# Covariant Gravitational Equations on Brane World with Gauss-Bonnet term

###### Abstract

We present the covariant gravitational equations to describe a four-dimensional brane world in the case with the Gauss-Bonnet term in a bulk spacetime, assuming that gravity is confined on the symmetric brane. It contains some components of five-dimensional Weyl curvature () which describes all effects from the bulk spacetime just as in the case of the Randall-Sundrum second model. Applying this formalism to cosmology, we derive the generalized Friedmann equation and calculate the Weyl curvature term, which is directly obtained from a black hole solution.

## I Introduction

A brane is now one of the most important ideas in particle physics[1]. It may provide us a new solution for the so-called hierarchy problem and a new mechanism for compactification of extra dimensions. Since the fundamental scale could be TeV in some models[2], a gravitational effect is not ignored even at much lower energy scale than the Planck mass. For example, a black hole formation in the next generation particle collider could be observed[3]. It should be also stressed that we could come across the first experimental evidence of quantum gravity. It may also change our view of the universe: we live in a 4-dimensional (4-D) hypersurface embedded in a higher-dimensional bulk spacetime[4]. By these reasons, the brane world scenarios attract many attention.

Among many brane models, ones proposed by Randall and Sundrum are very important[5, 6]. They are motivated by superstring/M-theory, i.e., the orbifold compactification of higher-dimensional string theory by the dimensional reduction of 11-dimensional supergravity in [7]. The standard-model particles are confined in a 4-D brane world while gravity accesses extra dimensions like a string/D-brane system. In their first model (RS I)[5], they proposed a mechanism to solve the hierarchy problem with two branes, whereas in their second model (RS II)[6], they considered a single brane with a positive tension, where 4-D Newtonian gravity is recovered at low energies even if the extra dimension is not compact. This mechanism provides us an alternative compactification of extra dimensions. However, those models may inevitably expect a singular spacetime just as in general relativity, although they are based on a string theory. In fact, Maldacena and Nunez showed no-go theorem[8], which states that there are no non-singular warped compactifications in a large class of supergravity theories including 11-dimensional supergravity, IIB, IIA and massive IIA. One of the ways to evade this argument is adding the higher curvature corrections to the bulk Lagrangian. The higher curvature terms naturally arise as a next leading order of the -expansion of a superstring theory[9]. One may expect that they are described by the so-called Gauss-Bonnet combination, which is shown to be a ghost-free combination[10]. It also plays a fundamental role in Chern-Simon theories[11]. It was shown that the graviton zero mode is localized at law energies in the Gauss-Bonnet brane system as in the RS II model[12, 13, 14] and that the correction of the Newton’s law becomes milder by including the Gauss-Bonnet term [15].

As for cosmology with a brane world, there has been a lot of works over the last several years[16, 17, 18]. In particular, based on the RS II brane model, which is one of most popular ones, some interesting properties such as “dark radiation” or quadratic density term in the Friedmann equation have been found, assuming a simple bulk metric[18]. Since gravity is confined on a brane, the induced metric describes gravity on a brane. Hence the geometrical reduction gives a covariant form of the basic equations for brane gravity[19, 20, 21, 22]. Applying this formalism, we find the Friedmann equation easily.

As we discussed above, since including the Gauss-Bonnet term is important, such models are also extensively studied[23, 24, 25, 26, 27, 28]. Many authors so far studied mainly in the contexts of a resolution of initial singularity, inflation and a self-tuning mechanism of cosmological constant. In these analysis, a simple bulk metric is assumed just as in Ref. [18].

