Energy-Momentum Problem of Bell-Szekeres Metric in General Relativity and Teleparallel Gravity

0.15 MB·English

by  M. Sharif

Energy-Momentum Problem of
Bell-Szekeres Metric in General
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0 Relativity and Teleparallel Gravity
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2

n
a M. Sharif and Kanwal Nazir
∗
J
Department of Mathematics, University of the Punjab,
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1 Quaid-e-Azam Campus, Lahore-54590, Pakistan.

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Abstract
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Thispaperisdevotedtotheinvestigationoftheenergy-momentum
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v problem in two theories, i.e., General Relativity and teleparallel grav-
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ity. WeuseEinstein,Landau-Lifshitz,Bergmann-ThomsonandM¨oller’s
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prescriptionstoevaluateenergy-momentumdistributionofBell-Szekeres
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1 metric in both the theories. It is shown that these prescriptions give
.
1 thesame energy-momentum density components in both General Rel-
0 ativity and teleparallel theory. M¨oller’s prescription yields constant
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energy in both the theories.
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:
v
PACS: 04.20.-q, 04.20.Cv
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X
Key Words: Energy-Momentum, Bell-Szekeres Metric
r
a
1 Introduction
One of the most interesting problems which remains unsolved, since the birth
of General Theory of Relativity (GR), is the energy-momentum localization.
To find a generally accepted expression, there have been different attempts.
However, someattemptstodefinetheenergy-momentumdensityforthegrav-
itationalfield lead to prescriptions that arenot truetensors. The first of such
∗[email protected]
1
attemptswasmadebyEinstein[1]himselfwhoproposedanexpressionforthe
energy-momentum distribution of the gravitational field. Following Einstein,
many scientists like Landau-Lifshitz [2], Papapetrou [3], Bergmann-Thomson
[4] and M¨oller [5] introduced their own energy-momentum complexes. All
these prescriptions, except M¨oller, are restricted to do calculations only in
Cartesian coordinates.
The notion of energy-momentum complexes was severely criticized for a
number of reasons. Firstly, the nature of a symmetric and locally conserved
object is non-tensorial and thus its physical interpretation appeared obscure
[6]. Secondly, different energy-momentum complexes could yield different
energy-momentum distributions for the same gravitational background [7,8].
Finally, energy-momentum complexes were local objects while it was usually
believed that the suitable energy-momentum of the gravitational field cannot
belocalized[9]. Analternateconcept ofenergy, calledquasi-localenergy, was
developed by Penrose and many others [10,11]. Although, these quasi-local
masses are conceptually very important, yet these definitions have serious
problems. Chang et al. [12] showed that energy-momentum complexes are
actually quasi-local and legitimate expressions for the energy-momentum.
Virbhadra [13-15] and his collaborator [16] showed that different energy-
momentum complexes yield the same results for a general non-static spher-
ically symmetric metric of the Kerr-Schild class. These definitions comply
withthequasi-localmassdefinitionofPenrose forageneralnon-staticspheri-
cally symmetric metric of the Kerr-Schild class. However, these prescriptions
disagreeinthecase ofthemost generalnon-staticspherically symmetric met-
ric. Aguirregabiria et al. [17] proved the consistency of the results using dif-
ferent energy-momentum complexes for any Kerr-Schild class metric. Xulu
[18,19] extended this investigation and found the same energy distribution
in the Melvin magnetic and Bianchi type I universes. Chamorro and Virb-
hadra [20] and Xulu [21] studied the energy distribution of the charged black
holes with a dilaton field. Ramdinschi and Yang [22], Vagenas [23], Gad
[24] and Xulu [25] investigated the energy distribution of the string black
holes using different prescriptions. Using Einstein and Landau-Lifshitz com-
plexes, Cooperstock [26] and Rosen [27] found that total energy of the closed
Friedmann-Robertson-Walker (FRW) spacetime vanishes everywhere. With
the help of Einstein energy-momentum complex, Banerjee and Sen [28]found
that the total energy density of Bianchi type-I universe is zero everywhere.
However, there exist examples [29-32] which do not support this viewpoint.
It has been argued [33,34] that the energy-momentum problem can also
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be localized in the framework of the teleparallel theory (TPT) of gravity.
