Energy and Momentum Distributions of Kantowski and Sachs Space-time
arXiv:0704.1567v2  [gr-qc]  15 Apr 2007
Energy and Momentum Distributions of
Kantowski and Sachs Space-time
Ragab M. Gad1 and A. Fouad
Mathematics Department, Faculty of Science,
Minia University, 61915 El-Minia, EGYPT.
Abstract
We use the Einstein, Bergmann-Thomson, Landau-Lifshitz and Pa-
papetrou energy-momentum complexes to calculate the energy and mo-
mentum distributions of Kantowski and Sachs space-time. We show
that the Einstein and Bergmann-Thomson definitions furnish a consis-
tent result for the energy distribution, but the definition of Landau-
Lifshitz do not agree with them. We show that a signature switch
should affect about everything including energy distribution in the
case of Einstein and Papapetrou prescriptions but not in Bergmann-
Thomson and Landau-Lifshitz prescriptions.
1 Introduction
One of the most interesting and intricate problems still unsolved since the
outset of general relativity is the energy-momentum localization. Einstein
himself proposed the first energy-momentum complex in an attempt to de-
fine the local distribution of energy and momentum [1]. After this attempt, a
plethora of different energy-momentum complexes were proposed, including
formulations by Tolman [2], Papapetrou [3], Møller [4], Landau and Lifshitz
[5], Weinberg [6], Bergmann-Thomson [7] and others. This approach was
abandoned for a long time due to severe criticism for a number of reasons.
Recently, Virbhadra re-opened the subject of energy-momentum com-
plexes [8]. He pointed out that though these complexes are non-tensors,
they yield reasonable and consistent results for a given space-time. Aguir-
regabiria et al [9] found that for any metric of the Kerr-Schild class, sev-
eral different definitions of the energy-momentum complex yield precisely
the same results.Virbhadra [10] investigated whether or not these energy
momentum complexes lead to the same results for the most general non-
static spherically symmetric metric and found that they disagree. He noted
that the energy-momentum complexes of Landau and Lifshitz, Papapetrou
1Email Address: ragab2gad@hotmail.com
http://arxiv.org/abs/0704.1567v2
and Weinberg give the same results as in the Einstein definition if the cal-
culations are performed in Kerr-Schild Cartesian coordinates. However,
these energy-momentum complexes disagree if computations are done in
”Schwarzschild Cartesian” coordinates2. In a detail study of the question,
Xulu [11] has confirmed this suggestion. He obtained the energy distribu-
tion for the most general non-static spherically symmetric using Møller’s
definition and found different results in general from those obtained us-
ing Einstein’s definition. These results agree for the Schwarzschild, Vaidya
and Janis-Newmann-Winicour space-times, but disagree for the Reissner-
Nordström space-time. Many authors had similarly successfully applied the
aforementioned energy-momentum complexes to various black hole configu-
rations [12].
It has been remained a controversial problem whether or not energy and
momentum are localizable. There are different opinions on this subject.
Contradicting the viewpoint of Misner et al. [13] that the energy is local-
izable only for spherical systems, Cooperstock and Sarracino [14] argued
that if the energy localization is meaningful for spherical systems then it
is meaningful for all systems. Bondi [15] expressed that a non-localizable
form of energy is inadmissible in relativity and its location can in prin-
ciple be found. These contradictory viewpoints bear significantly on the
study of gravitational waves. It is an interesting question whether or not
gravitational waves have energy and momentum content. In a series of pa-
pers, Cooperstock [16] hypothesized that in a curved space-time energy and
momentum are confined to the region of non-vanishing energy-momentum
tensor T ab and consequently the gravitational waves are not carriers of energy
and momentum in vacuum space-times. This hypothesis has neither been
proved nor disproved. There are many results supporting this hypothesis
(see for example, [17, 18]).
In this paper we evaluate the energy and momentum distributions of
the Kantowski and Sachs space-time, using Einstein, Bergmann-Thomson,
Landau-Lifshitz and Papapetrou energy-momentum complexes.
Through this paper we use G = 1 and c = 1 units and follow the conven-
tion that Latin indices take value from 0 to 3 and Greek indices take value
from 1 to 3.
2Schwarzschild metric in “Schwarzschild Cartesian coordinates” is obtained by trans-
forming this metric (in usual Schwarzschild coordinates {t, r, θ, φ}) to {t, x, y, z} using
x = r sin θ cosφ, y = r sin θ sinφ, z = r cos θ.
2 Kantowski and Sachs Space-time
The standard representation of Kantowski and Sachs space-times are given
by [19]
dS2 = dt2 −A2(t)dr2 −B2(t)(d2θ + sin2 θd2φ), (2.1)
where the functions A(t) and B(t) are to be determined from the field equa-
tions.