In order to understand those problems further, it may be convenient for us to extend the covariant gravitational equations on a brane to the case with the Gauss-Bonnet term. This is the purpose of the present paper. To find such equations, we first have to prove a consistency with a thin-wall ansatz. When we have a system with quadratic curvature terms in a bulk spacetime, we will be soon faced with an obstacle. In general, we expect terms such as in the field equations, where is the extrinsic curvature of a brane. If a brane is an infinitely thin singular wall, which could be described by the -function, the extrinsic curvature must have a jump at a brane. However, if is proportional to the -function, a term of makes troubles because it gives a square of the -function. The reason for this breakdown is our thin-wall ansatz. We have to find other relevant junction condition which may require information about an internal structure of a brane, that is, we have to discuss a thick brane model. The basic equations may not be described only by a geometric reduction.

In the case with the Gauss-Bonnet term, however, the situation changes completely. The basic equations show a quasi-linear property pointed out by Deruelle and Madore[29], which guarantees a thin-wall ansatz because it contains only linear terms of . Using this fact, some authors derived the generalized Friedmann equation with a simple bulk metric. With this fact, here we derive the covariant gravitational equations on a brane in the case with the Gauss-Bonnet term. The basic equations are described by 4-D brane variables except for the 5-dimensional (5-D) Weyl curvature tensor . Although our system is not closed because of the existence of , for a cosmological setting, we recover the generalized Friedmann equation which contains one integration constant and then it gives a closed form just as in the case of Ref. [19]. This generalized Friedmann equation is the same as that obtained by the previous authors[30]. In this formulation, we need not to assume any functional form for the brane action. We can add any curvature terms in four dimensions, which may be induced by quantum effects of matter fields. These brane-induced gravity models were investigated mainly in the cosmological aspect [31, 32, 33, 34].

## Ii The effective gravitational equations

We consider a 5-D bulk spacetime with a single 4-D brane world, on which gravity is confined. We assume the 5-D bulk spacetime , whose coordinates are , is described by the Einstein-Gauss-Bonnet action:

(1) |

where

(2) |

is the 5-D gravitational constant, , , and are the 5-D scalar curvature, Ricci tensor, Riemann curvature and the matter Lagrangian in the bulk, respectively. is a coupling constant. The 4-D brane world is located at a hypersurface () in the 5-D bulk spacetime and the induced 4-D metric is defined by

(3) |

where is the spacelike unit-vector field normal to the brane hypersurface . The action is assumed to be given by the most generic action:

(4) |

where are the induced 4-D coordinates on the brane,

(5) |

is the surface term[35, 36, 37], and is the effective 4-D Lagrangian, which is given by a generic functional of the brane metric and matter fields . , , and in the surface term are the extrinsic curvature of , its trace, its cubic combination defined later, and the Einstein tensor of the induced metric , respectively.

The total action () gives our basic equations as

(6) |

where

(7) | |||||

(8) |

and

(9) |

is the energy-momentum tensor of bulk matter fields, while is the “effective” energy-momentum tensor localized on the brane which is defined by

(10) |

The denotes the localization of brane contributions. It is worth noting that may include curvature contributions from induced gravity[31, 21]. In that term, we can also include “non-local” contributions such as a trace anomaly[33, 34], although those contributions are not directly derived from the effective Lagrangian .

The basic equations in the brane world are obtained by projecting the variables onto the brane world because we assume that the gravity on the brane is confined. We then project the 5-D Riemann tensor onto the brane spacetime as

(11) | |||||

(12) | |||||

(13) |

where is the Riemann tensor of the induced metric , is the covariant differentiation with respect to , and denotes the Lie derivative in the -direction. The first equation is called the Gauss equation. Using this projection, the 5-D Riemann curvature and its contractions (the Ricci tensor and scalar curvature) are described by the 4-D variables on the brane with the normal as

(14) | |||||

(15) | |||||

(16) |

As was shown by Deruelle and Madore[29], the Einstein-Gauss-Bonnet equation is quasi-linear, which means that apart from non-singular terms given by the 4-dimensional variables, it contains only linear terms of but no quadratic terms appear. In fact, inserting these relations into the basic equation (8), we find the effective equations on the brane as

(17) | |||

(18) | |||

(19) |

where

(20) | |||||

(21) | |||||

(22) | |||||

(23) |

Note that the linear terms of appear only in Eq. (17) but not in Eqs. (18) and (19). This is consistent with our ansatz that contribution from a brane is given by Eq. (10) because its contraction by vanishes.

