untitled

406

Brindusa Ciobanu, Irina Radinschi

Hertz potential) formalism [5]. Their method can be extended to space-times
with local rotational symmetry, the Gödel universe being a specific example of
this class [5, 6]. We have considered the orthonormal tetradic formalism for our
investigations, while Cohen, Vishveshwara and Dhurandhar adopted the Newman-
Penrose method.

Most of the astronomical objects in the universe (planets, stars or galaxies)

have some form of rotation (differential or uniform). Hence the possibility that
the universe itself could be rotating has attracted a lot of attention. But even
though observational evidence of cosmological rotation has been reported, it is
still a controversial subject [7–11]. Our present day universe is rotating very slowly,
if at all. However, the existence of such a small rotation, when extrapolated to
the early stages of the universe, could have played a major role in the dynamics
of the early universe, and possibility also in the processes involving galaxy
formation [9]. Recently, Nodland and Ralson reported to have a discovered a
cosmic axis. Kühne argues that their axis is supported by an earlier independent
observation on the spin axis of galaxies in the Perseus- Pisces supercluster. The
large alignment of this supercluster (over a distance of at least 130 million light
years) cannot be explained within the framework of conventional models of
galaxy formation. He explaines this approach of the subject within the frame-
work of Gödel’s cosmology [10, 11].

The absence of an explicit cosmological solution of the Einstein’s field

equations that can describe both the expansion and the rotation of the universe
requires some approximations. The Friedmann models (those with a Robertson-
Walker metric) describe accurately the universe expansion, but they cannot explain
the observed rotation of this one (

= 10

–13

rad/year), [7, 12]. The cosmological

problem of the rotation of the universe has been studied also by Ellis, Olive and
Grøn [13, 14], but the first important model of the universe in rotation, to which
corresponds a new cosmological solution of the Einstein’s equations is the model
proposed in 1949 by Kurt Gödel [15].

Gödel universe has a very unusual property. In the Gödel universe closed

timelike curves are formated. Therefore it allows traveling backwards in time. If
you enter in some of these timelike curves, you can return to the past. “

You enter

a rocket, and you take a journey in the universe along a certain path. Then you
will return to the starting point before you started

” [16]. The Gödel model is

perhaps the best known example of a solution of Einstein’s field equations in
which causality may be violated. It thus became a paradigm for causality
violation in gravitational theory. In their work [17], Gleiser, Gürses, Karasu and
Sariou

lu verified that “

the spacetimes described by Gödel-type metrics with

both flat and non-flat backgrounds always have closed timelike or null curves,
provided that at least one of the u

(

)

≠

constant

.” They also showed “

that the

geodesics of Gödel-type metrics with constant

u are characterized by the (D-1)-

Electric and magnetic fields in a rotating universe

407

dimensional Lorentz force equation for a charged point particle formulated in
the corresponding Riemannian background

.” Recently, there has been renewed

interest in this kind of space-time because of the discovery of a supersymmetric
solution of

= 1 five dimensional supergravity with very similar features [3],

[15–19]. Another essential part of Gödel’s relativistic model is isometrically
immersed into a six- dimensional pseudo-Euclidean space [20].

On the other hand, field quantization in a Gödel-type space-time has received

scant attention in the literature. The difficulties in the standard field quantization
in Gödel universe have been explicitly pointed out by Leahy and consists
maintly of the absence of a complete Cauchy surface and of the incompleteness
of the mode solutions to the field equations. Despite some attempts, the meaning
of a quantum field theory in this background is still unclear [18, 21].

Recently, Caldarelli and Klemm gave a particular solution to the resulting

that describes a Gödel-type universe preserving one quarter of the supersymme-
tries. They showed that external dust sources (as well as a negative cosmological
constant) are necessary ingredients to obtain the Gödel universe or its genera-
lizations [3, 22].

It was shown that low energy string theory admits supersymmetric solu-

tions of the Gödel-type [19, 23, 24].

