BORN-INFELD COSMOLOGIES

Ricardo García-Salcedo and Nora Bretón

Depto. de Física, Cinvestav

A.P. 14-740, 07000 México D.F., México

Received

We present a model for an inhomogeneous and anisotropic early universe filled with a nonlinear electromagnetic field of Born-Infeld (BI) type. The effects of the BI field are compared with the linear case (Maxwell). Since the curvature invaria nts are well behaved then we conjecture that our model does not present an initial big bang singularity. The existence of the BI field modifies the curvature invariants at as well as sets bounds on the amplitude of the conformal metric function.

## 1 Introduction

The advantage in exploring a cosmological model with nonlinear electrodynamics resides in approaching with a classical model but taking into account some quantical features. It is well known that nonlinear electrodynamics (NLE) is the best classical solution to the self-energy problem of charged particles. The aim of nonlinear electrodynamics of Born and Infeld (BI) was to establish a model of a finite classical field theory without divergences. However, NLE can also be a model of quantum electrodynamics in the classical limit of high occupation numbers. As demonstrated by Stehle and De Baryshe , theories with Lagrangians similar to the Heisenberg-Euler effective Lagrangians (the weak field expansion of the Born-Infeld theory is of this form) are more accurate classical approximations of QED than Maxwell’s theory in the case of fields with high intensities at a fixed frequency. This fact makes NLE particularly interesting in general relativity, specialy in cosmological theories, as a simple classical model to explain vacuum polarization processes- a possible influence on the mechanism of the evolution of the early universe. The processes called scattering of light by light can be expressed in terms of a kind of electric and magnetic permeability of vacuum. From this point of view, the exact solutions of the Einstein-Born-Infeld (EBI) equations are worth to study since they may indicate the physical relevance of nonlinear effects in strong gravitational and strong electromagnetic fields.

The Born-Infeld theory is self-consistent and satisfies all natural requirements. The BI Lagrangian depends on a constant , that is a maximum field strenght, and also depends on the invariants of the field. Recently BI has recover interest in the context of string theory, since the low energy action Lagrangian corresponding to the objects called branes can be written as a term that is BI plus a Weiss-Zumino or Chern-Simmons term . Furthermore, the BI strings are solitons which represent strings ending on branes.

In relation to exact solutions of the EBI equations we can mention that there are exact solutions for spaces possessing one timelike and one spacelike Killing vector fields. They correspond to spherical static spacetimes and also to axisymmetric stationary spacetimes .

In the literature cosmological models have been proposed for the source-free Einstein-Maxwell equations , . Charach and Malin presented a cosmological model of Einstein-Rosen with gravitational, electromagnetic and scalar waves, this work generalized the electromagnetic Gowdy model. In all of these models the matter fields cause an evolution significantly different from that of the vacuum model. The interest is to get insight on the coupled electromagnetic and gravitational radiation at early epochs of the universe. Since at that epoch the fields must had been very strong, the nonlinear electromagnetic effects could be of importance. These fields could originate from processes near the Planck era ().

The spacetime we address in this work possesses two Killing vectors, both spacelike. It is not immediate to conclude that this spacetime is also compatible with a BI source. For instance it does not admit a perfect fluid source together with BI, which is the case in the stationary axisymmetric spacetime. This is the first atempt, as far as we know, of exploring BI early universes. Our aim is to determine to what extent the BI field can prevent or modify the initial big bang singularity. Spatially inhomogeneous cosmological models with perfect fluid which do not originate in an initial singularity have been previously found by Senovilla (see also , ). For the solutions derived in the present work the initial big bang singularity appears only as a coordinate singularity, since the invariants do not diverge at the origin. It turns out that the existence of the BI field set bounds to the amplitude of the conformal metric function and also affects quantitatively the curvature invariants at . In this work we do not treat the possible transition of our model to a homogeneous, isotropic one. In Sec. 2 the Einstein-Born-Infeld equations are established. In Sec. 3 the solution is presented and analyzed in the linear limit as well as the asymptotic behavior. In Sec. 3.4 we remark the coordinate transformations relating the spacetime studied and the Einstein-Rosen metrics. Concluding remarks are given at the end.

