Relative Energy Associated with a White Hole Model of the Big Bang
ABSTRACT This paper has been removed by arXiv administrators because it plagiarizes gr-qc/9803014, "A White Hole Model of the Big Bang," by Philip Gibbs.
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ABSTRACT: Energy-momentum is an important conserved quantity whose definition has been a focus of many investigations in general relativity. Unfortunately, there is still no generally accepted definition of energy and momentum in general relativity. Attempts aimed at finding a quantity for describing distribution of energy-momentum due to matter, non-gravitational and gravitational fields resulted in various energy-momentum complexes whose physical meaning have been questioned. The problems associated with energy-momentum complexes resulted in some researchers even abandoning the concept of energy-momentum localization in favour of the alternative concept of quasi-localization. However, quasi-local masses have their inadequacies, while the remarkable work of Virbhadra and some others, and recent results of Cooperstock and Chang {\it et al.} have revived an interest in various energy-momentum complexes. Hence in this work we use energy-momentum complexes to obtain the energy distributions in various space-times. We elaborate on the problem of energy localization in general relativity and use energy-momentum prescriptions of Einstein, Landau and Lifshitz, Papapetrou, Weinberg, and M{\o}ller to investigate energy distributions in various space-times. It is shown that several of these energy-momentum complexes give the same and acceptable results for a given space-time. This shows the importance of these energy-momentum complexes. Our results agree with Virbhadra's conclusion that the Einstein's energy-momentum complex is still the best tool for obtaining energy distribution in a given space-time. The Cooperstock hypothesis for energy localization in GR is also supported.09/2003; - SourceAvailable from: Gamal G. L. Nashed[Show abstract] [Hide abstract]
ABSTRACT: I find the most general spherically symmetric nonsingular black hole solution in a special class of teleparallel theory of gravitation. If r is large enough, the general solution coincides with the Schwarzschild solution. Whereas, if r is small, the general solution behaves in a manner similar to that of a de Sitter solution. Otherwise it describes a spherically symmetric black hole singularity free everywhere. Moreover, the energy associated with the general solution is calculated using the superpotential given by Møller.Physical review D: Particles and fields 09/2002; 66(6). - SourceAvailable from: ArXiv[Show abstract] [Hide abstract]
ABSTRACT: In this paper, we observe that the brane functional studied in hep-th/9910245 can be used to generalize some of the works that Schoen and I [4] did many years ago. The key idea is that if a three dimensional manifold M has a boundary with strongly positive mean curvature, the effect of this mean curvature can influence the internal geometry of M. For example, if the scalar curvature of M is greater than certain constant related to this boundary effect, no incompressible surface of higher genus can exist.Advances in Theoretical and Mathematical Physics 10/2001; · 1.07 Impact Factor
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arXiv:gr-qc/0607116v2 15 Dec 2006
ENERGY ASSOCIATED WITH A WHITE HOLE MODEL OF THE BIG BANG
Figen Binbay∗and Irfan Acikgoz†
Department of Physics, Faculty of Art and Science, Dicle University, 21280, Diyarbakir-Turkey
Mustafa Saltı‡§
Department of Physics, Faculty of Art and Science,
Middle East Technical University, 06531, Ankara-Turkey
A specific Lemaˆ ıtre-Tolman model of a spherically symmetric non-rotating white hole model with
a few adjustable parameters is investigated to calculate the momentum four-vector distribution
(due to matter plus fields including gravity) in the teleparallel gravity. The energy-momentum
distributions (due to matter and fields including gravity) of a model are found to be zero. The
result that the total energy and momentum components of a white hole model of the big bang
are zero supports the viewpoints of Albrow and Tryon. It is also independent of the teleparallel
dimensionless coupling constant, which means that it is valid in any teleparallel model. The results
we obtained support the viewpoint of Lessner that the Møller energy-momentum formulation is
powerful concept to calculate energy and momentum distributions associated with the universe,
and sustains the importance of the energy-momentum definitions in the evaluation of the energy-
momentum distribution of a given space-time. Also the results obtained here are also agree with
ones calculated in literature by Cooperstock-Israelit, Rosen, Johri et al., Banerjee-Sen, Vargas and
Saltı et al..
