# On the relative energy associated with space-times of diagonal metrics

**ABSTRACT** In order to evaluate the energy distribution (due to matter and fields including gravitation) associated with a space-time

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).

We find same energy distribution 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 space-time

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 Bianchi Type-I models 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.

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**ABSTRACT:**This paper has been removed by arXiv administrators because it plagiarizes gr-qc/0301090, gr-qc/0303034, gr-qc/0212018, and gr-qc/9601044.Foundations of Physics Letters 01/2006; - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper has been removed by arXiv administrators because of overlap with gr-qc/9910015 and hep-th/0308070. It also has excessive overlap with the following papers also written by the authors or their collaborators: gr-qc/0509047, gr-qc/0607110, gr-qc/0602012, gr-qc/0606028, and others.Progress of Theoretical Physics 03/2006; · 2.48 Impact Factor

Page 1

PRAMANA

— journal of

physics

c ? Indian Academy of SciencesVol. 68, No. 5

May 2007

pp. 735–748

On the relative energy associated with space-times

of diagonal metrics

MURAT KORUNUR1, MUSTAFA SALTI2and ALI HAVARE3

1Department of Physics, Dicle University, 21280, Diyarbakir-Turkey

2Department of Physics, Middle East Technical University, 06531, Ankara-Turkey

3Department of Physics, Mersin University, 33343, Mersin-Turkey

E-mail: mkorunur@dicle.edu.tr; musts6@yahoo.com; alihavare@mersin.edu.tr

MS received 5 October 2006; accepted 28 February 2007

Abstract.

ing gravitation) associated with a space-time model of generalized diagonal metric, we

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

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

(the tetrad theory of gravitation). We find same energy distribution using Einstein and

Bergmann–Thomson formulations, but we also find that the energy–momentum prescrip-

tion of Landau–Lifshitz disagree in general with these definitions. We also give eight

different well-known space-time models as examples, and considering these models and

using our results, we calculate the energy distributions associated with them. Further-

more, we show that for the Bianchi Type-I models 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.

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

Keywords. Energy; diagonal space-times; teleparallel theory.

PACS Nos 04.40.-q; 04.20.Jb; 04.50.+h; 04.70.-s

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 formulation was

written by Einstein who proposed an expression for the energy–momentum of the

gravitational field. After this pioneering work, there have been many attempts to re-

solve 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 four-momentum def-

inition, these complexes give meaningful results when we transform the line-element

735

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Murat Korunur, Mustafa Salti and Ali Havare

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 same and acceptable results

for a given space-time model [12–23]. Virbhadra [15], using the general relativity

versions of energy and momentum complexes of Einstein, Landau–Lifshitz, Papa-

petrou 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 that the teleparallel

gravitational energy–momentum definitions give the same results as obtained by

using the general relativistic ones [10,24–27]. In his recent paper, Vargas [10] us-

ing 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. The present paper

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

model to be considered and give eight special space-time models as examples for our

model. Next, in §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 §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 §3. Section 5 gives us some special cases of our energy solutions.

Finally, we summarize and discuss our results.

Throughout this paper 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.

2. Space-time models 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,

(2)

736

Pramana – J. Phys., Vol. 68, No. 5, May 2007

Page 3

Relative energy in diagonal metrics

C2(x,y) =

1

M(x + y)√K,

D2(x,y) =

√W

M(x + y),

(3)

where

K(y) = −Λ + 3M2

3M2

+ y2− 2mMy3+ q2M2y4,

(4)

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 inter-

preted as electromagnetic charge of the non-accelerated black holes.

(B) The spatially self-similar locally rotationally symmetric models: These mod-

els can be written under the following definitions [30]:

(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 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 below.

1. 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 at-

tracted considerable attention in the relativistic cosmology literature. One

of the most important properties of this model may be, as predicted by in-

flation [31], the flatness, which agrees with the observed cosmic microwave

background radiation.

(11)

Pramana – J. Phys., Vol. 68, No. 5, May 2007

737

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Murat Korunur, Mustafa Salti and Ali Havare

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 in-

vestigate, 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-wavelength gravitational waves, Yang–Mills fields [32].

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

the well-known Bianchi Type-I metric.

3. Writing A = tp1, B = tp2and C = tp3(where p1, p2and p3are constants),

then we obtain the well-known viscous Kasner-type model.

4. When the functions A(t), B(t) and C(t) are given by the following equations

[33]

A(t) =

?N

H2

?

0

sinh3H0t

?1/3?

1 +3H0t0

2

?k1−(1/3)?

cosh3H0t

sinh3H0t

2

?k1

×

2

+3H0t0

22

?(2/3)−k1

,

(12)

B(t) =

?N

H2

?