## 2 Einstein-Born-Infeld Equations

The line element considered corresponds to a spacetime, given by

(1) |

where , are ignorable coordinates (i.e. the spacetime has two spacelike Killing vectors, ), the metric functions depend on and : , , . and are separable functions of and . The metric (1) corresponds to a space with gravitational waves propagating in the direction, with their wavefronts on the plane surfaces spanned by the Killing vectors. The function describes what is called the “” polarization of gravitational waves. describes the transverse scale expansion created by the energy density of the waves. The function is playing the role of unpolarizer for the propagating gravitational waves that constitute the background. Making the metric recover the form of a space of linearly polarized gravitational waves (diagonal metric).

To interpret line element (1) as cosmological, we refer to the Einstein-Rosen metric , Eq. (32), Sec. (3.3). The local behavior of these spacetimes is defined by the gradient of the metric function , i.e. w hich can be spacelike, null or timelike. The globally spacelike case corresponds to cylindrical spacetimes. The case of being globally null corresponds to the plane symmetric waves. The other case of being globally timelike as well as cases in which the sign of the gradient can vary from point to point are used to describe cosmological models and also colliding gravitational waves . It must also be remarked that due to the presence of an Abelian subgroup in the Bianchi models of types I-VII, the Einstein-Rosen metrics include these models as particular cases . Also the rotationally invariant Bianchi models of types VIII and IX belong to the generalized Einstein-Rosen spacetimes .

We note that in the case of Einstein-Rosen spacetimes, the interpretation as gravitational waves propagating in the direction arises directly from the equation satisfied by ,

(2) |

For vacuum or when , satisfies the usual wave equation. In our case, the B-I field is not traceless and then is some propagating wave with a source (the B-I field).

Turning back to the spacetime (1), it is filled with a BI field with energy-momentum tensor given by:

(3) |

where is the Born-Infeld structural function and . and are the invariants associated with the Born-Infeld field, given by

(4) |

denotes the dual tensor of ,defined by . The antisymmetric tensor is the generalization of the electromagnetic tensor . The structural function is constrained to satisfy some physical requirements: (i) the correspondence to the linear theory ; (ii) the parity conservation ; (iii) the positive definiteness of the energy density () and the requirement of the timelike nature of the energy flux vector (). Such structural function for the BI field is given by

(5) |

where is a maximum field strenght of the BI field. In the linear limit, which is obtained by taking , then and , coinciding with the Maxwell electromagnetism. and are related through the material or constitutive equations:

(6) |

We have worked out the problem in the null tetrad formalism . The function in Eq. (5) involves a square root that complicates the solving of the system of the Einstein-Born-Infeld equations. To surmount this difficulty, first of all we have aligned the two non-null components of along the two principal directions of the tetrad vectors and (the metric is type D); second, we parametrize the BI field in terms of the electric and magnetic fields: and where and stand for the usual electric displacement and magnetic field intensity, respectively. Also, we introduce the function ,

(7) |

If the previous Eq. (7) has to make sense, it demands that . From the structural Eqs. (6) we obtain the expressions that clearly manifest the vacuum polarization, in this case depending on the function :

(8) |

Besides the Einstein field equations we must consider the BI electrodynamical equations that amount to the closure condition of the electromagnetic two form, ,

(9) |

This formalism of the Born-Infeld field was introduced by Plebañski et al. We have assumed that , where can take the values . For each value of we obtain different curvatures in the direction. For the metric (1) and with the alignment chosen the electromagnetic equations are

(10) | |||||

(11) |

Eq.(9) tells us that neither nor depend on . Eq.(10) and Eq. (15), quoted below, lead to infer that neither nor depend on . Therefore, for this spacetime, the BI is a spatially homogeneous field. The Einstein equations for the BI energy-momentum tensor (3) amount to

(12) | |||||

(13) | |||||

(14) |

It is a system of equations for the functions , and , while the rest of the metric functions are and . Eq. (11) for can be solved immediately, giving , with constants. Eqs. (12)-(13) can be decoupled and it is obtained the solution for in terms of as well as Eq. (16) fo r :

(15) |

(16) |

In order that , given in Eq. (15), be a real function, must be such that . This means that the existence of the BI field restrains the conformal metric function in its amplitude because it bounds the values of the constants and . Demanding that

(17) |

fulfils the requirement . Putting the NLE functions, and , Eqs. (8), in terms of

(18) |

with the corresponding coordinate components of the electromagnetic field ,

(19) |

The previous expressions show that the BI field modulates the amplitude and frequency of the electromagnetic field. The fields and are shown in Fig. 1. They are oscillating functions depending on the constant through the function .