PACS numbers: 04.20.-q; 04.20.Jb; 04.50.+h.
Keywords: Energy; Lemaˆ ıtre-Tolman; White hole; teleparallel gravity.
I. ON THE ENERGY-MOMENTUM
PRESCRIPTIONS
It is usually accepted that in the context of general
relativity, despite the existence of some controversial
points related the formulation of the equivalence principle
[1], no tensorial expression for the gravitational energy-
momentum distribution can exist.
The problem of calculating energy, momentum and
angular momentum distributions has been a subject of
many research activities dating back to the very onset
of the theory of general relativity but it still remains an
open question. The numerous attempts aimed at finding
a more suitable quantity for describing distribution of en-
ergy, momentum and/or angular momentum due to mat-
ter, non-gravitational and gravitational fields resulted
in more energy-momentum prescriptions, notably those
proposed by Einstein, Landau-Lifshitz, Møller, Wein-
berg, Papapetrou, Bergmann-Thomson, Tolman and
Qadir-Sharif [2, 3, 4, 5, 6, 7, 8, 9]. The physical meaning
of these non-tensorial (under general coordinate trans-
formations) complexes have been questioned by some re-
searchers [10, 11]. Except for the Møller definition these
formulations only give meaningful results if the calcu-
lations are performed in Cartesian coordinates. Møller
proposed a new expression for energy-momentum com-
plex which could be utilized to any coordinate system.
‡Corresponding author.
∗Electronic address: figenb@dicle.edu.tr
†Electronic address: iacikgoz@dicle.edu.tr
§Electronic address: musts6@yahoo.com
Next, Lessner [12] argued that the Møller prescription is
a powerful concept for energy-momentum in general rela-
tivity. This approach was abandoned for a long time due
to severe criticism for a number of reasons [10, 13]. Virb-
hadra and collaborators revived the interest in this ap-
proach [14] and since then numerous works on evaluating
the energy and momentum distributions of several grav-
itational backgrounds have been completed [15]. Later
attempts to deal with this problematic issue were made
by proposers of quasi-local approach. The determination
as well as the computation of the quasi-local energy and
quasi-local angular momentum of a (2+1)-dimensional
gravitational background were first presented by Brown,
Creighton and Mann [16]. A large number of attempts
since then have been performed to give new definitions
of quasi-local energy in Einstein’s theory of general rel-
ativity [17]. Furthermore, according to the Cooperstock
hypothesis [18], the energy is confined to the region of
non-vanishing energy-momentum tensor of matter and
all non-gravitational fields.
The energy-momentum localization problem has also
been considered in teleparallel gravity [19].
showed that a tetrad description of a gravitational field
equation allows a more satisfactory treatment of the
energy-momentum complex than does general relativ-
ity. Therefore, we have also applied the super-potential
method by Mikhail et. al. [20] to calculate the energy
of the central gravitating body. In Gen. Relat. Gravit.
36, 1255 (2004); Vargas, using the definitions of Einstein
and Landau-Lifshitz in teleparallel gravity, found that
the total energy is zero in Friedmann-Robertson-Walker
space-times. There are also several papers on the energy-
momentum problem in teleparallel gravity. The authors
Møller
Page 2
2
obtained the same energy-momentum for different for-
mulations in teleparallel gravity [21, 22, 23, 24].
In his new paper, Vagenas hypothesized [25] that there
is a connection between the coefficients of the expression
for the energy (when the energy-momentum complex of
Einstein is employed) of the form
E(r) =
+∞
?
n=0
α(Einstein)
n
r−n
(1)
and those of the expression for the energy (when the
energy-momentum complex of Møller is employed) of the
form
E(r) =
+∞
?
n=0
α(Møller)
n
r−n.
(2)
The relation that materializes this connection between
the aforementioned coefficients is given by
α(Einstein)
n
=
1
n + 1α(Møller)
n
.
(3)
Considerable efforts have also been performed in con-
structing super-energy tensors [26].
works of Bel [27] and independently of Robinson [28],
many investigations have been carried out in this field
[30].
In the next section, we introduce the Lemaˆ ıtre-Tolman
model. In section 3, we calculate energy-momentum both
in general relativity and the tetrad theory of gravity using
Møller’s formulations. Finally, section 4 is devoted to
conclusions.