The singular behavior in 5-D bulk spacetime appears in the 5-D gravitational equations (6) as the -function. Then, has the -functional singularity to balance to the energy-momentum tensor of the brane world. Integrating Eq. (17) in the -direction, we obtain the generalized Israel’s junction condition[38, 39];

(24) |

where

(25) | |||||

(26) |

We have introduced

(27) |

where are ’s evaluated either on the or side of the brane and is the divergence free part of the Riemann tensor, i.e.

(28) |

Because of the -symmetry, we have

(29) |

then the extrinsic curvature of the brane is uniquely determined by the junction condition as

(30) |

where

(31) |

In what follows, we omit the indices below for brevity.

The above quasi-linearity guarantees the ansatz of an infinitely thin brane. The obtained equations for induced metric is described by geometrical quantities and does not depend on microphysics of the brane. This situation will be changed when we discuss other curvature-squared terms. On the other hand, if we include the Lovelock Lagrangian which is higher than Gauss-Bonnet one but does not contain higher-derivatives, we can assume that a brane is infinitely thin because it is also quasi-linear and then extend the present approach.

In order to find the effective equations on the brane, we have to replace the terms of in Eq. (17) with the 4-D variables on the brane. The singular part in Eq. (17) has been evaluated by the junction condition. Hence we have to evaluate either on the or side of the brane, which is nonsingular. From Eqs. (13), (15) and (16) with the decomposition of the Riemann tensor as

(32) |

where is the 5-D Weyl curvature, we find

(33) |

where

(34) |

However, because Eq. (33) is a trace free equation, we cannot fix by Eq. (33). We have to find from other independent equation. We shall take a trace of our basic equation (6), finding

(35) |

Inserting Eqs. (11)-(13) with Eq. (33) into Eq. (35), we find

(36) |

where

(37) |

From Eq. (33) with Eq. (36), we then find

(38) |

Inserting Eq. (38) into Eq. (17), we obtain the effective gravitational equations on the brane as

(39) |

where

(40) |

Eq. (19) is automatically satisfied when we take a trace of Eq. (39), which means that it is not independent. We have then two basic equations (39) and (18). Eq. (18) is rewritten as

(41) |

which gives the constraint on the brane matter fields through the junction condition (30), i.e.

(42) |

If there is no energy-momentum transfer from the bulk, we find the energy-momentum conservation of brane matter fields as

(43) |

In Eq. (39), we have so far three unknown variables; , , and . The first two variables are described by bulk information, whereas the extrinsic curvature is related to the brane “energy-momentum” tensor as Eq. (30). Hence Eqs. (30) and (39) with the energy momentum conservation (43) give the effective gravity theory on the brane.

It may be better to rewrite Eq. (39) to the Einstein-type equations with “correction” terms. From Eqs. (19) and (39), we find

(44) |

where

(45) | |||||

As for the junction condition, we find

(46) |

If , we find two equations:

(47) | |||

(48) |

which are exactly the same as those found in Ref. [19], which gives the Einstein gravitational theory in the 4-D brane world. However, if the Gauss-Bonnet term appears, gravitational interaction on the brane will be modified in the effective theory.

The gravity on the brane is described by Eq. (39) with Eq. (30), or equivalently by Eq. (44) with Eq. (46). Just as the case of the RS II model, this system is not closed because of appearance of the terms with , which is some component of the 5-D Weyl curvature. Although we have to solve a bulk spacetime as well as a brane world, we know that any contribution from a bulk spacetime to a brane world is described only by the tidal force .

## Iii Friedmann equation

We apply the present reduction to the FRW cosmology. We assume spacetime as

(56) |

where denotes the metric of maximally symmetric 3-dimensional space. This gives