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arXiv:gr-qc/0603044 v2 8 Aug 2006

On the Gravitational Energy Associated with

Spacetimes of Diagonal Metric

Murat Korunur‡

Department of Physics, Faculty of Art and Science,

Dicle University, 21280, Diyarbakir-Turkey

mkorunur@dicle.edu.tr

Ali Havare

Department of Physics, Faculty of Art and Science,

Mersin University, 33343, Mersin-Turkey

alihavare@mersin.edu.tr

Mustafa Salti

Department of Physics, Faculty of Art and Science,

Middle East Technical University, 06531, Ankara-Turkey

musts6@yahoo.com, e145447@metu.edu.tr

In order to evaluate the energy distribution (due to matter and fields including

gravitation) associated with a spacetime model of generalized diagonal metric,

we consider the Einstein, Bergmann-Thomson and Landau-Lifshitz energy and/or

momentum definitions both in Einstein’s theory of general relativity and the teleparallel

gravity (the tetrad theory of gravitation).

using Einstein and Bergmann-Thomson formulations, but we also find that the energy-

momentum prescription of Landau-Lifshitz disagree in general with these definitions.

We also give eight different well-known spacetime models as examples, and considering

these models and using our results, we calculate the energy distributions associated with

them. Furthermore, we show that for the Bianci type-I all the formulations give the

same result. This result agrees with the previous works of Cooperstock-Israelit, Rosen,

Johri et al., Banerjee-Sen, Xulu, Vargas and Saltı et al. and supports the viewpoints of

Albrow and Tryon.

We find the same energy distribution

Energy; diagonal spacetimes; general relativity; teleparallel gravity.

‡ Corresponding Author.

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1. Introduction

In the gravitation theories of general relativity and the tetrad theory of gravity (the

teleparallel gravity), the formulation of energy and/or momentum distributions is one

of the oldest, interesting and controversial problems. The first of such attempts was

made by Einstein who proposed an expression for the energy-momentum distribution

of the gravitational field. After this pioneering work, there have been many attempts

to resolve the energy-momentum problem; e.g. Tolman [1], Papapetrou[2], Bergmann-

Thomson [3], Møller [4, 5], Weinberg [6], Qadir-Sharif [7], Landau-Liftshitz [8] and the

teleparallel gravity analogs of the Møller [9] and Einstein, Landau-Lifshitz, Bergmann-

Thomson [10] prescriptions. Except for the Møller definition, these complexes give

the meaningful results when we transform the line-element into the quasi-Cartesian

coordinates. The energy and momentum complex of Møller gives the possibility to

make the calculations in any coordinate system [11].

Virbhadra and his collaborators have considered many space-time models and

showed that several energy-momentum complexes give the same and acceptable

results for a given space-time model [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

Virbhadra[15], using the general relativity versions of energy and momentum complexes

of Einstein, Landau-Lifshitz, Papapetrou and Weinberg’s for a general non-static

spherically symmetric metric of the Kerr-Schild class, showed that all of these energy-

momentum formulations give the same energy distribution as in Penrose energy-

momentum complex. And in teleparallel gravity, there have been some attempts to

show the teleparallel gravitational energy-momentum definitions give the same results

as obtained by using the general relativistic ones [10, 24, 25, 26, 27]. In his recent paper,

Vargas [10] using the Einstein and Landau-Lifshitz complexes, calculated the energy-

momentum density of the Friedman-Robertson-Walker space-time and showed that the

result is the same as obtained in general relativity.

We hope to find the same and acceptable energy distribution both in Einstein’s

theory of general relativity and the tetrad theory of gravity.

organized as follows: in the next section, we introduce our general space-time model to

be considered and give eight special space-time mode as examples for our model. Next,

in section 3, we give the energy and/or momentum definitions of Einstein, Bergmann-

Thomson and Landau-Lifshitz both in general relativity and the tetrad theory of gravity.

In section 4, we calculate the energy distributions (due to matter and fields including

gravitation) associated with our general model both in Einstein’s theory of general

relativity and the tetrad theory of gravity using the formulas which we give in section

3. Section 5 gives us some special cases of our energy solutions. Finally, we summarize

and discuss our results.

Throughout this paper we use that the Greek indices take values from 0 to 3, Latin

indices take values from 1 to 3 and G = 1, c = 1 units as a convention.