Notations and conventions: We use the Greek alphabet
(µ, ν, α ...=0,1,2,3) to denote indices related to space-
time and to represent the vector components, and the
Latin alphabet (i, j, k ...=1,2,3) to denote indices num-
ber the vectors. Throughout this paper, c = h = 1 con-
vention and metric signature (−,+,+,+) will be used.
Motivated by the
II.THE LEMAˆITRE-TOLMAN SPACE-TIME
Now lets give a brief information for the Lemaˆ ıtre-
Tolman model [29]. The model is a spherically symmet-
ric, pressureless dust solution for the gravitational field.
The model will be considered only the case of zero cosmo-
logical constant. The solution of the Einstein equation
in this model co-moving coordinates with the dust gives
the following line-element
ds2= −dt2+[1+2E(r)]−1R2
here dΩ2= dθ2+sin2θdϕ2. E(r) is an arbitrary function
for the total energy per unit mass of the dust shell as r.
The area and circumference of the mass shell at r are
that of a sphere of radius R(t,r) but the actual distance
to the center of symmetry may be more or less that,
,r(t,r)dr2+R2(t,r)dΩ2(4)
depending on whether E(r) is positive or negative. The
radial parameter r can be re-scaled using any monotonic
function without changing the form of the line-element
[29].
R(t,r) must satisfy the equation,
R2
,r(t,r) = 2E(r) +2M(r)
r
(5)
in these equations, M(r) is another arbitrary function
and is interpreted as κ times the total dust mass within
the sphere out to r. This equation is identical to the
energy equation for a test particle traveling radially out
from a sphere of mass
κ. The particle will escape if
E ≥ 0 or collapse back if E < 0. The mater density is
defined by,
M(r)
ρ(t,r) =2
κ
M(r)
R2R,r,κ = 8π
(6)
Next, for an exploding white hole out of which all matter
will eventually escape, E(r) is positive everywhere and
the solution is given by,
R(t,r) =M
2Ecych
?(2E)3/2
M
(t − t0(r))
?
(7)
here cych(x) is the hyperbolic cycloid function defined
by
cych(sinhη − η) = coshη − 1(8)
cych′(x) =
?
1 +
2
cych(x)
(9)
cych(x) =
3
?
9
4x
2
3+3
10
3
?
3
4x
4
3−
27
1400x2+ O(x
8
3) (10)
t0(r) is an arbitrary function describing the location of
the initial singularity, and is known as the bang time.
At large time this solution describes a dust sphere with
shells expanding at velocity [29],
dR
dt
→
?
2E(r).
(11)
For the line element (4), gµνand gµνare defined by
gµν= −δ0
µδ0
ν+ [1 + 2E(r)]−1R2
+R2(t,r)(δ2
,r(t,r)δ1
ν+ sin2θδ3
µδ1
µδ3
ν
µδ2
ν)(12)
gµν= −δµ
0δν
+R−2(t,r)(δµ
0+ [1 + 2E(r)]R−2
,r(t,r)δµ
2+ sin−2θδµ
1δν
3δν
1
2δν
3) (13)
Page 3
3
III. ENERGY-MOMENTUM IN
TELEPARALLEL GRAVITY
In the context of teleparallel gravity, curvature and
torsion are able to provide each one equivalent descrip-
tions of the gravitational interactions. According to gen-
eral relativity, curvature is used to geometrize space-time,
and in this way successfully describe the gravitational in-
teraction. Teleparallelism, on the other hand, attributes
gravitation to torsion, but in this case torsion accounts
to gravitation not by geometrizing the interaction, but
by acting as a force. This means that, in the teleparallel
equivalent of general relativity, there are no geodesics,
but force equations quite analogous to the Lorentz force
equation of the electrodynamics [31]. Thus we can say
that the gravitational interaction can be described al-
ternatively in terms of curvature, as is usually done in
general relativity, or in terms of torsion, in which case
we have the so called teleparallel gravity. Whether grav-
itation requires a curved or a torsioned space-time, there-
fore, turns out to be a matter of convention.