0

sinh3H0t

?1/3?

1 +3H0t0

2

?k2−(1/3)?

cosh3H0t

sinh3H0t

2

?k2

×

2

+3H0t0

22

?(2/3)−k2

,

(13)

C(t) =

?N

H2

?

0

sinh3H0t

?1/3?

1 +3H0t0

2

?k3−(1/3)?

cosh3H0t

sinh3H0t

2

?k3

×

2

+3H0t0

22

?(2/3)−k3

,

(14)

then the line-element describes the Heckmann–Schucking solution of the

Bianchi Type-I space-time. Here H0 =

meter for the de 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 para-

(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)

738

Pramana – J. Phys., Vol. 68, No. 5, May 2007

Page 5

Relative energy in diagonal metrics

(E) The Bianchi Type-V space-time [24] is defined under the case given 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, we give the Einstein, Bergmann–Thomson and Landau–Lifshitz

energy–momentum definitions.

Einstein’s formulation: The energy and momentum formulation of Einstein [4,5]

in general relativity is given by

Θν

μ=

1

16πHνα

μ,α,

(17)

where

Hνα

μ

=

gμβ

√−g

?−g(gνβgαξ− gαβgνξ)?

αare the momentum density components, and Θα

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

densities satisfy the local conservation laws

,ξ.

(18)

Θ0

0is the energy density, Θ0

0are

∂Θν

∂xν= 0.

μ

(19)

and energy–momentum components are given by

? ? ?

Bergmann–Thomson’s formulation:

Bergmann–Thomson [3] is given by

Pμ=Θ0

μdxdydz.

(20)

The energy–momentum prescription of

Ξμν=

1

16πΠμνα

,α,

(21)

where

Πμνα= gμβVνα

β

(22)

with

Vνα

β

= −Vαν

β

=

gβξ

√−g

?−g?gνξgαρ− gαξgνρ??

,ρ.

(23)

Pramana – J. Phys., Vol. 68, No. 5, May 2007

739

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Murat Korunur, Mustafa Salti and Ali Havare

The Bergmann–Thomson’s energy–momentum prescription satisfies the following

local conservation law

∂Ξμν

∂xν= 0(24)

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

? ? ?

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

by Ξ00and Ξa0, respectively.

Landau–Lifshitz’s formulation: The energy–momentum prescription of Landau–

Lifshitz [8] is

Pμ=Ξμ0dxdydz.

(25)

Υμν=

1

16πSμναβ

,αβ,

(26)

where

Sμναβ= −g(gμνgαβ− gμαgνβ).

Υμνis symmetric in its indices. Υ00is the energy density and Υa0are the mo-

mentum (energy current) density components. Sμναβhas symmetries of Riemann

curvature tensor. The energy–momentum of Landau and Lifshitz satisfies the local

conservation laws:

(27)

∂Υμν

∂xν

= 0 (28)

with

Υμν= −g(Tμν+ tμν).

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 component is

given by

? ? ?

(29)

Pμ=Υμ0dxdydz.

(30)

3.2 In the teleparallel gravity

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

tive 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,

740

Pramana – J. Phys., Vol. 68, No. 5, May 2007

Page 7

Relative energy in diagonal metrics

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 non-trivial

tetrad field, which gives rise to the metric as a by-product.

gauge potentials appear as the non-trivial item of the tetrad field, and so induce on

space-time a teleparallel structure which is directly related to the presence of the

gravitational field. The interesting quality of teleparallel theory is that, due to its

gauge structure, it can reveal a more appropriate approach for some specific prob-

lems. This is the situation, for example, in the energy and momentum problem,

which becomes more transparent [10].

Møller modified general relativity by constructing a new field theory in telepar-

allel 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 Poincar´ e

gauge theory [40,41].

The energy–momentum complexes of Einstein,

Landau–Lifshitz in the teleparallel gravity [10] are given by the following equa-

tions, respectively:

The translational

Bergmann–Thomson and

hEμ

ν=

1

4π∂λ(Uμλ

ν),

(31)

hBμν=

1

4π∂λ(gμβUνλ

β),

(32)

hLμν=

1

4π∂λ(hgμβUνλ

β),

(33)

where h = det(ha

μ) and Uνλ

β

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

Uνλ

β

= hSνλ

β.

(34)

Here Sμνλis the tensor where

Sμνλ= m1Tμνλ+m2

2(Tνμλ− Tλμν) +m3

2(gμλTβν

β

− gνμTβλ

β) (35)

with m1, m2 and m3 the 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,m2=1

2,m3= −1.