Since the metric is of Petrov type D, the only nonvanishing Weyl scalar is given by

(20) |

This scalar is not real and therefore the magnetic part of the Weyl tensor does not vanish. We also note that it is homogeneous, i. e. it depends only on time. The invariants are given by ; from their expressions it is clear that they share (if they exist) the singularities of . On its turn depends on the behaviour of the functions and . is a periodic function while , governed by Eq. (16), carries much of NLEBI information. In the next section we analyze the behavior of .

## 3 The BI solution and the linear case

The solution to Eq. (16) for , in terms of is given by

(21) |

where is an integration constant related to initial or boundary conditions for vacuum. It is interesting now to analyze graphics of the function . Fig. 2 shows for several values of the BI constant . For values of the function becomes positive for all the range and reduces its oscillations; for larger t he maximums become greater. The transverse scale expansion is an inhomogeneous function expanding in the direction, it is shown in Fig.3.

Fig. 4 shows the plots of the real and imaginary parts of ; they give us qualitative information with respect to the singularities of the invariants and . Both Re and Im are continuous functions that do not present infinities at all.

In Fig. 5, the function displays different behaviours at depending if or . In the former case and consequently there appears a coordinate singularity at ; however the corresponding Weyl scalar is not singular there. We can see from Fig. 5 that this class of solutions admits both behaviors at , with coordinate singularity or without it. To a linear combination corresponds .

### 3.1 Linear limit

In order to compare the effects of the BI field with the Maxwell case, we compute the linear limit. The Maxwell electrodynamics is recovered when . The BI Eqs. (9) and (11) in the linear limit correspond to

(22) |

whose solution is

(23) |

where and are constants. In terms of the coordinate components of the electromagnetic field the expressions are

(24) |

This electromagnetic field is homogeneous of intensity . In accordance with the parametrization (7), for , Eqs. (12)-(13) become

(25) |

Solving for in terms of we have

(26) |

expression that also can be obtained from Eq.(21) taking the limit ,

Fig. 6 shows the comparison between corresponding to the linear electromagnetic field, Eq. (26), and of the BI field, Eq. (21). The presence of BI field smooths the function respect of the linear version. Making we get the vacuum limit of the spacetime (1), by this meaning that only gravitational waves remain.

Comparing Eq.(21) for corresponding to the BI field with Eq.(26) for the linear case we see that the first term in the integral is directly associated with the NLEBI effects. To show the dependence on we write as

(27) |

where , in accordance with the requirement in Eq. (7),the range of is . If one plots vs. , it can be seen that the NLEBI effects are sensible for values of near , while for larger , generically it tends to the constant linear Maxwell field .

### 3.2 Behavior at early times

For it is reasonable to approximate as

(28) |

Since const. implies , we take along with the restrictions (17) imposed by the BI field. The constant is suposed not to be large since we are in a low frequency regime. To solve Eq. (21) we also approximate , obtaining

(29) |

predominance of one term over the others depend on the relative values between and . The term in brackets represents the BI contribution at early times. It is expected to be large if the electromagnetic field is strong, then the BI contributi on can be of importance. The first term corresponds to the contribution of the space curvature, determined by the value of The Weyl scalar approaches as

(30) |

The contribution of the BI field goes as and higher orders in . In the magnetic part of the contribution is not important at very early times since comes from the term with and goes as .

We now mention the case , which corresponds to a spacetime of propagating linearly polarized gravitational plane waves (diagonal metric). Eq. (11) for become , with solution , being constants. The refore, the behavior of in the case resembles the one for . Consequently, we assert that this spacetime approaches the origin in time as propagating linearly polarized waves.

### 3.3 Behavior at

To complete the previous analysis, the question arises as how does the model evolve at , or which is the asymptotic behavior of the solution at , i. e. for times when the universe is already a causally connected one.