2. THE GENERAL EQUATIONS OF THE GÖDEL UNIVERSE

Gödel universe is one of the most intriguing solutions of the Einstein field

equations [25]. In this section of our paper we briefly review some important
properties of the Gödel universe [1, 2, 3, 26, 27]. We present the general-
relativistic equations that have been used to describe this model.

The Gödel metric can be written in the form

(

)

( )

( ) ( )

( )

1
2

e dx

e dx dx

⎡

⎤

−

≡

⎢

⎥

⎣

⎦

⎡

⎤

≡

−

⎢

⎥

⎣

⎦

(1)

where

(

)

/ , / , / , /

ct a x a y a z a

are the adimensional variables and

is a

nonvanishing constant with dimension of length, namely

=χρ =

⋅ρ

(2)

where

represents the constant of the universal attraction and

is the mass

density of the matter, which creates the gravific field.

The gravitic field is created by the energy- momentum tensor of a perfect fluid

408

Brindusa Ciobanu, Irina Radinschi

1
2

u u

αβ

β β

αβ

⎛

⎞

′

= ρ +

−

≡ ρ

− ρ

⎜

⎟

⎝

⎠

(3)

1 ,

′

ρ = ρ

= ρ

(4)

1 , u

ρ =

(5)

where

ρ′

is the rest density of energy (in rest)/

The coordinates

are comoving coordinates (the model of the comoving

fluid), so that the quadrispeed has the form

(

)

(

)

1 , 0, 0, 0

, 0,

, 0

≡

(6)

The movement of the comoving particles (the observers) in this system of

reference is determined by the following kinematical parameters:
– the acceleration vector

;

≡

(7)

– and the vorticity tensor

;

αβ

α β

β α

αβ

ω =

−

≠

(8)

According to the formula (9) of the quadrispeed, the only nonvanishing

components of the angular speed tensor are given by

ω = ω = −

(9)

The scalar expansion

and the shear tensor

αβ

are like-wise null. This fact

means that the particles of reference (the observers) are in free falling and they
turn round one in comparison with the other rigidly and uniformly.

The angular speed vector may be written in the form

( )

1 2

;

u u

−

αβγδ

β γ δ

ω = − η

= −

−

(10)

Therefore the only nonnull component of the angular speed is

ω =

(11)

and it results that the angular speed of matter is given by

(

)

1/ 2

Ω = −ω ω

(12)

Electric and magnetic fields in a rotating universe

409

The rotation is of a dynamical importance when the ratio between the

rotation period

2 /

π Ω

and the free-fall time 1

is of the order unity. This

condition is satisfied in the Gödel universe for

Ω = π ρ

For each reference

particle, the Gödel universe turns round uniformly as a solid-rigid body with
constant angular speed, around the compass of inertia of each observer at rest in
the substance (there is no difference between these observers in the rotation
center) [2, 3].

It is well known that in the classical case, the thermodynamic equilibrium

exists only for the systems, which move with uniform speed (with the center of
mass in rest in some frame), but for the systems when all their components also
have a common constant angular speed around an axis [2, 28]. This last condition
is satisfied in the Gödel universe.

In their paper [25], Barow and Tsagas used covariant techniques to describe

the properties of the Gödel universe and considered its linear response to a
variety of perturbation. They showed that

the stability of the Gödel model depends

primarily upon the presence of gradients in the centrifugal energy, and secon-
darily on the equation of state of the fluid.

In the case of the Gödel fluid, the equations

p c

Tds

⎛ ⎞

′

= ρ

= ε + ⎜ ⎟

⎝ ⎠

(13)

lead to following results

ε = −

(14)

and

(

)

log

(15)

where

= the proper temperature of the fluid

= the specific proper entropy

= the coefficient of specific heat.

This fact means that the Gödel fluid behaves like a fluid with a coefficient

= 2.

For a perfect fluid we can write the equation of state

(

)

p c

′

−

= − γ ρ

(16)

where 1

≤

2. The superior limit

= 2 implies the fact that the sound speed

dp d

′

(17)

becomes equal to the light speed

[26].