The solutions of the Einstein field equations for the above metric were con-
sidered with dust source [19], but they were generalized (in fact earlier) to
the general perfect fluid source [21].
From the geometrically point of view, this line element admits a four
parameter continuous group of isometries which acts on space-like hypersur-
face, and has no three parameter subgroup that would be simply transitive
on the orbits (for more detailed description see Kantowski and Sachs [19]
and Collins [20]).
From the physical point of view, the metric (2.1) automatically defines an
energy-momentum tensor of a fluid with anisotropic pressure, and the coor-
dinates of (2.1) are comoving. The rotation and acceleration are zero, but
if the source is to be a perfect fluid, then the shear is necessarily non-zero.
It is well known that if the calculations are performed in quasi-Cartesian
coordinates, all the energy-momentum complexes give meaningful results.
According to the following transformations
x2 + y2 + z2, φ = arctan(
the line element (2.1) written in terms of quasi-Cartesian coordinates reads:
dS2 = dt2+
(dx2+ dy2+ dz2)−
(xdx+ ydy+ zdz)
. (2.2)
For the above metric the determinant of the metric tensor and the con-
travariant components of the tensor are given, respectively, as follows
det(g) = −A
g00 = 1,
g11 = x2
g12 = xy
g13 = xz
g22 = y2
g23 = yz
g33 = z2
(2.3)
3 Einstein’s Energy-momentum Complex
The energy-momentum complex as defined by Einstein [1] is given by
θki =
Hkli,l, (3.4)
where the Einstein’s superpotential Hkli is of the form
Hkli = −H
[− g(gknglm − glngkm)],m. (3.5)
and θ0α are the energy and momentum density components, respectively.
The Einstein energy-momentum satisfies the local conservation law
The energy and momentum in the Einstein’s prescription are given by
∫ ∫ ∫
θ0i dx
1dx2dx3. (3.6)
Using the Gauss theorem we obtain
H0αi nαds, (3.7)
where nα = (
) are the components of a normal vector over an in-
finitesimal surface element ds = r2 sin θdθdφ.
The required non zero components of Hkli for the line element (2.1) are
given by
A2r2 +B2
A2r2 +B2
A2r2 +B2
(ȦB −AḂ)− (ȦB +AḂ)
= H02
= 2xyB
(ȦB −AḂ),
= H03
= 2xzB
(ȦB −AḂ),
(ȦB −AḂ)− (ȦB +AḂ)
= H03
= 2yzB
(ȦB −AḂ),
(ȦB −AḂ)− (ȦB +AḂ)
(3.8)
Using the components (3.8) we obtain the components of energy and mo-
mentum densities in the form
8πAr4
(A2r2 −B2),
(ȦB +AḂ),
(ȦB +AḂ),
(ȦB +AḂ).
(3.9)
Using equations (3.8) in equation (3.7), the energy and momentum distri-
butions are the following
EEin = P0 =
(A2r2 +B2),
P1 = P2 = P3 = 0.
We notice that if the signature of the space-time under study is changed
from +2 to -2, we find that the values of energy and momentum densities
as well as the energy distribution are changed from positive to negative.
4 The Energy-Momentum Complex of Bergmann-
Thomson
The Bergmann-Thomson energy-momentum complex [7] is given by
[gilBkml ],m, (4.1)
where
Bkml =
gkngmp − gmngkp
B00 and B0α are the energy and momentum density components.
The energy and momentum are given by
P i =
∫ ∫ ∫
Bi0dx1dx2dx3. (4.2)
Using the Gauss theorem we have
P i =
Bi0αnαdS. (4.3)
In order to calculate the energy and momentum distributions for the
space-time under consideration, using Bergmann-Thomson energy-momentum
complex, we require the following non-vanishing components of Bkml
A2r2 +B2
A2r2 +B2
A2r2 +B2
(ȦB −AḂ)− (ȦB +AḂ)
= B02
(ȦB −AḂ),
= B03
= 2xzB
(ȦB −AḂ),
(ȦB −AḂ)− (ȦB +AḂ)
= B03
= 2yzB
(ȦB −AḂ),
(ȦB −AḂ)− (ȦB +AḂ)
(4.4)
Using the components (4.4) in (4.1), the components of energy and mo-
mentum densities are as follows
00 = 1
8πAr4
(A2r2 −B2),
01 = − x
Ȧr2 + B
(2AḂ − ȦB
02 = − y
Ȧr2 + B
(2AḂ − ȦB
03 = − z
Ȧr2 + B
(2AḂ − ȦB
(4.5)
Using equations (4.4) in equation (4.3), we obtain the energy and momentum
distributions in the following form
EBerg = P
(A2r2 +B2),
P1 = P2 = P3 = 0.