The present paper is

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2. Space-time to be considered

The general diagonal line-element can be given as

ds2= −A2(t,x,y,z)dt2+B2(t,x,y,z)dx2+C2(t,x,y,z)dy2+D2(t,x,y,z)dz2.(1)

This space-time model can be reduced to some well-known space-times under special

choices of A, B, C and D. One can easily find lots of models which are special cases of

our general line-element. Here we give a few of them as examples

A. The de Sitter C-Space-time; the massive charged de Sitter C-metric has been

found by Plebanski and Demianski [28], and its gravitational field can be written by the

conditions (see e.g. [29])

√K

M(x + y),

A2(x,y) =

B2(x,y) =

1

M(x + y)√W,

√W

M(x + y),

(2)

C2(x,y) =

1

M(x + y)√K,

D2(x,y) =(3)

where

K(y) = −Λ + 3M2

W(x) = 1 − x2− 2mMx3− q2M2x4.

This solution depends on four parameters namely, the cosmological constant Λ, M > 0

which is the acceleration of the black holes, and m and q which are the interpreted as

electro magnetic charge of the non-accelerated black holes.

B. The spatially self similar locally rotationally symmetric models; these models

can be written under the following definitions [30]

3M2

+ y2− 2mMy3+ q2M2y4,

(4)

(5)

A2(t,x,y,z) = 1,B2(t,x,y,z) = D2

1(t),

(6)

C2(t,x,y,z) = D2

2(t)e−2ax,D2(t,x,y,z) = D2

2(t)k−1sin

?

ky,

(7)

where a and k are parameters describing symmetry groups of the various models.

C. The general Bianchi type-I space-times; this model is given by the following

conditions [24];

A2(t,x,y,z) = 1,B2(t,x,y,z) = B2(t),

(8)

C2(t,x,y,z) = C2(t),D2(t,x,y,z) = D2(t).

(9)

Also, under special choices of A(t), B(t) and C(t) functions, one can define some well-

known closed universes as given blow.

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(i) Defining A = B = C = R(t) and transforming the line element (1) to t,x,y,z

coordinates according to

x = rsinθcosφ,

y = rsinθsinφ,

z = rcosθ,

(10)

gives

ds2= −dt2+ R2(t)[dr2+ r2(dθ2+ sin2θdφ2)],

which describes the well-known spatially flat Friedmann-Robertson-Walker space-

time. The Friedmann-Robertson-Walker cosmological model has attracted

considerable attention in the relativistic cosmology literature. Maybe one of the

most important properties of this model is, as predicted by inflation [31], the

flatness, which agrees with the observed cosmic microwave background radiation.

In the early universe the sorts of the matter fields are uncertain. The existence

of anisotropy at early times is a very natural phenomenon to investigate, as an

attempt to clarify among other things, the local anisotropies that we observe

today in galaxies, clusters and superclusters. So, at the early time, it appears

appropriate to suppose a geometry that is more general than just the isotropic and

homogenous Friedmann-Robertson-Walker geometry. Even though the universe, on

a large scale, appears homogenous and isotropic at the present time, there are no

observational data that guarantee this in an epoch prior to the recombination. The

anisotropies defined above have many possible sources; they could be associated

with cosmological magnetic or electric fields, long-wave length gravitational waves,

Yang-Mills fields [32].

(ii) Defining A = el(t), B = em(t)and C = en(t), then the line element describes the

well-known Bianchi-I type metric.

(iii) Writing A = tp1, B = tp2and C = tp3(where p1, p2and p3are constants), then we

obtain the well-known viscous Kasner-type model.

(11)

(iv) When the functions A(t), B(t) and C(t) are given by the following equations [33]

A(t) =

?N

H2

0

?1

3?

sinh3H0t

1 +3H0t0

2

?k1−1

3?

cosh3H0t

sinh3H0t

2

?k1

3−k1

×

?

?1

2

+3H0t0

22

?2

,

(12)

B(t) =

?N

H2

0

3?

sinh3H0t

1 +3H0t0

2

?k2−1

3?

cosh3H0t

sinh3H0t

2

?k2

3−k2

×

?

?1

2

+3H0t0

22

?2

,

(13)

C(t) =

?N

H2

0

3?

1 +3H0t0

2

?k3−1

3?

sinh3H0t

2

?k3

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×

?

sinh3H0t

2

+3H0t0

2

cosh3H0t

2

?2

3−k3

,

(14)

then the line element describes the Heckmann-Schucking solution of the Bianchi-I

type space-time. Here H0 =

Sitter universe with a cosmological constant Λ, while the constant N characterizes

the quantity of the dust in the universe.