Teleparallel theories of gravity, whose basic entities are
tetrad fields haµ(a and µ are SO(3,1) and space-time in-
dices, respectively) have been considered long time ago
by Møller [32] in connection with attempts to define the
energy of the gravitational field. Teleparallel theories of
gravityare defined on Weitzenb¨ ock space-time [33], which
is endowed with the affine connection Γλ
and where the curvature tensor, constructed out of this
connection, vanishes identically. This connection defines
a space-time with an absolute parallelism or teleparal-
lelism of vector fields [34]. In this geometrical framework
the gravitational effects are due to the space-time torsion
corresponding to the above mentioned connection.
As remarked by Hehl [35], by considering Einstein’s
general relativity as the best available alternative theory
of gravity, its teleparallel equivalent is the next best one.
Therefore it is interesting to perform studies of the space-
time structure as described by the teleparallel gravity.
In teleparallel gravity, the super-potential of Møller is
given by Mikhail et al. [20] as
µν= haλ∂µhaν
Σνβ
µ =(−g)1/2
2κ
Pτνβ
χρσ[Φρgσχgµτ− λgτµξχρσ
−(1 − 2λ)gτµξσρχ] (14)
where
ξαβµ= hiαhi
β;µ
(15)
is the con-torsion tensor and h
defined uniquely by
µ
i
is the tetrad field and
gαβ= hα
ihβ
jηij
(16)
here ηijis the Minkowski space-time. κ is the Einstein
constant and λ is free-dimensionless coupling parameter
of teleparallel gravity. For the teleparallel equivalent of
general relativity, there is a specific choice of this con-
stant.
Φµis the basic vector field given by
Φµ= ξρ
µρ
(17)
and Pτνβ
χρσcan be found by
Pτνβ
χρσ= δτ
χgνβ
ρσ+ δτ
ρgνβ
σχ− δτ
σgνβ
χρ
(18)
with gνβ
ρσbeing a tensor defined by
gνβ
ρσ= δν
ρδβ
σ− δν
σδβ
ρ.
(19)
The energy-momentum density is defined by
Ξβ
α= Σβλ
α,λ
(20)
where comma denotes ordinary differentiation. The en-
ergy is expressed by the surface integral;
E = lim
r→∞
?
r=constant
Σ0ζ
0ηζdS
(21)
where ηζ(with ζ = 1,2,3) is the unit three-vector normal
to surface element dS.
The general form of the tetrad, hµ
symmetry was given by Robertson [36]. In the Cartesian
form it can be written as
i, having spherical
h
0
0 = iW,h
α
0
a = Zxa,
= Kδα
h
α
0
= iHxα,
h
aa+ Sxaxα+ ǫaαβGxβ
(22)
where W,K,Z,H,S, and G are functions of t and r =
√xaxa, and the zeroth vector hµ
to preserve Lorentz signature. We impose the boundary
condition that in the case of r → ∞ the tetrad given
above approaches the tetrad of Minkowski space-time,
hµ
a).
Using the general coordinate transformation
0has the factor i2= −1
a= diag(i,δα
haµ=∂Xν′
∂Xµhaν
(23)
where {Xµ} and
and Schwarzschild coordinates (t,r,θ,φ). In the spher-
ical, static and isotropic coordinate system X1
rsinθcosφ, X2= rsinθsinφ, X3= rcosθ. We obtain
the tetrad components of h
a
?
Xν′?
are, respectively, the isotropic
=
µ
as
io
00
0
√
√
1+2E(r)
R,r(t,r)sθcφ
1+2E(r)
R,r(t,r)sθsφ
√
1+2E(r)
R,r(t,r)cθ
1
R(t,r)cθcφ −
1
R(t,r)cθsφ
1
R(t,r)
sφ
sθ
0
1
R(t,r)
cφ
sθ
0
−
1
R(t,r)sθ
0
(24)
where i2= −1. Here, we have introduced the follow-
ing notation: sθ = sinθ, cθ = cosθ, sφ = sinφ and
cφ = cosφ. Now, we are interested in to find the total
Page 4
4
energy distribution. Since the intermediary mathemati-
cal exposition are lengthy, we give only the final result.