410

Brindusa Ciobanu, Irina Radinschi

Zel’dovich [29] has motivated that an equation of state with 4/3

≤

2 is

possible and can be available for an extremely dense matter, like that from the
final stage of a gravific collaps. The limit case

= 2 is the case of the “rigid

matter” (or the extreme fluid) [1–4].

3. THE TETRADIC FORM OF MAXWELL’S EQUATIONS

IN THE GÖDEL UNIVERSE

The main focus of our paper is to integrate the equations of the electro-

magnetic field in the Gödel universe background. In general, it is very difficult
to treat this problem. Therefore we will concentrate on the special case, namely
we will obtain the Maxwell’s equations within a tetrad frame associated with
Gödel metric.

In our previous works we have described how we can performe to compute

the components of an orthonormal tetradic system in the Gödel universe [1, 3, 27].

Processing the expression (1) such as

( )

(

)

( )

(

)

, ,

0, 1, 2, 3,

0, 1, 2, 3

αβ

= η

α β =

μ γ =

(18)

where the Minkowskian matrix has the diagonal terms: (1, –1, –1, –1), we have

established an ensemble of quadrivectors

( )

(of label

), named mark-vectors

or tetrads. This ensemble of quadrivectors defines a local frame. We have
obtained the mutual components of the tetrads

( )

(

)

( )

(

)

( )

(

)

, 0,

, 0 ,

0, 0,

, 0

0, , 0, , 0 ,

0, 0, 0,

⎛

⎞

= λ

= −λ

= ⎜

⎟

⎝

⎠

λ = −λ

= −λ

(19)

In accordance with the general formula, we have founded the nonvanishing

components of the objects of anholonomity in the Gödel universe [1, 3, 30]

( )( )

( )

( )( )

( )

( )( )

( )

( )( )

( )

1 2

2 1

1 2

2 1

= −Ω

(20)

It is very known that in an arbitrary coordinate frame, the Maxwell’s

equations are

;

αβ

(21)

;

αβ

(22)

where

αβ

represent the contravariant components of the electromagnetic field

tensor (

= 0, 1, 2, 3) and

αβ

are the components of the dual associated to

the tensor

αβ

which are calculated according to the formula

Electric and magnetic fields in a rotating universe

411

( )

1 2

1 1

αβ

αβγδ

γδ

⎡

⎤

= η

−

⎣

⎦

(23)

namely

a e

αβ

αβγδ

γδ

(24)

From the relations (21) and (22) we can establish the form of the

Maxwell’s equations, written in a tetradic coordinate frame

( )( )

( )

( )( )

( )( )( ) ( )( )

( )( )

( )( )( ) ( )( )

( )

n k

k n

n k

r k

k n

r k

n k

r k

− Ω

+ Ω

− Ω

(25)

We note with

(

)(

)

symbols the local components of the electromagnetic field

tensor and with

(

)

the tetradic components of the current density quadrivector.

The tetradic form of the Maxwell’s equations in vacuum becames [3, 4]:

(0)(0)

(0)(1)

(0)(2)

(0)(3)

(0)(1)

(1)(2)

(1)(0)

(1)(1)

(1)(2)

(1)(3)

(1)(1)

(2)(0)

(2)(1)

(2)(2)

−

(2)(3)

(3)(0)

(3)(1)

(3)(2)

(3)(3)

(3)(1)

(0)(0)

(2)(0)

(1)(0)

(2)(0)

(3)(0)

(1)(0)

(0)

(2)(1)

(0)(1)

(1)(1)

(2)(1)

−

(3)(3)

(1)(1)

(1)

(26)

Supposing the private case of an electromagnetic field characterized by the

quadrivectors:

(0, 0,

, 0),

(0, 0,

, 0)

with

= 0, 1, 2, 3 where

the current density quadrivector and

is the quadripotential, one can find the

solution of the Maxwell’s equations.