√Λ is nothing but a Hubble parameter for the de

D. The Kantowski-Sachs model [24]; we can write this space-time by choosing

A2(t,x,y,z) = B2(t,x,y,z) = C2(t,x,y,z) = 1,D2(t,x,y,z) = siny.(15)

E. The Bianchi type-V space-time [24] is defined under the case give below.

A2(t,x,y,z) = C2(t,x,y,z) = 1,B2(t,x,y,z) = D2(t,x,y,z) = e2y.(16)

3. Gravitational Energy and Momentum

3.1. In general relativity

In this section of the paper, we give the Einstein, Bergmann-Thomson, Landau-Lifshitz

energy-momentum definitions.

Einstein’s formulation; the energy and momentum prescription of Einstein [4, 5] in

general relativity is given by

1

16πHνα

where

gµβ

√−g

Θ0

components of energy current density. The Einstein energy and momentum densities

satisfy the local conservation laws

∂Θν

µ

∂xν= 0.

And energy-momentum components are given by

Θν

µ=

µ,α

(17)

Hνα

µ =

?

−g(gνβgαξ− gαβgνξ)

?

,ξ.

(18)

0is the energy density, Θ0

αare the momentum density components, and Θα

0are the

(19)

Pµ=

? ? ?

Θ0

µdxdydz.

(20)

Bergmann-Thomson’s formulation; the energy-momentum prescription of Bergmann-

Thomson [3] is given by

1

16πΠµνα

where

Ξµν=

,α

(21)

Πµνα= gµβVνα

β

(22)

with

Vνα

β

= −Vαν

β

=

gβξ

√−g

?

−g

?

gνξgαρ− gαξgνρ??

,ρ.

(23)

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The Bergmann-Thomson’s energy-momentum prescription satisfies the following local

conservation laws

∂Ξµν

∂xν= 0

in any coordinate system. The energy and momentum components are defined by

(24)

Pµ=

? ? ?

Ξµ0dxdydz.

(25)

The energy and momentum (energy current) density components are represented by Ξ00

and Ξa0, respectively.

Landau-Lifshitz’s formulation; the energy-momentum prescription of Landau-

Lifshitz [8] is

1

16πSµναβ

where

Υµν=

,αβ

(26)

Sµναβ= −g(gµνgαβ− gµαgνβ).

Υµνis symmetric in its indices. Υ00is energy density and Υa0are momentum (energy

current) density components. Sµναβhas symmetries of Riemann curvature tensor. The

energy-momentum of Landau and Lifshitz satisfies the local conservation laws:

∂Υµν

∂xν= 0

with

(27)

(28)

Υµν= −g(Tµν+ tµν).

(29)

Here Tµνis the energy-momentum tensor of the matter and all non-gravitational fields,

and tµνis known as Landau-Lifshitz energy-momentum pseudotensor. Thus the locally

conserved quantity Υµνcontains contributions from the matter, non-gravitational fields

and gravitational fields. The energy-momentum components is given by

Pµ=

? ? ?

Υµ0dxdydz.

(30)

3.2. In the teleparallel gravity

The teleparallel theory of gravity (the tetrad theory of gravitation) is an alternative

approach to gravitation and corresponds to a gauge theory for the translation group

based on Weitzenb¨ ock geometry [34]. In the theory of teleparallel gravity, gravitation is

attributed to torsion [35], which plays the role of a force [36], and the curvature tensor

vanishes identically. The essential field is acted by a nontrivial tetrad field, which gives

rise to the metric as a by-product. The translational gauge potentials appear as the

nontrivial item of the tetrad field, so induce on space-time a teleparallel structure which

is directly related to the presence of the gravitational field. The interesting place of

teleparallel theory is that, due to its gauge structure, it can reveal a more appropriate

approach to consider some specific problems. This is the situation, for example, in the

energy and momentum problem, which becomes more transparent.

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Møller modified general relativity by constructing a new field theory in teleparallel

space. The aim of this theory was to overcome the problem of the energy-momentum

complex that appears in Riemannian Space [37]. The field equations in this new theory

were derived from a Lagrangian which is not invariant under local tetrad rotation. Saez

[38] generalized Møller theory into a scalar tetrad theory of gravitation. Meyer [39]

showed that Møller theory is a special case of Poincare gauge theory [40, 41].

