Table Of ContentClassification of Spherically Symmetric Static
Spacetimes according to their Matter
Collineations
3
0 M. Sharif ∗and Sehar Aziz
0
Department of Mathematics, University of the Punjab,
2
Quaid-e-Azam Campus Lahore-54590, PAKISTAN.
n
a
J
0
1 Abstract
The spherically symmetric static spacetimes are classified according to
1
their matter collineations. These are studied when the energy-momentum
v
1 tensor is degenerate and also when it is non-degenerate. We have found
4 a case where the energy-momentum tensor is degenerate but the group
0 of matter collineations is finite. For the non-degenerate case, we ob-
1 tain either four, five, sixor ten independent matter collineations inwhich
0 four are isometries and the rest are proper. We conclude that the mat-
3 ter collineations coincide with the Ricci collineations but the constraint
0
equations are different which on solving can provide physically interesting
/
c cosmological solutions.
q
- Keywords : Matter symmetries, Spherically symmetric spacetimes
r
g
:
v
i
X
r
a
∗Present Address: Department of Mathematical Sciences, University of Aberdeen, Kings
College,AberdeenAB243UEScotland, UK.<[email protected]>
1
1 Introduction
Einstein’s field equations (EFEs) are given by
1
G R Rg =κT , (a,b=0,1,2,3), (1)
ab ab ab ab
≡ − 2
whereG arethecomponentsoftheEinsteintensor,R thoseoftheRicciand
ab ab
T of the matter (energy-momentum) tensor. Also, R = gabR is the Ricci
ab ab
scalar, κ is the gravitationalconstant and, for simplicity, we take Λ=0.
Let (M,g) be a spacetime, i.e., M is a four-dimensional, Hausdorff, smooth
manifold, and g is smooth Lorentz metric of signature (+ - - -) defined on M.
The manifold M and the metric g are assumed smooth (C ). We shall use
∞
the usual component notation in local charts, and a covariant derivative with
respect to the symmetric connection Γ associated with the metric g will be
denoted by a semicolon and a partial derivative by a comma.
Ingeneralrelativity(GR) theory,the EinsteintensorG playsasignificant
ab
role, since it relates the geometry of spacetime to its source. The GR theory,
however, does not prescribe the various forms of matter, and takes over the
energy-momentum tensor T from other branches of physics.
ab
Collineations are geometrical symmetries which are defined by a relation of
the form
£ Φ=Λ, (2)
ξ
where£istheLiederivativeoperator,ξaisthesymmetryorcollineationvector,
Φ is any of the quantities g ,Γa ,R ,Ra and geometric objects constructed
ab bc ab bcd
by them and Λ is a tensor with the same index symmetries as Φ. One can
find all the well-known collineations by requiring the particular forms of the
quantities Φ and Λ. For example if we take Φ = g and Λ = 2ψg ,
ab ab ab ab
whereψ(xa)isa scalarfunction, this definesa ConformalKillingvector(CKV)
and it specializes to a Special Conformal Killing vector (SCKAV) when ψ =
;ab
0, to a Homothetic vector field when ψ = constant and to a Killing vector
(KV) when ψ = 0. If we take Φ = R and Λ = 2ψR the symmetry
ab ab ab ab
vector ξa is called a Ricci inheritance collineation (RIC) and reduces to a Ricci
collineation (RC) for Λ = 0. When Φ = T and Λ = 2ψT , where T
ab ab ab ab ab ab
is the energy-momentum tensor, the vector ξa is called a Matter inheritance
collineation(MIC)anditreducestoaMattercollineation(MC)forΛ =0. In
ab
the case of CKVs, the function ψ is called the conformal factor and in the case
of inheriting collineations the inheriting factor.
