# Geodesically complete nondiagonal inhomogeneous cosmological
solutions

in dilatonic gravity with a stiff perfect fluid

###### Abstract

New nondiagonal inhomogeneous cosmological solutions are presented in a wide range of scalar-tensor theories with a stiff perfect fluid as a matter source. The solutions have no big-bang singularity or any other curvature singularities. The dilaton field and the fluid energy density are regular everywhere, too. The geodesic completeness of the solutions is investigated.

###### pacs:

04.50.+h, 04.20.Jb, 98.80.HwAll versions of string theory and higher dimensional gravity theories predict the existence of the dilaton field which determines the gravitational ”constant” as a variable quantity. The existence of a scalar partner of the tensor graviton may have a serious influence on the space-time structure and important consequences for cosmology and astrophysics. A large amount of research has been done in order to unveil the possible cosmological significance of the dilaton [1], [2]-[14](and references therein). With a few exceptions most of the cosmological studies within the scalar-tensor theories were devoted to the homogeneous case. The homogeneous models are good approximations of the present universe. There is, however, no reason to assume that such a regular expansion is also suitable for a description of the early universe. Moreover, as is well known, the present universe is not exactly spacially homogeneous. That is why it is necessary to study inhomogeneous cosmological models. They allow us to investigate a number of long standing questions regarding the occurrence of singularities, the behaviour of the solutions in the vicinity of a singularity and the possibility of our universe arising from generic initial data.

In this work we shall address the question of the occurrence of singularities in inhomogeneous cosmologies within the framework of scalar-tensor theories.

As is well known, most of the homogeneous models (both in general relativity and in scalar-tensor theories) predict a universal space-like big-bang singularity in a finite past. It was, therefore, believed that this would be the usual singularity in general. The inclusion of inhomogeneities drastically changes this point of view. There are inhomogeneous cosmological solutions in general relativity which have no bing-bang or any other curvature singularity. The first such solution was discovered by Senovilla in 1990 Ref.[15]. Senovilla’s solution represents a cylindrically symmetric universe filled with radiation. This solution has a diagonal metric and is also globally hyperbolic and geodesically complete [16]. Senovilla’s solution was generalized by Ruiz and Senovilla in Ref.[17] where a large family of singularity-free diagonal inhomogeneous perfect fluid solutions was found. Nonsingular diagonal inhomegeneous solutions in general relativity describing cylindrically symmetric universes filled with stiff perfect fluid were found by Patel and Dadhich in Ref.[18]. Other examples of diagonal nonsingular solutions in general relativity can be found in Refs.[19]- [25].

In Ref.[26], Mars found the first nondiagonal inhomogeneous cosmological solution of the Einstein equations with stiff perfect fluid as a source. This solution is globally hyperbolic and geodesically complete. Mars’s solution was generalized by Griffiths and Bicak in Ref.[27].

Within the framework of scalar-tensor theories there are also inhomogeneous cosmological solutions without big-bang or any other curvature singularity. In Ref.[28], Giovannini derived gravi-dilaton inhomogeneous cosmological solutions with everywhere regular curvature invariants and bounded dilaton in tree-level dilaton driven models. In a subsequent paper [29], it was shown that these solutions describe singularity-free dilaton driven cosmologies. A nondiagonal inhomogeneous cosmological solution with regular curvature invariants and unbounded dilaton in the tree level effective string models was found by Pimentel [30]. Very recently, inhomogeneous cosmological solutions without any curvature singularities were obtained by the author in a wide class of scalar-tensor theories with stiff perfect fluid as a source [31].

In this work we take a further step upwards and present new nondiagonal inhomogeneous cosmological stiff perfect fluid solutions with no curvature singularities in a wide range of scalar-tensor theories.

Scalar-tensor theories (without a cosmological potential) are described by the following action in Jordan (string) frame [32],[33]:

(1) | |||

Here, is the bare gravitational constant and is the Ricci scalar curvature with respect to the space-time metric . The dynamics of the scalar field depends on the functions and . In order for the gravitons to carry positive energy the function must be positive. The nonnegativity of the energy of the dilaton requires that . The action of matter depends on the material fields and the space-time metric but does not involve the scalar field in order for the weak equivalence principle to be satisfied.

As a matter source we consider a stiff perfect fluid with equation of state .