This theory has been considered long time ago in connection with attempts
to define the energy of the gravitational field. M¨oller [35] was probably the
first to notice that the tetrad description of the gravitational field allows a
moresatisfactorytreatmentofthegravitationalenergy-momentum thandoes
GR.UsingtheteleparallelversionofEinsteinandLandau-Lifshitzcomplexes,
Vargas [36] found that total energy of the closed FRW spacetimes vanishes
everywhere. This result agrees with the previous work of Cooperstock [26]
and Rosen [27]. It has been shown [37] that the results of Bianchi types
I and II in TPT are consistent with the results in GR. Recently, Salti et
al. [38,39] have calculated energy-momentum densities for some particular
spacetimes by using different prescriptions both in GR and TPT and found
the same results. However, there exist some spacetimes that do not provide
consistent results both in GR and TPT. Sharif and Jamil [40] considered
Lewis-Papapetrou metric and found that the results in TPT do not coincide
with those obtained in GR [41].
ThispaperexploresBell-Szekeresmetricbyevaluatingitsenergy-momentum
distributionusing different prescriptions bothinGRandTPT. Wewouldlike
to present results rather giving the details as these are available in the liter-
ature [40,41]. The paper has been organized as follows: Section 2 is devoted
to present the prescriptions of the Einstein, Landau-Lifshitz, Bergmann-
Thomson and M¨oller energy-momentum both in GR and TPT. In section
3, we find the energy-momentum distribution of Bell-Szekeres metric both in
GR and TPT using these prescriptions. Finally, section 4 provides a sum-
mary and discussion of the results obtained.
2 Energy-Momentum Complexes
Inthissection,weshallbrieflyoutlinetheEinstein, Landau-Lifshitz,Bergmann-
Thomson and M¨oller prescriptions used to calculate energy-momentum dis-
tribution of a metric both in GR and TPT.
2.1 Energy-Momentum Complexes in GR
ForEinsteinprescription, theenergy-momentumdensityisgivenintheform
[5,42]
1
Θb = Hbc , (a,b,c = 0,1,2,3), (1)
a 16π a ,c
3
where Hbc= -Hcb is given by
a a
g
Hbc = ad [ g(gbdgce gcdgbe)] . (2)
a √ g − − ,e
−
Here g is the determinant of the metric tensor g and comma denotes ordi-
µν
nary differentiation. Notice that Θ0 is the energy density, Θ0 (i = 1,2,3) are
0 i
the momentum density components and Θi are the energy current density
0
components. The momentum four-vector is defined as
p = Θ0dx1dx2dx3, (3)
a Z Z Z a
V
where p gives the energy and p , p and p are the momentum components
0 1 2 3
while the integration is taken over the hypersurface element described by
t = constant. Einstein’s conservation law becomes
∂Θb
a = 0. (4)
∂xa
Landau-Lifshitz energy-momentum density can be given in the form [2]
1
Lab = ℓabcd , (5)
,cd
16π
where
ℓabcd = ( g)(gabgcd gacgbd). (6)
− −
The quantities L00 and L0i represent the energy density and the total mo-
mentum (energy current) densities respectively.
The energy-momentum prescription of Bergmann-Thomson is given
by [4]
1
Bab = Mabc , (7)
,c
16π
where
Mabc = gadVbc, (8)
d
with
g
Vbc = de [ g(gbegcf gcegbf)] . (9)
d √ g − − ,f
−
The quantities B00 and Bi0 represent energy and momentum densities re-
spectively. The Bergmann-Thomson energy-momentum satisfies the follow-
ing local conservation law
∂Bab
= 0 (10)
∂xb
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in any coordinate system. The energy-momentum components are given by
pa = Ba0dx1dx2dx3. (11)
Z Z Z
V
Using Gauss theorem, the above integral takes the form
1
p = H0bn dS, (12)
a 16π Z Z a b
wheren istheoutwardunitnormalvectortoaninfinitesimalsurfaceelement
b
dS. The quantities p give momentum components while p gives the energy.
i 0
All the energy-momentum complexes mentioned above are coordinate de-
pendent and give meaningful results only when the calculations are carried
out in Cartesian coordinates. To overcome this deficiency, M¨oller [5] in-
troduced another energy-momentum pseudo-tensor Mb which is coordinate
a
independent given as
1
Mb = Kbc , (13)
a 8π a ,c
where
Kbc = √ g(g g )gbegcd, (14)
a − ad,e − ae,d
and Kbc is antisymmetric in its upper indices. This satisfies the conservation
a
law
∂Mb
a = 0, (15)
∂xb
where M0 is the energy density, M0 are momentum density components and
0 i
Mi are the components of energy current density.