Collineationscanbeproper(non-trivial)orimproper(trivial). Inthispaper,
we will define a proper MC to be an MC which is not a KV, or a HV. The MC
equation can be written as
£ T =0 £ G =0, (3)
ξ ab ξ ab
⇔
or in component form
T ξc+T ξc +T ξc =0. (4)
ab,c ac ,b cb ,a
2
Collineationsotherthanmotions(KVs)canbeconsideredasnon-Noetherian
symmetriesandcanbe associatedwithconstantsofmotionand,upto the level
of CKVs, they can be used to simplify the metric [1]. For example, Affine
vectors (AVs) are related to conserved quantities [2], RCs, are related to the
conservationofparticlenumberinFriedmannRobertson-Walkerspacetimes[3].
and the existence of Curvature collineations (CCs) implies conservation laws
for null electromagnetic fields [4]. The set of collineations of a spacetime can
be related with an inclusion relation leading to a tree like inclusion diagram[4]
which shows their relative hierarchy. A collineation of a given type is proper if
it does not belong to any of the subtypes in this diagram. In order to relate
a collineation to a particular conservation law and its associated constant(s) of
motion, the properness of the collineation must first be assured.
The motivation for studying MCs can be discussed as follows. When we
find exact solutions to the Einstein’s field equations, one of the simplifications
we use is the assumption of certain symmetries of the spacetime metric. These
symmetry assumptions are expressed in terms of isometries expressed by the
spacetimes,alsocalledKillingvectorswhichgiverise toconservationlaws[1,5].
Symmetries of the energy-momentumtensor provide conservationlaws onmat-
ter fields. These symmetries are calledmatter collineations. These enable us to
know how the physical fields, occupying in certain region of spacetimes, reflect
the symmetries of the metric [6]. In other words, given the metric tensor of a
spacetime,onecanfindsymmetryforthe physicalfieldsdescribingthe material
contentofthatspacetime. Thereisalsoapurelymathematicalinterestofstudy-
ing the symmetry properties ofa givengeometricalobject, namely the Einstein
tensor, which arises quite naturally in the theory of GR. Since it is related, via
the Einstein field equations, to the material content of the spacetime, it has an
important role in this theory.
Recently, there is a growinginterestinthe study ofMCs [7-13]. Carot,etal
[8] has discussed MCs from the point of view of the Lie algebra of vector fields
generating them and, in particular, he discussed spacetimes with a degenerate
T . Hall, et al [9], in the discussion of RC and MC, have argued that the
ab
symmetries of the energy-momentum tensor may also provide some extra un-
derstanding of the the subject which has not been provided by Killing vectors,
Ricci and Curvature collineations.
In this paper, we study the problem of calculating MCs for static spher-
ically symmetric spacetimes for both degenerate and non-degenerate energy-
momentumtensorsandestablishtherelationbetweenKVs,RCsandMCs. The
breakdown of the paper follows. In the next section we write down MC equa-
tionsforstaticsphericallysymmetricspacetimes. Insectionthree,weshallsolve
theseMCequationswhentheenergy-momentumtensorisdegenerateandinthe
nextsectionMCequationsaresolvedforthenon-degenerateenergy-momentum
tensor. Finally, a summary of the results obtained will be presented.
3
2 Matter Collineation Equations
Inthissection,wewritedowntheMCequationsforsphericallysymmetricstatic
spacetimes. The most general spherically symmetric metric is given as
ds2 =eν(t,r)dt2 eλ(t,r)dr2 eµ(t,r)dΩ2, (5)
− −
where dΩ2 =dθ2+sin2θdθ2. Since we are dealing with static spherically sym-
metric spacetimes, Eq.(5) reduces to
ds2 =eν(r)dt2 eλ(r)dr2 eµ(r)dΩ2, (6)
− −
We can write MC Eqs.(4) in the expanded form as follows
T ξ1+2T ξ0 =0, (7)
0,1 0 ,0
T ξ0 +T ξ1 =0, (8)
0 ,1 1 ,0
T ξ0 +T ξ2 =0, (9)
0 ,2 2 ,0
T ξ0 +sin2θT ξ3 =0, (10)
0 ,3 2 ,0
T ξ1+2T ξ1 =0, (11)
1,1 1 ,1
T ξ1 +T ξ2 =0, (12)
1 ,2 2 ,1
T ξ1 +sin2θT ξ3 =0, (13)
1 ,3 2 ,1
T ξ1+2T ξ2 =0, (14)
2,1 2 ,2
ξ2 +sin2θξ3 =0, (15)
,3 ,2
T ξ1+2T cotθξ2+2T ξ3 =0, (16)
2,1 2 2 ,3
whereT =sin2θT . ItistobenoticedthatweareusingthenotationT =T .