(36)

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

Γα

μν= hα

a∂νha

μ

(37)

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

Pramana – J. Phys., Vol. 68, No. 5, May 2007

741

Page 8

Murat Korunur, Mustafa Salti and Ali Havare

Tμ

νλ= Γμ

λν− Γμ

νλ.

(38)

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

relations

?

?

?

where Pigive momentum components P1, P2, P3while P0gives the energy and the

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

PE

μ=

Σ

hE0

μdxdydz,

(39)

PB

μ=

Σ

hB0

μdxdydz,

(40)

PL

μ=

Σ

hL0

μdxdydz,

(41)

4. Calculations

4.1 In general relativity

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

⎛

⎝

and its inverse matrix gμνis

⎛

⎝

⎜

−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)

⎜

−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=A

B(CD)x,

(44)

H02

0 = −Π002=A

C(BD)y,

(45)

H02

0 = −Π003=A

D(BC)z,

(46)

742

Pramana – J. Phys., Vol. 68, No. 5, May 2007

Page 9

Relative energy in diagonal metrics

S0011= C2D2,

(47)

S0022= B2D2,

(48)

S0033= B2C2.

(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, respec-

tively.

??A

+A

C(BD)yy+

D

8πΘ0

0=

B

?

x

(CD)x+A

B(CD)xx+

?A

?

?A

C

?

y

(BD)y

?

z

(BC)z+A

D(BC)zz

?

,

(50)

−8πΞ00=

?

1

AB(CD)x

?

x

+

1

AC(BD)y

?

y

+

?

1

AD(BC)z

?

z

,

(51)

8πΥ00=

??C2D2?

xx+?B2D2?

yy+?B2C2?

zz

?

.

(52)

4.2 In the teleparallel 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 hμ

aare

hμ

a= A−1δ0

aδμ

0+ B−1δ1

aδμ

1+ C−1δ2

aδμ

2+ D−1δ3

aδμ

3.

(54)

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

of the Weitzenb¨ ock connection

Γ0

00=At

A,

Γ0

01=Ax

A,

Γ0

02=Ay

A,

Γ0

03=Az

A

(55)

Γ1

10=Bt

B,

Γ1

11=Bx

B,

Γ1

12=By

B,

Γ1

13=Bz

B

(56)

Γ2

20=Ct

C,

Γ2

21=Cx

C,

Γ2

22=Cy

C,

Γ2

23=Cz

C

(57)

Γ3

30=Dt

D,

Γ3

31=Dx

D,

Γ3

32=Dy

D,

Γ3

33=Dz

D.

(58)

Pramana – J. Phys., Vol. 68, No. 5, May 2007

743

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Murat Korunur, Mustafa Salti and Ali Havare

The corresponding non-vanishing torsion components are found to be

T0

01= −Ax

10= −Bt

20= −Ct

30= −Dt

A,

T0

02= −Ay

12= −By

21= −Cx

31= −Dx

A,

T0

03= −Az

13= −Bz

23= −Cz

32= −Dy

A

(59)

T1

B,

T1

B,

T1

B

(60)

T2

C,

T2

C,

T2

C

(61)

T3

D,

T3

D,

T3

D.

(62)

Taking these results into eq. (35), the non-zero energy components of the tensor

Sνλ

μ

are found to be

1

2(AB)2

1

2(AC)2

1

2(AD)2

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

?A

U02

2

C

0 =1

2

D

Using eqs (31)–(33), the Einstein, Bergmann–Thomson and Landau–Lifshitz’s en-

ergy densities are found:

??A

+A

C(BD)yy+

D

z

??

S001= −

(CD)x

(CD),

(BD)y

(BD),

(BC)z

(BC).

(63)

S002= −

(64)

S003= −

(65)

U01

0 =1

2

B

?

?

?

(CD)x,

(66)

0 =1

?A

?A

(BD)y,

(67)

U03

(BC)z.

(68)

hE0

0=

1

8πB

?

x

(CD)x+A

B(CD)xx+

?A

C

?

y

(BD)y

?A

?

?

(BC)z+A

D(BC)zz

?

?

,

(69)

hB00= −1

8π

1

AB(CD)x

x

+

?

1

AC(BD)y

?

y

+

1

AD(BC)z

?

z

?

,

(70)

hL00=

1

8π

??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.

744

Pramana – J. Phys., Vol. 68, No. 5, May 2007

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Relative energy in diagonal metrics

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

1. The de Sitter C-space-time solutions: One can easily find the following result

for the given metric,

Θ0

0= hE0

=(x + y)−2

16πM2

0= Ξ0

0= hB0

?W,xx

+6W − 2K(x + y − 3)

(x + y)2

=(x + y)−4

48πM4[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

2

+(x + y − 2)K,y− 3W

x + y

?