The general form of the solutions is given by

(2) |

The solution depends on three parameters - , () and . The range of the coordinates is

(3) |

The explicit form of the functions and ,
and the range of the parameter depend on the
particular scalar tensor theory. These solutions can be
generated^{2}^{2}2Some of the solutions were first obtained by
solving the corresponding system of partial differential
equations for nondiagonal cosmologies [34]. from the
general relativistic Mars’s solution [26] using the solution
generating methods developed in Ref.[31].

Below we consider the explicit form of the general solution for some particular scalar-tensor theories.

## .1 Barker’s theory

Barker’s theory is described by the functions and .

In the case of Barker’s theory the explicit forms of the functions and are:

(4) | |||||

(5) |

where the range of is . This range can be extended to . For and we obtain the Mars’s solution and gravi-dilaton vacuum solution, respectively. That is why we consider only . It should be noted that the range of the parameter is crucial for the curvature invariants. It is easy to see that the gravi-dilaton vacuum solution corresponding to has divergent curvature invarinats because of the conformal factor .

## .2 Brans-Dicke theory

Brans-Dicke theory is described by the functions and where is a constant parameter. Here we consider the case . The explicit form of the functions and in the Brans-Dicke case is the following:

(6) | |||||

(7) |

Here the range of the parameter is . The solution exists for and , too. In these cases, however, we obtain a gravi-dilaton vacuum solution which is just the Pimentel’s solution [30]. That is the reason we do not consider these limiting values of . The solution is invariant under the trasformations and . In this generalized sense, we can say that the solution is even in time.

## .3 Theory with ”conformal” coupling

The theory with ”conformal” coupling is described by the functions and . In this case we have:

(8) | |||||

(9) |

The range of the parameter is . For we obtain a gravi-dilaton vacuum solution which is well-behaved and can be included as a limiting case.

## .4 theory

Here we consider the scalar-tensor theory described by the functions and where . The explicit forms of and are:

(10) | |||||

(11) |

Here, the range of the parameter is .

## .5 theory

The theory with and possesses the following solution:

(12) | |||||

(13) |

In order for the dilaton field in this solution to have positive energy we should restrict the range of the parameter to .

Using the solution generating methods developed in Ref.[31] we can generate nondiagonal inhomogeneous cosmological solutions in many other scalar-tensor theories different from those considered above. However, the solutions we have presented here are expressed in a closed analytic form and they are also representative and cover a wide range of the possible behaviors of the scalar-tensor solutions which can be generated from Mars’s solution.

Let us consider the main properties of the found solutions. The metric functions, the gravitational scalar (the dilaton) and the fluid energy density are everywhere regular. The space-times described by our solutions have no big-bang nor any other curvature singularity - the curvature invariants , , and are regular everywhere. The solution possesses a two dimensional abelian group of isometries inherited from the seed Mars’s solution and generated by the Killing vectors and . In addition, the metrics have a well defined axis of symmetry and the elementary flatness condition [35] is satisfied. Since the presented solutions are conformally related to the Mars’s solution, the spacetimes descibed by them are globally hyperbolic. In fact, the global hyperbolicity can be proved independently as a consequence of the poof of the geodesic completeness presented below.

The existence of two Killing vectors gives rise to two constants of motion along the geodesics:

(14) | |||

The affinely parameterized causal geodesics satisfy

(15) |

where and for null and timelike geodesics, respectively. Taking into account (.5) and (.5) the geodesic equations for and can be written in the following form:

(16) | |||

(17) | |||

where

(18) |

To demonstrate the geodesic completeness of our metric, we have to show that all non-spacelike (i.e. causal) geodesics can be extended to arbitrary values of the affine parameter. We shall consider only future directed geodesics. The past directed geodesics can be treated analogously.

First we consider null geodesics with . For them we have and

(19) |

After integrating we obtain

(20) |

where is a constant. Taking into account that for each of our solutions there exists a constant such that

(21) |

for arbitrary values of and fixed parameter , we obtain

(22) |

Therefore the geodesics under consideration are complete.

Now let us turn to the general case when at least one of the constants , or is different from zero. Here we shall use a method similar to that for diagonal metrics described in Ref.[16]. Let us parameterize and by writing :

(23) | |||

(24) |

Substituting these expressions in the equations for and we obtain

(25) |

or equivalently

(26) |

where

(27) | |||

In order for the geodesics to be complete and have to remain finite for finite values of the affine parameter. In fact, it is sufficient to consider only , since and are related via (.5). The derivatives and are regular functions of and , and the only problem we could have appear when approaches for . We shall show, however, that cannot become zero for .

