Energy Distribution associated with Static Axisymmetric Solutions

0.13 MB·English

by  M. Sharif

Energy Distribution associated with
Static Axisymmetric Solutions
6
0
0 M. Sharif and Tasnim Fatima
∗
2
Department of Mathematics, University of the Punjab,

n
a Quaid-e-Azam Campus, Lahore-54590, Pakistan.
J

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1

1 Abstract
v
0 This paper has been addressed to a very old but burning problem
6
of energy in General Relativity. We evaluate energy and momentum
0
1 densities for the static and axisymmetric solutions. This specializes
0 to two metrics, i.e., Erez-Rosen and the gamma metrics, belonging to
6
the Weyl class. We apply four well-known prescriptions of Einstein,
0
/ Landau-Lifshitz,PapaterouandMo¨llertocomputeenergy-momentum
c
q density components. We obtain that these prescriptions do not pro-
- vide similar energy density, however momentum becomes constant in
r
g each case. The results can be matched under particular boundary
:
v conditions.
i
X
Keywords: Energy-momentum, axisymmetric spacetimes.
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a
1 Introduction
The problem of energy-momentum of a gravitational field has always been
an attractive issue in the theory of General Relativity (GR). The notion
of energy-momentum for asymptotically flat spacetime is unanimously ac-
cepted. Serious difficulties in connection with its notion arise in GR. How-
ever, for gravitational fields, this can be made locally vanish. Thus one is
always able to find the frame in which the energy-momentum of gravita-
tional field is zero, while in other frames it is not true. Noether’s theorem
∗e-mail: [email protected]
1
and translation invariance lead to the canonical energy-momentum density
tensor, Tb, which is conserved.
a
Tb = 0, (a,b = 0,1,2,3). (1)
a;b
In order to obtain a meaningful expression for energy-momentum, a large
number of definitions for the gravitation energy-momentum in GR have been
proposed. The first attempt was made by Einstein who suggested an expres-
sion for energy-momentum density [1]. After this, many physicists including
Landau-Lifshitz [2], Papapetrou [3], Tolman [4], Bergman [5] and Weinburg
[6] had proposed different expressions for energy-momentum distribution.
These definitions of energy-momentum complexes give meaningful results
when calculations are performed in Cartesian coordinates. However, the
expressions given by Mo¨ller [7,8] and Komar [9] allow one to compute the
energy-momentum densities in any spatial coordinate system. An alternate
concept of energy, called quasi-local energy, does not restrict one to use par-
ticularcoordinatesystem. Alargenumber ofdefinitionsofquasi-localmasses
have been proposed by Penrose [10] and many others [11,12]. Chang et al.
[13] showed that every energy-momentum complex can be associated with
distinct boundary term which gives the quasi-local energy-momentum.
Thereisacontroversywiththeimportanceofnon-tensorialenergy-momentum
complexes whose physical interpretation has been a problem for the scien-
tists. There is a uncertainity that different energy-momentum complexes
would give different results for a given spacetime. Many researchers con-
sidered different energy-momentum complexes and obtained encouraging re-
sults. Virbhadra et al. [14-18] investigated several examples of the space-
times and showed that different energy-momentum complexes could provide
exactly the same results for a given spacetime. They also evaluated the
energy-momentum distribution for asymptotically non-flat spacetimes and
found the contradiction to the previous results obtained for asymptotically
flat spacetimes. Xulu [19,20] evaluated energy-momentum distribution using
the Mo¨ller definition for the most general non-static spherically symmetric
metric. Hefoundthattheresultisdifferentingeneralfromthoseobtainedus-
ing Einstein’s prescription. Aguirregabiria et al. [21] proved the consistency
of the results obtained by using the different energy-momentum complexes
for any Kerr-Schild class metric.
On contrary, one of the authors (MS) considered the class of gravitational
waves, Go¨del universe and homogeneous Go¨del-type metrics [22-24]and used
2
the four definitions of the energy-momentum complexes. He concluded that
thefourprescriptions differingeneralforthese spacetimes. Ragab[25,26]ob-
tained contradictory results for Go¨del-type metrics and Curzon metric which
is a special solution of the Weyl metrics. Patashnick [27] showed that dif-
ferent prescriptions give mutually contradictory results for a regular MMaS-
class black hole. In recent papers, we extended this procedure to the non-null
Einstein-Maxwell solutions, electromagnetic generalization of Go¨del solution,
singularity-free cosmological model and Weyl metrics [28-30]. We applied
four definitions and concluded that none of the definitions provide consistent
results for these models. This paper continues the study of investigation of
the energy-momentum distribution for the family of Weyl metrics by using
the four prescriptions of the energy-momentum complexes. In particular, we
would explore energy-momentum for the Erez-Rosen and gamma metrics.