(72)

and

Υ00= hL00

=

∂2

∂x2

?

1

M2(x + y)2

?

1 − x2− 2mMx3− q2M2x4

−Λ+3M2

?

3M2

+ y2− 2mMy3+ q2M2y4

?

+∂2

∂y2

?

1

M3(x + y)2

.

(73)

Here, for the energy in Landau–Lifshitz complex, since the intermediary math-

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

2. The solutions for spatially self-similar locally rotationally symmetric space-

times:

Θ0

0= hE0

0= Ξ0

0= hB0

0

=k2D2

1(t)[cos(2ky) − 3] + 32a2e−2axD2

64πk1/2D1(t)sin3/2(ky)

2(t)sin2(ky)

(74)

and

Υ00= hL00=

1

8πke−2axsin(ky)[16a2e−2axD2

2(t) − k2D2

1(t)].

(75)

Pramana – J. Phys., Vol. 68, No. 5, May 2007

745

Page 12

Murat Korunur, Mustafa Salti and Ali Havare

3. 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 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].

4. The Kantowski–Sachs solutions:

Θ0

0= hE0

0= Ξ0

0= hB0

0=sin(y)

8π

,

Υ00= hL00=cos(2y)

4π

.

(77)

5. The Bianchi Type-V solutions: We obtain the energy distributions as

Θ0

0= hE0

0= Ξ0

0= hB0

0=2

πe4y,

Υ00= hL00=8

πe8y.

(78)

6. Summary and final remarks

The main goal of the present paper is to show that it is possible to calculate the en-

ergy distribution by using the energy–momentum formulations in not only general

relativity but also the teleparallel gravity. To compute the energy density (due to

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.

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

the same result as its general relativity versions. Using these results, we also cal-

culated the energy distributions for some special space-times and showed that the

Einstein and Bergmann–Thomson formulations give the same results in these space-

times, 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 supports the viewpoints of Al-

brow 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.

746

Pramana – J. Phys., Vol. 68, No. 5, May 2007

Page 13

Relative energy in diagonal metrics

Acknowledgement

We would like to thank Dr Elias C Vagenas for his help.

References

[1] R C Tolman, Relativity, thermodynamics and cosmology (Oxford University Press,

London, 1934) p. 227

[2] A Papapetrou, Proc. R. Irish. Acad. A52, 11 (1948)

[3] P G Bergmann and R Thomson, Phys. Rev. 89, 400 (1953)

[4] C Møller, Ann. Phys. (NY) 4, 347 (1958)

[5] C Møller, Ann. Phys. (NY) 12, 118 (1961)

[6] S Weinberg, Gravitation and cosmology: Principle and applications of general theory

of relativity (John Wiley and Sons, Inc., New York, 1972)

[7] A Qadir and M Sharif, Phys. Lett. A167, 331 (1992)

[8] L D Landau and E M Lifshitz, The classical theory of fields, 4th edition (Pergamon

Press, Oxford, 1987) (Reprinted in 2002)

[9] F I Mikhail, M I Wanas, A Hindawi and E I Lashin, Int. J. Theor. Phys. 32, 1627

(1993)

[10] T Vargas, Gen. Relativ. Gravit. 36, 1255 (2004)

[11] I Radinschi, Mod. Phys. Lett. A15, 2171 (2000)

[12] I Radinschi, Fizika B9, 43 (2000); Chin. J. Phys. 42, 40 (2004); Fizika B14, 3

(2005); Mod. Phys. Lett. A17, 1159 (2002); U.V.T., Physics Series 42, 11 (2001);

Mod. Phys. Lett. A16, 673 (2001); Acta Phys. Slov. 49, 789 (1999); Acta Phys. Slov.

50, 609 (2000); Mod. Phys. Lett. A15, 803 (2000)

Th Grammenos and I Radinschi, gr-qc/0602105

I-Ching Yang and Irina Radinschi, Chin. J. Phys. 41, 326 (2003)

[13] K S Virbhadra, Phys. Rev. D41, 1086 (1990)

[14] K S Virbhadra, Phys. Rev. D42, 2919 (1990)

[15] K S Virbhadra, Phys. Rev. D60, 104041 (1999)

[16] F I Cooperstock and S A Richardson, in Proc. 4th Canadian Conf. on General Rela-

tivity and Relativistic Astrophysics (World Scientific, Singapore, 1991)

[17] N Rosen and K S Virbhadra, Gen. Relativ. Gravit. 25, 429 (1993)