First we consider geodesics with increasing (i.e. ). In this case it is not difficult to see that the term

(28) |

in eqn. (23) can not become singular (for increasing ). Therefore, could become singular only for . We shall show, however, that can not become singular for finite values of the affine parameter. For increasing , cannot diverge since for large (large ) the derivative becomes negative. Indeed, for all exact solutions presented here, there exists a constant such that

(29) |

for arbitrary and fixed .

Therefore, as can be seen from eqns. (.5), the terms
associated with the constant , and are all
positive for large values of .As a consequence we obtain that
the functions and are
positive, i.e. for large (large
).^{3}^{3}3In fact, the function is
exponentially small compared with and may not
be considered.

In the second case,when decreases (), the problem comes from when . The geodesics with can reach the axis without problems and then continue with (). When , cannot diverge for finite values of the affine parameter. This follows from the fact that the derivative becomes positive for small (large ) as can be seen from Eqns. (.5) and (26), taking into account that is exponentially suppressed compared with . The positiveness of the derivative when the geodesics are close to the axis prevents the radial coordinate from collapsing too quickly and reaching the axis. The fact that can not become zero for may be seen more explicitly as follows. When approaches zero the dominant term is that associated with and the other terms can be ignored. So, for small the geodesics behaves as null geodesics with :

(30) | |||

(31) | |||

(32) |

where . Hence, we obtain the orbit equation

(33) |

Integrating, we have

(34) |

where is a constant. Since , can not become zero.

From the proof of the geodesic completeness it follows that every maximally extended null geodesic intersects any of the hypersurfaces . According to [36], this a sufficient condition that the hypersurfaces are global Cauchy surfaces. Therefore, the solutions are globally hyperbolic.

We have explicitly proven the geodesic completeness of the solutions using their particular properties. The geodesic completeness can be proved independently by considering the solutions from a more general point of view. In Ref.[37](see also Ref.[38]), Fernandez-Jambrina presented a general theorem providing wide sufficient conditions for an orthogonally transitive cylindrical space-time to be geodesically complete. It can be verified that the solutions presented here satisfy all conditions in the Fernandez-Jambrina’s theorem and therefore they are geodesically complete.

New diagonal solutions can be obtained from (2) as a limiting case. Taking and keeping fixed, we obtain the following diagonal inhomogeneous cosmological scalar-tensor solutions:

(35) | |||||

where should be replaced by in the explicit formulas for .

We have proven that the solutions presented in the present paper are geodesically complete. This result is not in contradiction with the well-known singularity theorems because in our case the strong energy condition is violated in the Jordan frame. This can be explicitly seen by calculating the components of the Ricci tensor. All components are bounded except for . Therefore, for large enough , one can always find timelike and null vectors such that i.e. the strong energy condition is violated. However, the situation is different in the Einstein frame. The Einstein frame metric is just the Mars’s metric and it is geodesically complete as we have already mentioned. Since the energy conditions are satisfied in the Einstein frame it remains to see which other conditions of the singularity theorems are violated. The space-time described by the metric does not contain closed trapped surfaces. In order to prove this we will employ the techniques of differential geometry described in Refs.[39] and [40]. Let us consider a closed spacelike surface and suppose that it is trapped. Since the surface is compact it must have a point where reaches its maximum. Let us denote on a constant time hypersurface . For the traces of both null second fundamental forms at , it can be shown that (see Refs.[39] and[40])

(36) |

The traces have opposite signs so that there are no trapped surfaces.

Our solutions are stiff perfect fluid cosmologies and, therefore, the natural question which arises is what happens if the fluid is not stiff. In this case,however, the situation is much more complicated. In contrary to the stiff fluid case, the dilaton-matter sector does not posses nontrivial symmetries which allow us to generate new solutions from known ones.The only way to find exact solutions is to attack directly the corresponding system of coupled partial differential equations. This question is currently under investigation.

Summarizing, in this work we have presented new nondiagonal inhomogeneous stiff perfect fluid cosmological solutions in a wide range of scalar-tensor theories. The found solutions have no big-bang nor any other curvature singularity. The gravitational scalar (dilaton) and fluid energy density (pressure) are regular everywhere, too. Moreover, the solutions are globally hyperbolic and geodesically complete. To the best of our knowledge, these solutions are the first examples of nonsingular inhomogeneous perfect fluid scalar-tensor cosmologies with a nondiagonal metric.