0
2.2 Energy-Momentum Complexes in TPT
TheteleparallelversionofEinstein, Bergmann-ThomsanandLandau-Lifshitz
energy-momentum complexes are given [36], respectively, as
1
hEµ = U µν , (16)
ρ 4π ρ ,ν
1
hBµρ = [gµθU ρν ], (17)
4π θ ,ν
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1
hLµρ = [hgµθU ρν ], (18)
4π θ ,ν
where h = det(ha ) and U µν is the Freud’s superpotential given as
µ ρ
U µν = hS µν. (19)
ρ ρ
Here Sρµν is the tensor
m m
Sρµν = m Tρµν + 2(Tµρν Tνρµ)+ 3(gρνTθµ gρµTθν] (20)
1 2 − 2 θ − θ
with m , m and m as the three dimensionless coupling constants of the
1 2 3
teleparallel gravity. For the teleparallel equivalent of GR, the specific choice
of these three constants are
1 1
m = , m = , m = 1. (21)
1 2 3
4 2 −
To calculate this tensor, we evaluate Weitzenb¨ock connection [43]
Γθ = h θ∂ ha (22)
µν a ν µ
which is used to find the corresponding torsion [44]
Tθ = Γθ Γθ . (23)
µν νµ µν
−
Thusthemomentumfour-vectorforEinstein,Bergmann-ThomsanandLandau-
Lifshitz energy-momentum complexes will be
pE = hE0dxdydz, (24)
µ Z µ
Σ
pB = hB0dxdydz, (25)
µ Z µ
Σ
pL = hL0dxdydz. (26)
µ Z µ
Σ
Nowwediscuss M¨ollerenergy-momentum complexinthecontext ofTPT.
Mikhail et al. [33] defined the superpotential (which is antisymmetric in its
last two indices) of the M¨oller tetrad theory as
√ g
Uνβ = − Pτνβ[φρgσχg λg Kχρσ g (1 2λ)Kσρχ], (27)
µ 2κ χρσ µτ − τµ − τµ −
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where
Pτνβ = δτgνβ +δτgνβ δτgνβ, (28)
χρσ χ ρσ ρ σχ − σ χρ
and gνβ is a tensor quantity defined by
ρσ
gνβ = δνδβ δνδβ. (29)
ρσ ρ σ − σ ρ
Here Kσρχ is a contortion tensor, λ is a free dimensionless coupling constant
of TPT, κ is the coupling constant and φ is the basis vector field given by
µ
φ = Tν . (30)
µ νµ
The energy-momentum density is defined as
Mν = U νρ . (31)
µ µ ,ρ
The energy E contained in a sphere of radius R is expressed by the volume
integral as
p (R) = U0ρ dx3, (32)
µ Z Z Z µ ,ρ
r=R
p (R) = M0dx3. (33)
µ Z Z Z µ
r=R
3 Bell-Szekeres Metric
It is well-known that exact plane gravitational waves are simple time de-
pendent plane symmetric solutions of the Einstein field equations [45]. The
colliding planewave spacetimes have beeninvestigated extensively inGR[46]
due to their interesting behaviour. The first exact solution of the Einstein-
Maxwell equations representing colliding plane shock electromagnetic waves
with co-linear polarizations was obtained by Bell and Szekeres [47]. This
solution is conformally flat in the interaction region and is represented by
the metric
ds2 = 2dudv+e−U(eVdx2 +e−Vdy2), (34)
where the metric functions U and V depend on the null coordinates u and
v. The complete solution of the Einstein-Maxwell equations is
U = log(f(u)+g(u)),
−
V = log(rw pq) log(rw+pq), (35)
− −
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where
1 1 1 1
r = ( +f)2, p = ( f)2,
2 2 −
1 1 1 1
w = ( +g)2, q = ( g)2 (36)
2 2 −
with
1 1
f = sin2P, g = sin2Q. (37)
2 − 2 −
Here P = auθ(u), Q = bvθ(v), where θ is the Heaviside unit step function,
a and b are arbitrary constants.
The Cartesian form of the metric is found by substituting t = u+v and
z = v u
−
1 1 t z 1 t+z
ds2 = dt2 cos2 b(t z)θ( − )+ a(t+z)θ( ) dx2
2 − {2 − 2 2 2 }
1 t z 1 t+z 1
cos2 b(z t)θ( − )+ a(t+z)θ( ) dy2 dz2.
− {2 − 2 2 2 } − 2
(38)
3.1 Energy-Momentum Distribution in GR
In this section, we find the energy-momentum distribution of Bell-Szekeres
metric using Einstein, Landau-Lifshitz, Bergmann-Thomson and M¨oller’s
prescriptions in GR.