Thepaperhasbeendistributedasfollows. Inthenextsection,weshallde-
scribe the Weyl metrics and its two family members Erez-Rosen and gamma
metrics. Section3isdevotedtotheevaluationofenergy-momentum densities
for the Erez-Rosen metric by using the prescriptions of Einstein, Landau-
Lifshitz, Papapetrou and Mo¨ller. In section 4, we shall calculate energy-
momentum density components for the gamma metric. The last section
contains discussion and summary of the results.
2 The Weyl Metrics
Static axisymmetric solutions to the Einstein field equations are given by the
Weyl metric [31,32]
ds2 = e2ψdt2 e−2ψ[e2γ(dρ2 +dz2)+ρ2dφ2] (2)
−
in the cylindrical coordinates (ρ, φ, z). Here ψ and γ are functions of
coordinates ρ and z. The metric functions satisfy the following differential
equations
1
ψ + ψ +ψ = 0, (3)
ρρ ρ zz
ρ
γ = ρ(ψ2 ψ2), γ = 2ρψ ψ . (4)
ρ ρ − z z ρ z
3
It is obvious that Eq.(3) represents the Laplace equation for ψ. Its general
solution, yielding an asymptotically flat behaviour, will be
∞
a
n
ψ = P (cosθ), (5)
X rn+1 n
n=0
wherer = √ρ2 +z2, cosθ = z/r areWeylsphericalcoordinatesandP (cosθ)
n
are Legendre Polynomials. The coefficients a are arbitrary real constants
n
which are called Weyl moments. It is mentioned here that if we take
m m2ρ2
ψ = , γ = , r = ρ2 +z2 (6)
q
− r − 2r4
then theWeyl metric reduces to special solution ofCurzon metric [33]. There
are more interesting members of the Weyl family, namely the Erez-Rosen
and the gamma metric whose properties have been extensively studied in the
literature [32,34].
The Erez-Rosen metric [32] is defined by considering the special value of
the metric function
x 1 1 x 1 3
2ψ = ln( − )+q (3y2 1)[ (3x2 1)ln( − )+ x], (7)
2
x+1 − 4 − x+1 2
where q is a constant.
2
3 Energy and Momentum for the Erez-Rosen
Metric
In this section, we shall evaluate the energy and momentum density compo-
nents for the Erez-Rosen metric by using different prescriptions. To obtain
meaningful results in the prescriptions of Einstein, Ladau-Lifshitz’s and Pa-
papetrou, it is required to transform the metric in Cartesian coordinates.
This can be done by using the transformation equations
x = ρcosθ, y = ρsinθ. (8)
The resulting metric in these coordinates will become
e2(γ−ψ) e−2ψ
ds2 = e2ψdt2 (xdx+ydy)2 (xdy ydx)2 e2(γ−ψ)dz2. (9)
− ρ2 − ρ2 − −
4
3.1 Energy and Momentum in Einstein’s Prescription
The energy-momentum complex of Einstein [1] is given by
1
Θb = Hbc, (10)
a 16π a,c
where
g
Hbc = ad [ g(gbdgce gbegcd)] , a,b,c,d,e = 0,1,2,3. (11)
a √ g − − ,e
−
Here Θ0 is the energy density, Θi (i = 1,2,3) are the momentum density
0 0
componentsandΘ0 aretheenergycurrent density components. TheEinstein
i
energy-momentum satisfies the local conservation laws
∂Θb
a = 0. (12)
∂xb
The required components of Hbc are the following
a
4y 4x
H01 = e2γ(yψ xψ )+ (xψ +yψ )
0 ρ2 ,x − ,y ρ2 ,x ,y
x x
2xψ2 + e2γ, (13)
− ρ2 − ,ρ ρ2
4x 4y
H02 = e2γ(xψ yψ )+ (xψ +yψ )
0 ρ2 ,y − ,x ρ2 ,x ,y
y y
2yψ2 + e2γ. (14)
− ρ2 − ,ρ ρ2
Using Eqs.(13)-(14) in Eq.(10), we obtain the energy and momentum densi-
ties in Einstein’s prescription
1
Θ0 = [e2γ ρ2ψ2 +2(x2ψ +y2ψ xψ yψ )
0 8πρ2 { ,ρ ,yy xx − ,x − ,y }
+ 2 x2ψ +y2ψ +xψ +yψ ρ2ψ (ψ +ρψ ) ]. (15)
,xx ,yy ,x ,y ,ρ ,ρ ,ρρ
{ − }
All the momentum density components turn out to be zero and hence mo-
mentum becomes constant.