I would like to thank V. Rizov for discussions and especially L. Fernandez-Jambrina for his valuable comments on the geodesic completeness of the orthogonally transitive cylindrical spacetimes. My thanks also go to J. Senovilla for sending me some valuable papers. This work was supported in part by Sofia University Grant No 459/2001.

## References

- [1] M. Gasperini’s web page, http://www.to.infn.it gasperin
- [2] D. La, P. Steinhardt, Phys. Rev. Lett. 62, 376 (1989)
- [3] L. Pimentel, J. Stein-Schabes, Phys. Lett. B 216, 25 (1989)
- [4] J. Barrow, K. Maeda, Nucl. Phys. B 341, 294 (1990)
- [5] J. Barrow, Phys. Rev. D 47, 5329 (1993)
- [6] J. Barrow, J. Mimoso, Phys. Rev. D 50, 3746 (1994)
- [7] C. Will, P. Steinhardt, Phys.Rev. D 52, 628 (1995)
- [8] J. Mimoso, D. Wands, Phys. Rev. D 51, 477 (1995)
- [9] J. Mimoso, D. Wands, Phys. Rev. D 52, 5612 (1995)
- [10] J. Barrow, P. Parsons , Phys. Rev. D 55, 1906 (1997)
- [11] O. Bertolami, P. Martins, Phys. Rev. D 61, 064007-1 (2000)
- [12] N. Bartolo, M. Pietroni, Phys. Rev. D 61, 023518 (2000)
- [13] G. Esposito-Farese, D. Polarski, Phys. Rev. D 63, 063504 (2001)
- [14] J. Morris, Class.Quant.Grav. 18 2977 (2001)
- [15] J. Senovilla, Phys. Rev. Lett. 64, 2219 (1990)
- [16] F. Chinea, L. Fernandez-Jambrina, J. Senovilla, Phys. Rev D45, 481 (1992)
- [17] E.Ruiz, J. Senovilla, Phys.Rev.D45, 1995 (1992)
- [18] L. Patel, N. Dadhich, Report No. IUCAA-1/93; gr-qc/ 9302001
- [19] L. Patel, N. Dadhich, Class. Quant. Grav. 10, L85 (1993)
- [20] N. Dadhich, L. Patel, R Tikekar, Current Science 65, 694 (1993)
- [21] J. Senovilla, Phys. Rev. D53, 1799 (1996)
- [22] N. Dadhich, J. Astrophys. Astr. 18, 343 (1997)
- [23] N. Dadhich, A. Raychaudhuri, Mod. Phys. Lett A14, 2135 (1999).
- [24] M. Mars, Class. Quant. Grav.12, 2831 (1995)
- [25] L. Fernandez-Jambrina, Class. Quant. Grav.14, 3407 (1997)
- [26] M. Mars, Phys. Rev. D51, R3989 (1995)
- [27] J. Griffiths, J. Bicak, Class. Quant. Grav.12, L81 (1995)
- [28] M. Giovannini, Phys. Rev. D57, 7223 (1998)
- [29] M. Giovannini, Phys. Rev. D59, 083511 (1999)
- [30] L. Pimentel, Mod. Phys. Lett A, Vol 14, 43 (1999)
- [31] S. Yazadjiev, Phys. Rev. D65, 084023 (2002)
- [32] C. Will, The Confrontation between General Relativity and Experiment,Living Rev.Rel. 4 (2001) 4; E-print gr-qc/0103036
- [33] C. Will Theory and experiment in graviattional physics (Cambridge University Press, Cambridge, 1993)
- [34] S. Yazadjiev, Talk given at the Meeting of the researchers in physics, Sofia University, Bulgaria, 2002
- [35] D. Kramer, H. Stephani, E. Herlt, M. MacCallum, Exact Solutions of Einstein’s Field Equations(Cambridge University Press, Cambridge, 1980)
- [36] R. Geroch, J. Math. Phys. 11, 437 (1970)
- [37] L. Fernandez-Jambrina, J. Math. Phys. 40, 4028 (1999)
- [38] L. Fernandez-Jambrina, L. M. Gonzalez-Romero, Class.Quant.Grav. 16, 953, (1999)
- [39] J. Senovilla, Gen. Relativ. Gravit. 30, 701 (1998)
- [40] J. Senovilla, ”Trapped surfaces, horizons, and exact solutions in higher dimensions”, hep-th/0204005