[18] K S Virbhadra, Pramana – J. Phys. 45, 215 (1995)

[19] A Chamorro and K S Virbhadra, Pramana – J. Phys. 45, 181 (1995)

[20] J M Aguirregabiria, A Chamorro and K S Virbhadra, Gen. Relativ. Gravit. 28, 1393

(1996)

[21] A Chamorro and K S Virbhadra, Int. J. Mod. Phys. D5, 251 (1996)

[22] S S Xulu, Int. J. Mod. Phys. A15, 4849 (2000)

[23] E C Vagenas, Int. J. Mod. Phys. A18, 5781 (2003); Int. J. Mod. Phys. A18, 5949

(2003); Mod. Phys. Lett. A19, 213 (2004); Int. J. Mod. Phys. D14, 573 (2005); Int.

J. Mod. Phys. A21, 1947 (2006)

[24] H V Fagundes, Gen. Relativ. Gravit. 24(2), (1992)

M Saltı and A Havare, Int. J. Mod. Phys. A20, 2169 (2005)

M Saltı, Astrophys. Space Sci. 299, 159 (2005); Mod. Phys. Lett. A20, 2175 (2005);

Acta Phys. Slov. 55, 563 (2005); Czech. J. Phys. 56, 177 (2006)

[25] M Saltı and O Aydogdu, Found. Phys. Lett. 19, 269 (2006)

[26] O Aydogdu and M Saltı, Prog. Theor. Phys. 115, 63 (2006)

Pramana – J. Phys., Vol. 68, No. 5, May 2007

747

Page 14

Murat Korunur, Mustafa Salti and Ali Havare

[27] O Aydogdu, M Saltı and M Korunur, Acta Phys. Slov. 55, 537 (2005)

A Havare, M Korunur and M Saltı, Astrophys. Space Sci. 301, 43 (2006)

[28] J A Plebanski and M Demianski, Ann. Phys. (NY) 98, 98 (1976)

[29] O J C Dias and J P S Lemos, Phys. Rev. D67, 064001 (2003)

[30] C Wu, Gen. Relativ. Gravit. 3, 625 (1969)

[31] A H Guth, Phys. Rev. D23, 347 (1981)

A D Linde, Phys. Lett. B108, 389 (1982)

A D Linde, Phys. Lett. B129, 177 (1983)

A Albrecht and P J Steinhardt, Phys. Rev. D48, 1220 (1982)

[32] J D Barrow, Phys. Rev. D55, 7451 (1997)

[33] O Heckmann and E Schucking, Newtonsche and Einsteinsche Kosmologie, Handbuch

der Physik 53, 489 (1959)

[34] R Weitzenb¨ ock, Invariantten theorie (Gronningen, Noordhoff, 1923)

[35] K Hayashi and T Shirafuji, Phys. Rev. D19, 3524 (1978)

[36] V V de Andrade and J G Pereira, Phys. Rev. D56, 4689 (1997)

[37] C Møller, Mat. Fys. Medd. K. Vidensk. Selsk. 39, 13 (1978); 1, 10 (1961)

[38] D Saez, Phys. Rev. D27, 2839 (1983)

[39] H Meyer, Gen. Relativ. Gravit. 14, 531 (1982)

[40] K Hayashi and T Shirafuji, Prog. Theoret. Phys. 64, 866 (1980); 65, 525 (1980)

[41] F W Hehl, J Nitsch and P von der Heyde, General relativity and gravitation edited

by A Held (Plenum, New York, 1980)

[42] M G Albrow, Nature (London) 241, 56 (1973)

[43] E P Tryon, Nature (London) 246, 396 (1973)

[44] F I Cooperstock, Gen. Relativ. Gravit. 26, 323 (1994)

F I Cooperstock and M Israelit, Found. Phys. 25, 631 (1995)

[45] N Rosen, Gen. Relativ. Gravit. 26, 319 (1994)

V B Johri, D Kalligas, G P Singh and C W F Everitt, Gen. Relativ. Gravit. 27, 323

(1995)

N Banerjee and S Sen, Pramana – J. Phys. 49, 609 (1997)

S S Xulu, Int. J. Theor. Phys. 30, 1153 (2000)

M Saltı, Nuovo Cimento B120, 53 (2005)

[46] G Gamow, Nature (London) 158, 549 (1946)

[47] K G¨ odel, Rev. Mod. Phys. 21, 447 (1949)

[48] O Gron and H H Soleng, Acta Phys. Pol. B20, 557 (1989)

748

Pramana – J. Phys., Vol. 68, No. 5, May 2007

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