5
3.2 Energy and Momentum in Landau-Lifshitz’s Pre-
scription
The Landau-Lifshitz [2] energy-momentum complex can be written as
1
Lab = ℓacbd, (16)
16π ,cd
where
ℓacbd = g(gabgcd gadgcb). (17)
− −
Lab is symmetric with respect to its indices. L00 is the energy density and
L0i are the momentum (energy current) density components. ℓabcd has sym-
metries of the Riemann curvature tensor. The local conservation laws for
Landau-Lifshitz energy-momentum complex turn out to be
∂Lab
= 0. (18)
∂xb
The required non-vanishing components of ℓacbd are
y2 x2
ℓ0101 = e4γ−4ψ e2γ−4ψ, (19)
−ρ2 − ρ2
x2 y2
ℓ0202 = e4γ−4ψ e2γ−4ψ, (20)
−ρ2 − ρ2
xy xy
ℓ0102 = e4γ−4ψ e2ψ−4γ. (21)
ρ2 − ρ2
Using Eqs.(19)-(21) in Eq.(16), we get
e2γ−4ψ
L00 = [e2γ 2ρ2ψ2 8(y2ψ2 +x2ψ2)+2(x2ψ +y2ψ
8πρ2 { ,ρ − ,x ,y ,xx ,yy
xψ yψ )+16xyψ ψ 4xyψ
,x ,y ,x ,y ,xy
− − − }
ρ2ψ (3ψ +2ρ2ψ3 +2ρψ ) 8ρ2ψ2(xψ +yψ )
− ,ρ ,ρ ,ρ ,ρρ − ,ρ ,x ,y
8(x2ψ2 +y2ψ2)+2(x2ψ +y2ψ
− ,x ,y ,xx ,yy
+ xψ +yψ ) 16xyψ ψ +4xyψ ]. (22)
,x ,y ,x ,y ,xy
−
The momentum density vanishes and hence momentum becomes constant.
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3.3 Energy and Momentum in Papapetrou’s Prescrip-
tion
We can write the prescription of Papapetrou [3] energy-momentum distribu-
tion in the following way
1
Ωab = Nabcd, (23)
16π ,cd
where
Nabcd = √ g(gabηcd gacηbd +gcdηab gbdηac), (24)
− − −
and ηab is the Minkowski spacetime. It follows that the energy-momentum
complex satisfies the following local conservation laws
∂Ωab
= 0. (25)
∂xb
Ω00 and Ω0i represent the energy and momentum (energy current) density
components respectively.
The required components of Nabcd are
y2 x2
N0011 = e2γ e2γ−4ψ, (26)
−ρ2 − ρ2 −
x2 y2
N0022 = e2γ e2γ−4ψ, (27)
−ρ2 − ρ2 −
xy xy
N0012 = e2γ . (28)
−ρ2 − ρ2
Substituting Eqs.(26)-(28) in Eq.(23), we obtain the following energy density
e2γ
Ω00 = [ψ2 e−4ψ ψ2 +2ρ2ψ4 +2ρψ ψ
8π ,ρ − { ,ρ ,ρ ,ρ ,ρρ
8ψ2(xψ +yψ )+8(ψ2 +ψ2) 2(ψ +ψ ) ]. (29)
− ,ρ ,x ,y ,x ,y − ,xx ,yy }
The momentum density vanishes.
3.4 Energy and Momentum in M¨oller’s Prescription
The energy-momentum density components in M¨oller’s prescription [7,8] are
given as
1
Mb = Kbc, (30)
a 8π a,c
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where
Kbc = √ g(g g )gbegcd. (31)
a − ad,e − ae,d
Here Kbc is symmetric with respect to the indices. M0 is the energy den-
a 0
sity, Mi are momentum density components, and M0 are the components of
0 i
energy current density. The M¨oller energy-momentum satisfies the following
local conservation laws
∂Mb
a = 0. (32)
∂xb
Notice that M¨oller’s energy-momentum complex is independent of coordi-
nates.
The components of Kbc for Erez-Rosen metric is the following
a
K01 = 2ρψ . (33)
0 ,ρ
Substitute Eq.(33) in Eq.(30), we obtain
1
M0 = [ψ +ρψ ]. (34)
0 4π ,ρ ,ρρ
Again, we get momentum constant.