3 2 aa a
We solve these equations for the degenerate as well as the non-degenerate case.
The nature of the solutionof these equations changes when one (or more)T is
a
zero. The nature changes even if T =0 but T =0.
a a,1
6
3 Matter Collineations in the Degenerate Case
In this section we solve MC equations (7)-(16) when at least one of T = 0.
a
First, we consider the trivial case, where T = 0. In this case, Eqs.(7)-(16) are
a
identically satisfied and thus every vector field is an MC.
The other possibilities can be classified in three main cases:
(1) when only one of the T =0,
a
6
(2) when exactly two of the T =0,
a
6
(3) when exactly three of the T =0.
a
6
4
Case (1): In this case, there could be only two possibilities:
(1a):T =0, T =0 (i=1,2,3), (1b):T =0, T =0 (j =0,2,3).
0 i 1 j
6 6
The case (1a) is trivial and we get either (i) T = constant = 0 or (ii) T =
0 0
6 6
constant. For the first possibility, we have
ξ =c ∂ +ξi(xa)∂ , (17)
0 t i
where c is a constant. For the second possibility, we obtain
0
f(˙t)
ξ =f(t)∂ ∂ +ξℓ(xa)∂ , ℓ=2,3. (18)
t r ℓ
− [ln√T ]
0 ′
In the case (1b), MC Eqs.(7), (9), (10) and (14)-(16) are identically satisfied
while Eqs.(8), (12), (13) yield that ξ1 =ξ1(r) only. Using Eq.(11), we have the
following solution
c
ξ = 0 ∂ +ξj(xa)∂ . (19)
r j
√T
1
We have seen that in all subcases of the case (1), we obtain infinite number of
MCs.
Case (2): This case implies the following two possibilities:
(2a):T =0, T =0, (2b):T =0, T =0,
k ℓ k ℓ
6 6
where k =0,1 and ℓ=2,3 which are valid through this case.
The case (2a) explores further two possibilities i.e. (i) T = constant = 0 and
2
6
(ii)T =constant. Inthefirstoption,wehavethefollowingsolutionoftheMC
2
6
equations
ξk =ξk(t,r,θ,φ), ξ2 =c cosφ+c sinφ, ξ3 = cotθ(c sinφ c cosφ)+c ,
1 2 1 2 0
− −
(20)
where c , c , c are constants. Thus we can write
0 1 2
ξ =ξk(xa)∂ +c (cosφ∂ cotθsinφ∂ )+c (sinφ∂ +cotθcosφ∂ )+c ∂ . (21)
k 1 θ φ 2 θ φ 0 φ
−
WeseethatthiscontainstheKVsassociatedwiththeusualsphericalsymmetry
given in Appendix B.
In the case (2aii), MCs turn out to be
ξ =ξ0(xa)∂ +c (cosφ∂ cotθsinφ∂ )+c (sinφ∂ +cotθcosφ∂ )+c ∂ . (22)
t 1 θ φ 2 θ φ 0 φ
−
We againhave the usualthree KVs in addition to arbitraryMCs in the t direc-
tion.
Forthe case(2b), Eqs.(14)-(16)areidentically satisfiedand fromthe remaining
Eqs.(7)-(13), we have
A¨ = T0 T0,1 ′ =α, (23)
A T 2T √T
1 (cid:18) 0 1(cid:19)
where α is a separation constant. From Eq.(22), three possibilities arise
(i) α>0, (ii) α=0, (iii) α<0.