After making the required calculations [37, 38], the re-
quired non-vanishing component of Σνβ
µ is
Σ01
0(t,r,θ,ϕ) = 0(25)
while the momentum density distributions take the form
Ξ0
1(t,r,θ,ϕ) = 0,
(26)
Ξ0
2(t,r,θ,ϕ) = 0,
(27)
Ξ0
3(t,r,θ,ϕ) = 0.
(28)
Hence, we find the following energy
ETP(t,r,θ,ϕ) = 0(29)
here TP means Teleparallel Gravity, and one can easily
see that the momentum components are
− →
P
TP(t,r,θ,ϕ) = 0.
(30)
It is evident that the teleparallel gravitational results
are independent of teleparallel dimensionless coupling pa-
rameter λ which means that these results are valid not
only in teleparallel equivalent of general relativity but
also in any teleparallel model.
IV.ENERGY-MOMENTUM IN GENERAL
RELATIVITY
The aim of this section is to evaluate energy distribu-
tion associated with the white hole model of big bang by
using Møller’s formulation in general relativity.
In general relativity, Møller’s energy-momentum com-
plex is given by [4]
Ων
µ=
1
8π
∂χνα
∂xα
µ
(31)
satisfying the local conservation laws:
∂Ων
∂xν= 0
µ
(32)
where the antisymmetric super-potential χνα
µ is
χνα
µ =√−g
?∂gµβ
∂xγ−∂gµγ
∂xβ
?
gνγgαβ.
(33)
The
Ων
µ
gravitational and gravitational fields. Ω0
density and Ω0
aare the momentum density components.
locally
contains contributions from the matter,
conserved energy-momentumcomplex
non-
0is the energy
The momentum four-vector definition of Møller is given
by
Pµ=
? ? ?
Ω0
µdxdydz.
(34)
Using Gauss’s theorem, this definition transforms into
Pµ=
1
8π
? ?
χ0a
µµadS
(35)
where µa (where a = 1,2,3) is the outward unit nor-
mal vector over the infinitesimal surface element dS. Pi
give momentum components P1, P2, P3and P0gives the
energy.
The required non-zero component of the super-
potential of Møller, for the general line-element defines
the spherically symmetric black holes, is
χ01
0(t,r,θ,ϕ) = 0(36)
while the momentum density distributions take the form
Ω0
1(t,r,θ,ϕ) = 0 (37)
Ω0
2(t,r,θ,ϕ) = 0(38)
Ω0
3(t,r,θ,ϕ) = 0(39)
Therefore, if we substitute these results into equations
(34) and (35), we get the total energy that is contained
in a sphere of radius r
EGR(t,r,θ,ϕ) = 0(40)
which are also the energy (mass) of the gravitational field
that a neutral particle experiences at a finite distance r,
and here GR means General Relativity. Additionally, we
can find the momentum components which are given by
− →
P
GR(t,r,θ,ϕ) = 0.
(41)
It is evident that the energy-momentum distributions for
a given model in the formulations of Møller agree with
each other.
V. SUMMARY AND CONCLUSIONS
The notion of energy-momentum localization in the
general theory of relativity and tele-parallel gravity has
been very exciting and interesting; however, it has been
associated with some debate. Recently, some researchers
interested in studying the energy content of the universe
in various models.
The localization of gravitational energy-momentum
still remains one of the distinguished problems and this
subject continuous to be one of the most active areas of
Page 5
5
research in both general relativity and teleparallel grav-
ity (the tetrad theory of gravity). Many attempts have
been performed to obtain local or quasi-local energy-
momentum.However, there is no generally accepted
definition. The fundamental difficulty with these defini-
tions is that they are coordinate dependent. Therefore,
if the calculations are carried out in ”Cartesian” coordi-
nates, these complexes can give a reasonable and mean-
ingful result. Several researcher supposed that energy-
momentum complexes weren’t well-defined measures be-
cause of variety of such once. Recently, however, the
subject of the energy-momentum definition has been re-
opened by Virbhadra and his collogues.
The Møller energy-momentum prescription does not
necessitate carrying out calculation in ”Cartesian” coor-
dinates, while the others do. Therefore, we can calculate
the energy density in any coordinate system. Lessner
argued that the Møller prescription is a powerful con-
cept of energy-momentum in general relativity. Telepar-
allel version of this complex was obtained by Mikhail et
al, and Vargas, using the Einstein and Landau-Lifshitz
complexes in teleparallel gravity, calculated the energy-
momentum density of the Friedman-Robertson-Walker
space-time.