The energy-momentum complexes of Einstein, Bergmann-Thomson and Landau-

Lifshitz in the teleparallel gravity [10] are given by the following equations, respectively,

1

4π∂λ(U

1

4π∂λ(gµβU

1

4π∂λ(hgµβU

hEµ

ν=

µλ

ν

),

(31)

hBµν=

νλ

β

),

(32)

hLµν=

νλ

β

),

(33)

where h = det(ha

µ) and U

νλ

β

is the Freud’s super-potential, which is given by:

U

νλ

β

= hS

νλ

β

.

(34)

Here Sµνλis the tensor

Sµνλ= m1Tµνλ+m2

2(Tνµλ− Tλµν) +m3

2(gµλTβν

β− gνµTβλ

β) (35)

with m1, m2and m3the three dimensionless coupling constants of teleparallel gravity

[40]. For the teleparallel equivalent of general relativity the specific choice of these three

constants are:

m1=1

4,

To calculate this tensor firstly we must calculate Weitzenb¨ ock connection:

m2=1

2,m3= −1.

(36)

Γα

µν= h

α

a∂νha

µ

(37)

and after this calculation we get torsion of the Weitzenb¨ ock connection:

Tµ

νλ= Γµ

λν− Γµ

νλ.

(38)

For the Einstein, Bergmann-Thomson and Landau-Lifshitz complexes, we have the

relations,

PE

µ=

?

?

?

ΣhE0

µdxdydz,

(39)

PB

µ=

ΣhB0

µdxdydz,

(40)

PL

µ=

ΣhL0

µdxdydz,

(41)

where Pi give momentum components P1, P2, P3 while P0 gives the energy and the

integration hyper-surface Σ is described by x0= t =constant.

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4. Calculations

4.1. In general Relativity

The matrix form of the metric tensor gµνfor the line-element (1) is defined by

−A2(t,x,y,z)

0

0

0

00

0

0

0

0

B2(t,x,y,z)

0

0

C2(t,x,y,z)

0

D2(t,x,y,z)

,

(42)

and its inverse matrix gµνis

−A−2(t,x,y,z)

0

0

0

00

0

0

0

0

B−2(t,x,y,z)

0

0

C−2(t,x,y,z)

0

D−2(t,x,y,z)

, (43)

and the required components of Hνα

µ, Πµναand Sµναβare

H01

0= −Π001= 2A

B(CD)x,

(44)

H02

0= −Π002= 2A

C(BD)y,

(45)

H03

0 = −Π003= 2A

S0011= −C2D2,

S0022= −B2D2,

S0033= −B2C2.

D(BC)z,

(46)

(47)

(48)

(49)

Here the sub index defines derivative with respect to that coordinate. Next, using these

results we obtain the following energy distributions in the general relativity versions of

Einstein, Bergmann-Thomson and Landau-Lifshitz complexes, respectively.

8πΘ0

0=

??A

B

?

?A

x(CD)x+A

B(CD)xx+

?A

C

?

?

?

y(BD)y

+A

C(BD)yy+

D

?

z(BC)z+A

D(BC)zz

?1

,

(50)

− 8πΞ00=

?1

AB(CD)x

?

x+

AC(BD)y

y+

?1

AD(BC)z

?

z,

(51)

− 16πΥ00=

??

C2D2?

xx+

?

B2D2?

yy+

?

B2C2?

zz

?

.

(52)

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4.2. In the teleparalel gravity

In this section, using the teleparallel gravity analogs of the Einstein, Bergmann-Thomson

and Landau-Lifshitz energy and/or momentum formulations, we evaluate the energy

density.

The tetrad components of ha

µfor the line-element (1) are given by

ha

µ= Aδa

0δ0

µ+ Bδa

1δ1

µ+ Cδa

2δ2

µ+ Dδa

3δ3

µ

(53)

and for h

µ

a

are

h

µ

a

= A−1δ0

aδµ

0+ B−1δ1

aδµ

1+ C−1δ2

aδµ

2+ D−1δ3

aδµ

3.

(54)

Using equation (37), (53) and (54), we obtain the following non-vanishing components

of the Weitzenb¨ ock connection

00=At

A,

10=Bt

B,

20=Ct

C,

30=Dt

D,

The corresponding non-vanishing torsion components are found:

Γ0

Γ0

01=Ax

A,

Γ0

02=Ay

A,

Γ0

03=Az

A

(55)

Γ1

Γ1

11=Bx

B,

Γ1

12=By

B,

Γ1

13=Bz

B

(56)

Γ2

Γ2

21=Cx

C,

Γ2

22=Cy

C,

Γ2

23=Cz

C

(57)

Γ3

Γ3

31=Dx

D,

Γ3

32=Dy

D,

Γ3

33=Dz

D.