In what follows, and mostly for sake of completeness, we give a physical

meaning of the parameters. We observe that the norm of the quadrivector current
density is positive, therefore this current is a conduction one. We specify that the
Gödel universe cannot admit conduction current along

-axis, but only convection

current along this axis (or the convection current is prevalent along

-axis)

412

Brindusa Ciobanu, Irina Radinschi

( )

⋅

(27)

The covariant components of the quadripotential are

, 0,

, 0

a e A

⎛

⎞

= ⎜

⎟

⎝

⎠

(28)

( )

−

(29)

where

(2)

is the tetradic component of the quadripotential.

The tetradic components of the quadricurrent

(

)

are given by

( )

, 0,

, 0

ae J

⎛

⎞

⎜

⎟

⎝

⎠

(30)

According with the assumptions regarding the electromagnetic field, the

relations (26) lead to a large system of partial differential equations [4]

(

)

(2)

(

x z

z x

z y

y z

z x

x z

z x

x y

y x

x x

−

∂

∂ ∂

⎛

⎞

∂

−

⎜

⎟

∂ ∂

⎝

⎠

∂

−

∂ ∂

∂

⎛

⎞

∂

−

⎜

⎟

∂ ∂

∂

⎝

⎠

∂

(2)

y x

−

∂

⎛

⎞

∂

−

⎜

⎟

∂

∂ ∂

⎝

⎠

= −

∂

−

∂

∂ ∂

∂

⎛

⎞

∂

−

⎜

⎟

∂

∂ ∂

⎝

⎠

(2)

2 (2)

(2)

a J

y z

∂

∂ ∂

(31)

Electric and magnetic fields in a rotating universe

413

4. THE ELECTRIC AND MAGNETIC FIELDS IN THE GÖDEL UNIVERSE

In order to establish the expressions for electric and magnetic parts of field

we impose to the components of the quadripotential

the gauge-conditions.

And by solving the equations (31) of the electromagnetic field one obtains

( )

/ 2

−

(32)

Knowing the relations

u G

βα

(33)

where

(

)

1 , 0, 0, 0

is the quadrispeed of the element of fluid, the compo-

nents of electric and magnetic fields quadrivectors have the expressions

(

)

/ 2

: 0,2 2

, 0, 0

(34)

(

)

/ 2

: 0, 0, 0,

(35)

The electric and magnetic fields depend on the

-axis coordinate and

parameter, which depends also on the mass density

. Plotting (34) and (35) with

the

and

respectively on

-axis, against

-axis coordinate and

⋅

–3

) we obtain the graph from Fig. 1.

CONCLUSIONS

The aim of this work is the study of the effect of rotation of universe on

electromagnetic field. In the first and second sections of our paper, after a brief
introduction to our problem, we brought out some important properties of the
Gödel universe model. The main focus of our paper was to integrate the
equations of the electromagnetic field in the Gödel universe background. In
general, it is very difficult to treat this problem. Therefore we concentrated on a
special case, and the Maxwell’s equations are derived using the orthonormal
tetradic formalism. In other words, in the section 3, the components of an
orthonormal tetradic system and the objects of anholonomity in the Gödel uni-
verse was computed. Next, we established the form of the Maxwell’s equations,
written in a tetradic coordinate frame. The private case of an electromagnetic
field in vacuum which interacts with the gravitational field of the Gödel universe
is presented. We conclude that the quadrivectors of electric field and magnetic
field respectively do not have temporal components, but only spatial orthogonal

414

Brindusa Ciobanu, Irina Radinschi

components. In this context, the Gödel universe does not admit conduction
current along

-axis, but only convection current, or the convection current is

prevalent along

-axis. The physical meaning of the electromagnetic field

properties is also derived from the graphs of electric and magnetic components
against

-axis coordinate and mass density respectively. As a final remark, the

dependence of the components on the single spatial

-coordinate is an expo-

nential one.

Acknowledgment.

Research for B. Ciobanu and I. Radinschi was supported by MEdC-

CERES in the framework of the grant CEX05-D10-08/ 03.10.2005.

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