The dynamics of our model is driven by the function . For both and become rapidly oscillating periodic functions that can be approximated by a constant. With this, the metric function (Eq. (21) becomes a linear function on . Absorbing constants and transforming ( being constants), the line element can be written as

(31) |

### 3.4 Relation to Einstein-Rosen Universes

By means of a coordinate transformation, the metric (1) can be put in the form of an Einstein-Rosen line element

(32) |

The coordinate transformation for depends on the value of as follows

(33) | |||||

(34) | |||||

(35) |

We also note that for , the functions in terms of the spatial coordinate become periodic, then the topology of the universe in this case can be a closed one. The coordinate transformation for , , is not a simple one in the general case due to the form of , Eq. (21). It involves elliptic integrals which lead to Jacobi family of elliptic functions and there is no analytical expression in terms of elementary functions. However, for particular cases such transformation can be given in a simple form. An example is for early times. In this case the relation between and is .

Another example is the case (vacuum) and . In this case the line element (32) takes the form

(36) |

Gowdy constructed exact vacuum solutions of the Einstein field equations which represent inhomogeneous closed universes. These models possesses compact spacelike hypersurfaces as well as invariance. The line element (36) can be written as a Gowdy model with thre-torus topology performing the coordinate transformation

(37) |

then one obtains the three-torus Gowdy line element

(38) |

If one arrives to a Gowdy model for vacuum, one can speculate about that a geometry so simple as Gowdy does not admit a Born-Infeld field.

## 4 Concluding Remarks

In this work we have presented a family of solutions to the Einstein-Born-Infeld equations for a space-time which is a cosmological model. The spacetime describes propagating gravitational plane waves coupled with a nonlinear spatially homogeneous e lectromagnetic field of the Born-Infeld type. We also present the limit in which the BI field become Maxwell electromagnetism.

Some results are:

The presence of an initial coordinate singularity depends on the choice of the conformal metric function , for the solution exhibites initial singularity while if there is no such coordi nate singularity. However, from the smooth behavior of the Weyl scalar, , we guess that there are no singularities at all.

The presence of the BI field sets bounds on the amplitude of the periodic function . In this sense BI modifies the global expansion of the spacetime. If compared with the effect of linear electrodynamics, the presence of the BI field smooths the metric function (see Fig. 6). Since is involved in the expression of the scale expansion and of the Weyl scalars, then BI field smooths the curvature. The BI field also modifies quantitatively the curvature at , the effect being more sensible when the BI parameter approaches the magnitude of a critic magnetic field intensity . We conjecture that a less restrictive spacetime (not type D) shall permit the existence of an inhomogeneous nonlinear electromagnetic field, whose consequences on the space curvature could be more drastic.

For early times, , the spacetime approaches a space of unpolarized gravitational waves. For the spacetime becomes an inhomogeneous anisotropic spacetime with a homogeneous Maxwell field.

For vacuum one obtains a Gowdy model of three-torus topology.

It should be of interest to classify the spacetime in different regions according to the sign of the gradient of the function and see the distinct interpretations of the found solution according to each region in the plane.

Generalizing this solution to include a scalar field and to investigate the coupled effect with nonlinear electrodynamics for early universes could also lead to interesting results.

Acknowledgements

We gratefully ackowledge to the anonymous referee whose opinions help to improve this work. Partially supported by CONACyT (México), project No. 32086-E.

References

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Fig.1. These are plots of the nonlinear electromagnetic fields, electric displacement, , and magnetic induction, . The corresponding conformal metric function is .

Fig.2. It is displayed the metric function for different values of the BI constant , . For greater values of the maximums are higher but the function preserves the shape of . These plots correspond to , and .

Fig.3. It is shown the transverse scale expansion . It is increasing in and oscillating in . For this plot the values of the constants are .

Fig.4. The continuous plot corresponds to Re and the dashed one to Im. Both are continuous functions, on this basis we guess that the spacetime has no singularities. These graphics correspond to in Fig.2 with .

Fig.5. The function displays different behaviour at depending if (continuous curve) or (dashed curve).

Fig.6. The function for the linear limit corresponds to the dashed curve, the continuous line is in the presence of BI field with . The constant . of BI field is smoother than the one corresponding to linear electromagnetic field. The conformal metric function considered is .