generalized partial traces coming from (4) and thus to achieve the consistency conditions accordingly. Nevertheless, as we want to deal with non-conservative gravity theory in the bulk, we are going to use the standard protocol.

*D*-dimensional bulk. Besides it is indeed more profitable to change the notation a bit making explicit the separation between bulk, brane, and internal space. The line element reads

*W*(

*r*) contribution. Finally, the \((D-p-1)-\)dimensional internal space is described by \(k_{mn}\). Also, in order to make utterly clear the different geometrical quantities we denote by \(\tilde{\tilde{A}}\) internal space quantities, while \(\bar{\bar{A}}\) stands for a brane quantities. Thus it can be readily verified that

## 5 Conclusion

The study we have performed exhausted the formal approach concerning braneworld scenarios based upon non-conservative gravity. It is important to emphasize that even in the most rudimentary approach, the result encoded in Eqs. (11)–(12) is promising from the cosmological point of view. In fact, in a context in which the \(\pi _{\mu \nu }\) can be disregarded (notice the \(\kappa _5^4\) coefficient), the bulk has no additional stresses, and the geometrical setup carries symmetries enough to set \(E_{\mu \nu }=0\), the remaining effective field equation has some properties potentially interesting at cosmological level.

We have analyzed a possible braneworld setup based upon a gravitational theory recently proposed where dissipative effects are introduced in the least action principle. We used this framework to generalize the consistency conditions to be obeyed by any viable braneworld model. We have shown that these non-conservative terms appearing in the new consistency relations open the possibility of relaxing the negative tension condition verified in the Randall–Sundrum context, so avoiding an undesirable property which plagues some braneworld models. Besides, we have seen through Eq. (12) that this model of gravity provides a braneworld scenario with a running effective cosmological “constant”. As such this novel aspect is promising for cosmology as it can make feasible the emerging of interactions between dark energy and dark matter [17, 18, 19, 20, 21, 22, 23, 24, 25, 26].

Our study also shows that the model investigated is allowed to have a standard conservation law for the energy-momentum tensor on the brane even with a non-zero stress in the bulk. On the other hand, we have seen that it is also possible to exist exchange of energy between the brane and the bulk, even if there is no stress in the bulk. The cosmological consequences of the possibilities arising in the present study shall be investigated in a future opportunity.

## Footnotes

## Notes

### Acknowledgements

JCF thanks to CNPq and FAPES for the financial support. JMHS has been partially supported by CNPq and also thanks to the Cosmology group of UFES for the kind hospitality. TRPC is grateful to CAPES for the full financial support.

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