The above energy density and energy distribution are agreement with that
obtained before, using Einstein’s energy-momentum complex. In the case of
Bergmann’s energy-momentum complex, we notice that a signature switch
do not affect about everything including energy distribution. Consequently,
the Einstein and Bergmann-Thomson prescription do not give the same
results when the signature of the space-time under study is -2.
5 Landau-Lifshitz’s Energy-momentum Complex
Landau-Lifshitz’s energy-momentum complex [5] is given by
Lij =
, (5.1)
where
Sikjl = −g(gijgkl − gilgkj). (5.2)
Lij is symmetric in its indices, L00 is the energy density and L0α are the
momentum (energy current) density components. Sikjl has the symmetries
of the Riemann curvature tensor.
The energy and momentum are given by
P i =
∫ ∫ ∫
Li0dx1dx2dx3. (5.3)
Using the Gauss theorem we have
P i =
Si0αnαdS. (5.4)
The required non-vanishing components of Sikjl are
S0101 = B2
(A2r2 −B2)− A
S0102 =
B2(A2r2 −B2),
S0103 = xz
B2(A2r2 −B2),
S0202 = B2
(A2r2 −B2)− A
S0203 =
B2(A2r2 −B2),
S0303 = B2
(A2r2 −B2)− A
(5.5)
Using these components in equation (5.1), we obtained the energy and mo-
mentum densities are
L00 = − B
[A2r2 + 3B2],
L10 = − xB
[A(AḂ + ȦB)r2 + 2B2Ḃ],
L20 = − yB
[A(AḂ + ȦB)r2 + 2B2Ḃ],
L30 = − zB
[A(AḂ + ȦB)r2 + 2B2Ḃ].
(5.6)
Using equations (5.1) in equation (5.4), we obtain the energy and momentum
distributions
ELL =
(A2r2 +B2). (5.7)
P 1LL = P
LL = P
LL = 0. (5.8)
The above results do not agree with the results obtained before, using Ein-
stein and Bergmann-Thomson energy-momentum complexes. A signature
switch does not affect about everything including energy distribution.
6 Papapetrou’s Energy-momentum Complex
The symmetric energy-momentum complex of Papapetrou [3] is given by
Ωij =
, (6.1)
where
Υijkl =
−g(gijηkl − gikηjl + gklηij − gjlηik), (6.2)
and ηik is the Minkowski metric with signature +2.
Ω00 and Ωα0 are the energy and momentum density components.
The energy and momentum, using the Papapetrou prescription are given by
P i =
∫ ∫ ∫
Ωi0dx1dx2dx3. (6.3)
Using the Gauss theorem we obtain
P i =
Υi0αl,l nαdS. (6.4)
The non-vanishing components of Υijkl are as follows
Υ0011 = 1
(A2r2 −B2)−A2
Υ0012 = xy
(A2r2 −B2),
Υ0013 = xz
(A2r2 −B2),
Υ0022 = 1
(A2r2 −B2)−A2
Υ0023 =
(A2r2 −B2),
Υ0033 = 1
(A2r2 −B2)−A2
(6.5)
Using these components in (6.1), we get the following energy and momentum
density components
Ω00 = A
[r2 −B2],
Ω10 = − x
[r2Ȧ+B(ȦB + 2AḂ)],
Ω20 = − y
[r2Ȧ+B(ȦB + 2AḂ)],
Ω30 = − z
[r2Ȧ+B(ȦB + 2AḂ)].
(6.6)
Using equations (6.5) in equation (6.4), we obtain the energy and momentum
distributions
(r2 +B2) (6.7)
P 1 = P 2 = P 3 = 0 (6.8)
The above results do not agree with the results obtained before, using Ein-
stein, Bergmann-Thomson and Landau and Lifshitz energy-momentum com-
plexes. A signature switch should affect about everything including energy
distribution.
Discussion
We investigated the energy and momentum (due to matter plus fields in-
cluding gravity) distribution of the Kantowski and Sachs space-time using
the Einstein, Bergmann-Thomson, Landau-Lifshitz and Papapetrou energy-
momentum complexes. We found that the quantities of energy and mo-
mentum densities as well as energy distribution are well-defined and well-
behaved. For the space-time under consideration, we found that the energy-
momentum complexes of Einstein and Bergmann-Thomson give the same
results, while Landau-Lifshitz and Papapetrou do not give the same results
and not agree with the aforementioned complexes. We have shown that a
signature switch affects about every thing (by changing the sign of the values
of energy and momentum densities as well as energy distribution) including
energy distribution. These changes occur in the case of Einstein and Pa-
papetrou prescriptions but not in Bergmann-Thomson and Landau-Lifshitz
prescriptions.
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