In literature, Cooperstock hypothesized that the en-
ergy is confined to the region of non-vanishing energy-
momentum tensor of matter and all non-gravitational
fields and Vagenas show that there is a connection be-
tween the coefficients of the expression for the energy in
Møller and Einstein complex.
Albrow [39] and Tryon [40] suggested that in our uni-
verse, all conserved quantities have to vanish. Tryon’s big
bang model predicted a homogenous, isotropic and closed
universe including of matter and anti-matter equally.
They argue that any closed universe has zero energy.
The subject of the energy-momentum distributions of
the closed universes was opened by an interesting work
of Cooperstock and Israelit [41]. They found the zero
value energy for a closed homogenous isotropic universe
described by a Friedmann-Robertson-Walker metric in
the context of general relativity. After this interesting
result some general relativists studied the same problem,
for instance: Rosen [42], Johri et al. [43], Banerjee-Sen
[44], Vargas [19] and Saltı et al. [23]. Johri et al., us-
ing the Landau-Liftshitz’s energy-momentum complex,
found that the total energy of a Friedmann-Robertson-
Walker spatially closed universe is zero at all times.
Banerjee and Sen who investigated the problem of to-
tal energy of the Bianchi-I type space-times using the
Einstein complex, obtained that the total energy is zero.
This result agrees with the studies of Johri et al.. Be-
cause, the line element of the Bianchi-I type space-
time reduces to the spatially flat Friedmann-Robertson-
Walker line element in a special case. Next, Vargas us-
ing the definitions of Einstein and Landau-Lifshitz in
teleparallel gravity, found that the total energy associ-
ated with the Friedmann-Robertson-Walker space-time is
zero. These results extend the works by Rosen and Johri
et al.. After Vargas’s work, Saltı et al. considered various
complexes both in general relativity and teleparallel grav-
ity for the viscous Kasner-type metric and found that the
total energy and momentum are zero everywhere. This
results agree with the works of Rosen and Johri. Re-
cently, Radinshi [45] using the energy-momentum com-
plexes of Tolman, Bergmann-Thomson, Møller, Einstein,
Landau-Lifshitz in general relativity, showed that total
energy in a model of the universe based on Bianchi type
V10 universe is zero everywhere at all times.
In this study, we have calculated the energy and mo-
mentum components (due to matter plus fields includ-
ing gravity) associated with a white hole model of the
big bang in general relativity by using Møller energy-
momentum formulation and also in Møller’s tetrad the-
ory of gravity (the teleparallel geometry). We find that
ETP(t,r,θ,ϕ) = EGR(t,r,θ,ϕ) = 0,
(42)
− →
P
TP(t,r,θ,ϕ) =− →
P
GR(t,r,θ,ϕ) = 0.
(43)
Our results show that the Møller energy-momentum
formulation in general relativity and its teleparallel grav-
itational analog agree with each other, and the momen-
tum components are equal to zero in bpth of the for-
mulations. Next, the teleparallel gravitational energy is
also independent of the teleparallel dimensionless cou-
pling parameter, which means that it is valid in any tele-
parallel model.
Furthermore, this paper sustains (a) the importance of
the energy-momentum definitions in the evaluation of the
energy distribution of a given space-time, (b) the view-
point of Lassner that the Møller energy-momentum com-
plex is a powerful concept of energy and momentum, (c)
the hypothesis by Cooperstock that the energy is con-
fined to the region of non-vanishing energy-momentum
tensor of matter and all non-gravitational fields. More-
over; the result that the total energy and momentum
of a white hole model of the big bang are zero sup-
ports the viewpoints of Albrow and Tyron.
results obtained here are also agree with ones calculated
in literature by Cooperstock-Israelit, Rosen, Johri et al.,
Banerjee-Sen, Vargas and Saltı et al..
Also the
Acknowledgments
The work of MS was supported by the Turkish Scien-
tific and Technical Research Council (T¨ ubitak). MS also
would like to thank T¨ ubitak-Feza G¨ ursey Institute, Is-
tanbul, for the hospitality we received in summer terms
2002-2006.
Page 6
6
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