(58)

T0

01= −Ax

10= −Bt

20= −Ct

30= −Dt

A,

T0

02= −Ay

12= −By

T2

A,

T0

03= −Az

13= −Bz

T2

A

(59)

T1

B,

T1

B,

T1

B

(60)

T2

C,

21= −Cx

T3

C,

23= −Cz

T3

C

(61)

T3

D,

31= −Dx

D,

32= −Dy

D.

(62)

Taking these results into the equation (35), the non-zero energy components of the

tensor S

µ

are found as:

1

2(AB)2

1

2(AC)2

1

2(AD)2

Next, the non-vanishing components of Freud’s super-potential are

=1

2

B

=1

2

C

=1

2

D

νλ

S001= −

(CD)x

(CD),

(BD)y

(BD),

(BC)z

(BC).

(63)

S002= −

(64)

S003= −

(65)

U

01

0

?A

?A

?A

?

?

?

(CD)x,

(66)

U

02

0

(BD)y,

(67)

U

03

0

(BC)z.

(68)

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Using equations (31), (32) and (33), the Einstein, Bergmann-Thomson and Landau-

Lisfhitz’s energy densities are found:

hE0

0=

1

8π

??A

B

?A

??1

?

x(CD)x+A

B(CD)xx+

?A

C

?

?

y(BD)y

+A

C(BD)yy+

D

?

z(BC)z+A

D(BC)zz

(69)

hB00= −1

8π AB(CD)x

?

x+

?1

AC(BD)y

?

y+

?1

AD(BC)z

?

z

?

,

(70)

hL00= −

1

16π

??

C2D2?

xx+

?

B2D2?

yy+

?

B2C2?

zz

?

.

(71)

We obtain that for the Einstein, Bergmann-Thomson and Landau-Lifshitz energy

complexes, each complex’s value is the same evaluated either in general relativity or

in teleparallel gravity.

5. Energy distributions for the special cases of the general metric

(i) The de Sitter C-Space-time solutions; one can easily find the following result for

the given metric,

Θ0

0= hE0

=(x + y)−2

0= Ξ0

16πM2[W,xx

=(x + y)−4

48π M4[3y2M2− 6Λ2− 6q2M4y4− 45mM3x3− 36q2M4x4− 6mM3y4

+ 6q2M4y5+ 6xy2− 18M2xy + 12M3mx4+ 6M4q2x5− 21x2M2

+ 2(x + y)Λ2+ 6M2x3− 24mxM3y3+ 18xq2M4y4− 18myx2M3

+ 27mxy2M3− 36q2M4x3y − 18q2M4x2y2− 24M4xy3q2+ 12M2yx2

− 18M3x2my2+ 12M4x2q2y3+ 12M3mx3y + 6M4q2x4y]

0= hB0

0

2

+(x + y − 2)K,y− 3W

x + y

+6W − 2K(x + y − 3)

(x + y)2

]

(72)

and

Υ00= hL00

= −

1

16π

∂2

∂x2

1

1

M2(x + y)2

?

?

?

?

1 − x2− 2mMx3− q2M2x4

−Λ+3M2

3M2

+ y2− 2mMy3+ q2M2y4

+

∂2

∂y2

?

M3(x + y)2

?

.

(73)

Here,

mathematical exposition is lengthy, we give the energy in the form as given above.

for the energy in Landau-Lifshitz complex,since the intermediary

(ii) The solutions for spatially self similar locally rotationally symmetric space-times;

Θ0

0= hE0

=k2D2

0= Ξ0

1(t)[cos(2ky) − 3] + 32a2e−2axD2

64πk1/2D1(t)sin3/2(ky)

0= hB0

0

2(t)sin2(ky)

(74)

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and

Υ00= hL00= −

1

16πke−2axsin(ky)[16a2e−2axD2

2(t) − k2D2

1(t)].

(75)

(iii) The general Bianchi type-I solutions;

Θ0

0= hE0

0= Ξ0

0= hB0

0= Υ00= hL00= 0.

(76)

Albrow [42] and Tryon [43] assumed that the net energy of the universe may be

equal to zero. The subject of the energy-momentum distributions of closed and open

universes was initiated by an interesting work of Cooperstock and Israelit [44]. They

found the zero value of energy for any homogenous isotropic universe described by

a Friedmann-Robertson-Walker metric in the context of general relativity. This

interesting result influenced some general relativists [45].