Using Einstein prescription, the components of the energy-momentum
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density of Bell-Szekeres metric turn out to be
1 t z t z t z
Θ00 = [cos b(t z)θ( − ) 2bθ( − )+b(t z)θ′( − ) 2
32π { − 2 }{ 2 − 2 }
t z t z t z
′ ′′
+ bsin b(t z)θ( − ) 4θ ( − )+(t z)θ ( − )
{ − 2 }{ 2 − 2 }
t+z t+z
+ a acos a(t+z)θ( ) 2θ( )
{ { 2 }{ 2
t z t+z t+z
+ (t+z)θ′( − ) 2 +sin a(t+z)θ( ) 4θ′( )
2 } { 2 }{ 2
t+z
′′
+ (t+z)θ ( ) ],
2 }}
Θ10 = Θ20 = 0,
1 t z t z t z
Θ30 = [cos b(t z)θ( − ) 2bθ( − )+b(t z)θ′( − ) 2
32π { − 2 }{ 2 − 2 }
t z t z t z
′ ′′
+ bsin b(t z)θ( − ) 4θ ( − )+(t z)θ ( − )
{ − 2 }{ 2 − 2 }
t+z t+z
+ a acos a(t+z)θ( ) 2θ( )
{− { 2 }{ 2
t z t+z t+z
+ (t+z) θ′( − ) 2 sin a(t+z)θ( ) 4θ′( )
{ 2 } − { 2 }{ 2
t+z
′′
+ (t+z)θ ( ) ]. (39)
2 }}}
When we use Landau-Lifshitz complex, we obtain the following energy-
momentum density components
1 t z t z t z
L00 = [2cos 2b(t z)θ( − ) 2bθ( − )+b(t z)θ′( − ) 2
256π { − 2 }{ 2 − 2 }
t z t z t z
′ ′′
+ bsin 2b(t z)θ( − ) 4θ( − )+(t z)θ ( − )
{ − 2 }{ 2 − 2 }
t z t+z t z
+ 2bsin b(t z)θ( − ) 2asin(a(t+z)θ( ))(2θ( − )
{ − 2 }{ 2 2
t z t z t z
′ ′
+ (t z)θ ( − ))(2θ( − )+(t+z)θ ( − ))
− 2 2 2
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t+z t z t z
′ ′′
+ cos(a(t+z)θ( ))(4θ ( − )+(t z)θ ( − ))
2 2 − 2 }
t z t+z t+z
+ 2cos b(t z)θ( − ) cos a(t+z)θ( ) 4a2θ2( )
{ − 2 }{ { 2 { 2
t z t z t+z t+z
+ 2bθ( − )+b(t z)θ′( − ) 2 +4a2(t+z)θ( )θ′( )
{ 2 − 2 } 2 2
t+z t+z t+z
+ a2(t+z)2θ′( )2 +asin a(t+z)θ( ) 4θ′( )
2 } { 2 }{ 2
t+z t+z t+z
′′
+ (t+z)θ ( ) +a 2acos 2a(t+z)θ( ) 2θ( )
2 }}} { { 2 }{ 2
t+z t+z t+z
+ (t+z)θ′( ) 2 +sin 2a(t+z)θ( ) 4θ′( )
2 } { 2 }{ 2
t+z
′′
+ (t+z)θ ( ) ],
2 }}
L10 = L20 = 0,
1 t z t z t z
L30 = [2cos 2b(t z)θ( − ) 2bθ( − )+b(t z)θ′( − ) 2
256π { − 2 }{ 2 − 2 }
t+z t z t z
′
+ 2bcos a(t+z)θ( ) sin b(t z)θ( − )) 4θ( − )
{ 2 } { − 2 }{ 2
t z t z t z
′′ ′
+ (t z)θ ( − ) +bsin 2b(t z)θ( − ) 4θ ( − )
− 2 } { − 2 }{ 2
t z t z
′′
+ (t z)θ ( − ) +2cos b(t z)θ( − ) cos a(t
− 2 } { − 2 } {
t+z t z t+z t z
′
+ z)θ( ) 2bθ( − ) 2aθ( )+b(t z)θ ( − )
2 }{ 2 − 2 − 2
t+z t z t+z t z
′ ′
a(t+z)θ ( ) 2bθ( − )+2aθ( )+b(t z)θ ( − )
− 2 }{ 2 2 − 2
t+z t+z t+z
′ ′
+ a(t+z)θ ( ) asin a(t+z)θ( ) 4θ ( )
2 }− { 2 }{ 2
t+z t+z t+z
′′
+ (t+z)θ ( ) +a 2acos 2a(t+z)θ( ) 2θ( )
2 } {− { 2 }{ 2
t+z t+z t+z
+ (t+z)θ′( ) 2 sin 2a(t+z)θ( ) 4θ′( )
2 } − { 2 }{ 2
t+z
′′
+ (t+z)θ ( ) ]. (40)
2 }}
Theenergy-momentumdistributioninBergmann-Thomson’s prescrip-
tion will become
1 t z t z t z
B00 = [cos b(t z)θ( − ) 2bθ( − )+b(t z)θ′( − ) 2
32π { − 2 }{ 2 − 2 }
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