The partial derivatives of the function ψ are given by
1 q x 1 3x2 1
ψ = + 2(3y2 1)[3xln( − )+ − +3], (35)
,x
x2 1 4 − x+1 x2 1
− −
3yq x 1
ψ = 2[(3x2 1)ln( − )+6x], (36)
,y
4 − x+1
2x q x 1 3x2 5
ψ = − + 2(3y2 1)[3ln( − )+2x − ], (37)
,xx
(x2 1)2 4 − x+1 (x2 1)2
− −
3q x 1
ψ = 2[(3x2 1)ln( − )+6x], (38)
,yy
4 − x+1
3yq x 1 3x2 2
2
ψ = U = [3xln( − )+2 − ], (39)
,xy ,yx
4 x+1 x2 1
−
ρ ρq x 1
ψ = + 2[3x(3ρ2 2)ln( − )
,ρ
x(x2 1) 4x − x+1
−
(3x2 1)(3y2 1)
+ 2 − − +18x2], (40)
x2 1
−
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1 2ρ2 q x 1
ψ = + 2 (3y2 1)[3(ρ2 +x2)ln( − )
,ρρ
x(x2 1) − x(x2 1)2 4x2 − x+1
− −
2x ρ2(3x2 5) 3ρq ρ
+ (3x2 2+ − )]+ 2(1+ )
x2 1 − x2 1 4 y2
− −
x 1 3ρ2q x 1 3x2 1
[(3x2 1)ln( − )+6x]+ 2[3ln( − )+2 − ].(41)
× − x+1 x x+1 x2 1
−
4 Energy and Momentum for the Gamma Met-
ric
A static and asymptotically flat exact solution to the Einstein vacuum equa-
tions is known as the gamma metric. This is given by the metric [34]
2m 2m ∆ ∆γ2
ds2 = (1 )γdt2 (1 )−γ[( )γ2−1dr2+ dθ2+∆sin2θdφ2], (42)
− r − − r Σ Σγ2−1
where
∆ = r2 2mr, (43)
−
Σ = r2 2mr +m2sin2θ, (44)
−
m and γ are constant parameters. m = 0 or γ = 0 gives the flat spacetime.
For γ = 1 the metric is spherically symmetric and for γ = 1, it is axially
| | | | 6
symmetric. γ = 1 gives the Schwarzschild spacetime in the Schwarzschild
coordinates. γ = 1 gives the Schwarzschild spacetime with negative mass,
−
as putting m = M(m > 0) and carrying out a non-singular coordinate
−
transformation (r R = r + 2M) one gets the Schwarzschild spacetime
→
(with positive mass) in the Schwarzschild coordinates (t,R,θ,Φ).
InordertohavemeaningfulresultsintheprescriptionsofEinstein, Landau-
Lifshitz and Papapetrou, it is necessary to transform the metric in Cartesian
coordinates. We transform this metric in Cartesian coordinates by using
x = rsinθcosφ, y = rsinθsinφ, z = rcosθ. (45)
The resulting metric in these coordinates will become
2m 2m ∆ 1
ds2 = (1 )γdt2 (1 )−γ[( )γ2−1 xdx+ydy+zdz 2
− r − − r Σ r2{ }
∆γ2 xzdx+yzdy (x2 +y2)dz ∆(xdy ydx)2
+ − 2 + − . (46)
Σγ2−1{ r2√x2 +y2 } r2(x2 +y2)
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Now we calculate energy-momentum densities using the different prescrip-
tions given below.
4.1 Energy and Momentum in Einstein’s Prescription
The required non-vanishing components of Hbc are
a
x ∆ x m 2x
H01 = 4γm +( )γ2−1 (γ2 +1)(1 )
0 r3 Σ x2 +y2 − − r r2
m 2∆x 2∆x 2m2xz2
+ (γ2 1)(1 ) + +(γ2 1)
− − r Σr2 r4 − Σr4
xz2 x
+ , (47)
− r2(x2 +y2) r2
y ∆ y m 2y
H02 = 4γm +( )γ2−1 (γ2 +1)(1 )
0 r3 Σ x2 +y2 − − r r2
m 2∆y 2∆y 2m2yz2
+ (γ2 1)(1 ) + +(γ2 1)
− − r Σr2 r4 − Σr4
xyz2 y
+ , (48)
− r2(x2 +y2) r2
z m 2z m 2∆z
H03 = 4γm (γ2 +1)(1 ) +(γ2 1)(1 )
0 r3 − − r r2 − − r Σr2
2∆z 2m2z 2z
+ (γ2 1)(x2 +y2) + . (49)
r4 − − Σr4 r2
Using Eqs.(47)-(49) in Eq.(10), we obtain non-vanishing energy density in
Einstein’s prescription given as
1
Θ0 = [(γ2 1)Σ∆γ2−2r5(r m) (γ2 1)∆γ2−1r2
0 8πΣγ2r6 − − − −
r4 mr3 +m2r2 m2(x2 +y2) (γ2 +1)Σγ2r4
× { − − }−
+ 2(γ2 1)Σγ2−1r4(r m)2 +(γ2 1)m∆Σγ2−1r2
− − −
(γ2 1)∆γ2−2∆r4(r m)2 +(γ2 1)∆γ2−1
− − − −
∆r5(r m)+2Σγ2r3(r m) Σγ2∆r2 +Σγ2r4
× − − −
+ 3(γ2 1)Σγ2r2m2z2 (γ2 1)Σγ2−1r4m2
− − −
2(γ2 1)Σγ2−2m4z2(x2 +y2)]. (50)
− −
The momentum density components become zero and consequently momen-
tum is constant.
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