(iv) The Kantowski-Sachs solutions;

Θ0

0= hE0

0= Ξ0

0= hB0

0=sin(y)

8π

,

Υ00= hL00= −cos(2y)

8π

. (77)

(v) The Bianchi type-V solutions; we obtain the energy distributions as

Θ0

0= hE0

0= Ξ0

0= hB0

0=2

πe4y,

Υ00= hL00= −4

πe8y.

(78)

6. Summary and Final Remarks

The main object of the present paper is to show that it is possible to evaluate the

energy distribution by using the energy-momentum formulations in not only general

relativity but also the teleparallel gravity.

matter and fields including gravitation), we considered two different approaches of the

Einstein, Bergmann-Thomson and Landau-Lifshitz energy-momentum definitions such

as the general relativity and the teleparallel gravity analogs of them.

The subject of energy-momentum localization in gravitation theories has been

controversial, exciting and very interesting; however, it has been associated with some

debate.

We found the teleparallel gravity analog of the formulations considered give the

same result as its general relativity versions. Using these results we also calculated the

energy distributions for some special space-times. We showed that the Einstein and

Bergmann-Thomson formulations give the same results in these space-time model, but

the Landau-Lifshitz formulation does not.

In Bianchi type-I models all the formulations give the same zero value energy. This

result is the same as obtained by Cooperstock-Israelit, Rosen, Johri et al., Banerjee-

Sen, Xulu, Vargas and Saltı et al. and support the viewpoints of Albrow and Tryon.

Furthermore, if the calculations are performed for the Bianchi type-I model again, one

can easily show that the energy is also independent of the teleparallel dimensionless

coupling constants, which means that it is valid not only in the teleparallel equivalent

of general relativity, but also in any teleparallel model.

To compute the energy density (due to

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On the gravitational energy associated with...

12

Appendix-Kinematical quantities-

After the pioneering works of Gamow [46] and G¨ odel [47], the idea of global rotation of

the universe has became considerably important physical aspect in the calculations of

general relativity. Now, we introduce a tetrad basis by

θ0= Adt,θ1= Bdx,θ2= Cdy,θ3= Ddz.

(79)

By use of the co-moving tetrad formalism, the kinematics of this model can be expressed

solely in terms of the structure coefficients of the tetrad basis. We define the structure

coefficients by

dθα=1

2cα

By taking the exterior derivatives of the tetrad basis (79) and using the kinematics

formulas [48] given by

βγθβ∧ θγ.

(80)

Kinematical Quantity

Four-acceleration vector:

vorticity tensor:

expansion(deformation) tensor:

expansion scalar:

vorticity vector:

vorticity scalar:

shear tensor:

Definition

aµ= c0

ωµν=1

ξµν=1

ξ = c1

ω2=1

ω =1

4

(c0

σµν= ξµν−1

µ0

2c0

µν

2(cµ0ν+ cν0µ)

01+ c2

2c0

23)2+ (c0

02+ c3

31,

31)2+ (c0

3ξδµν

03

ω1=1

2c0

23,ω3=1

2c0

12

?

12)2

Table 1. List of kinematical quantities and their formulas.

we find the following quantities for the line-element (1)

A,i

A√gii,ai= 2 (81)

w10= −w01=A,x

wi= 0,

θ =2

AB

σ11=BB,t

A

σ22=CC,t

A

σ33=DD,t

A

AB,

w20= −w02=A,y

AC,

w30= −w03=A,y

AD,(82)

(83)

?B,t

+C,t

C

2

3A

2

3A

2

3A

+D,t

D

?

,

(84)

−

?B,t

B

?B,t

B

?B,t

+C,t

C

+D,t

D

?

?

?

,

(85)

−

+C,t

C

+D,t

D

,

(86)

−

B

+C,t

C

+D,t

D

.

(87)

From these results we see that the model given in (1) has expansion with non-zero shear.

We also note that this model has non-vanishing components of the four-acceleration

vector and vanishing vorticity vector.

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On the gravitational energy associated with...

13

Acknowledgement

We would like to thank Dr. E.C. Vagenas for his helps. MK and MS would like to thank

Turkish Scientific and Technical Research Council (T¨ ubitak)-Feza G¨ ursey Institute,

Istanbul, for the hospitality we received in summer terms 2002-2005. The work of

MS was supported by T¨ ubitak.

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