5
Solving MC equations for the case (2bi), we have the following set of MCs
T
ξ0 = 0,1 (c sinh√αt+c cosh√αt)+c , (24)
1 2 0
−2T √T √α
0 1
c cosh√αt+c sinh√αt
ξ1 = 1 2 , ξℓ =ξℓ(t,r,θ,φ). (25)
√T
1
In the case of (2bii), we obtain the following solution
t2 √T
ξ0 = β(c +c t) c 1dr+c , (26)
− 1 2 2 − 1 T 0
Z 0
c t+c
ξ1 = 1 2, ξℓ =ξℓ(t,r,θ,φ), (27)
√T
1
where β is an integration constant which can be zero or non-zero and is given
by
T
0,1 =β, (28)
2T √T
0 1
For the case of (2biii), solution of MC equations becomes
√T
ξ0 =√p(c sin√pt c cos√pt) 1dr c , (29)
1 − 2 T − 0
Z 0
ξ1 = c1cos√pt+c2sin√pt, ξℓ =ξℓ(t,r,θ,φ), (30)
√T
1
where α= p and p>0. When T and T do not satisfy Eq.(22), we have the
0 1
−
following solution
ξ =c ∂ +ξℓ(xa)∂ . (31)
0 t ℓ
Again we see that all the possibilities of the case (2) give infinite-dimensional
MCs.
Case (3): In this case, we have the following two possibilities:
(3a):T =0, T =0, (3b):T =0, T =0.
0 i 1 j
6 6
For the case (3a), Eq.(7) is identically satisfied and Eqs.(8)-(10) respectively
give ξi = ξi(r,θ,φ), (i = 1,2,3). From the remaining equations, we have the
following constraint
A,22 = T2 T2,1 ′ =α, (32)
A √T 2T √T
1 (cid:18) 2 1(cid:19)
where α is a separation constant. This implies that we have three different
possibilities:
(i) α>0, (ii) α=0, (iii) α<0.
The case (3ai) gives the same MCs as for the case (2aii).
6
For the case (3aii), we have two possibilities depending upon the value of con-
straint β = T2,1 . When β =0, we have the following MCs
2T2√T1
1
ξ =ξ0(xa)∂ +c ∂ +c (cosφ∂ cotθsinφ∂ )+c (sinφ∂
t 0 r 1 θ φ 2 θ
T (r) −
1
+cotθcosφ∂ )+c ∂ . (33)
p φ 3 φ
This again yields infinite-dimensional MCs in addition to the usual spherical
symmetry KVs. For β =0, MCs turn out to be of the case (2aii).
6
The case (3aiii) gives the following MCs
cosθ T (r)
ξ =ξ0∂ +c ( ∂ 2′ sinθ∂ )
t 0 r θ
T (r) − 2T (r) T (r)
1 2 2
sinθcosφ T (r)
+c [ ∂ + p2′ (cosθcospφ∂ +sinφ∂ )]
1 r θ φ
T (r) 2T (r) T (r)
1 2 2
cosθsinφ T (r)
+c [ p ∂ + 2′p (cosθsinφ∂ cosφ∂ )]
2 r θ φ
T (r) 2T (r) T (r) −
1 2 2
T (r)
p +c 2p′ (cosφ∂ cotθsinφ∂ )
3 θ φ
2T (r) T (r) −
2 2
T (r)
+c 2′p (sinφ∂ +cotθcosφ∂ )
4 θ φ
2T (r) T (r)
2 2
T (r)
p +c 2′ ∂ . (34)
5 φ
2T (r) T (r)
2 2
This again provides infinite-dimensional MCs in additiopn to the usual KVs.
The case (3b) deals with the constraints T = 0, T = 0 and T = 0. In
0 1 2
6 6
addition to these, we can have the following constraints
(i) T =0, (ii) T =0, T =0,
j,1 0,1 2,1
6 6
(iii) T =0, T =0, (iv) T =0, T =0.
0,1 2,1 0,1 2,1
6
In case (3bi), from MC Eqs.(7)-(16), we obtain the following MCs
ξ =c ∂ +c (cosφ∂ cotθsinφ∂ )+c (sinφ∂ +cotθcosφ∂ )+c ∂ (35)
0 t 1 θ φ 2 θ φ 3 φ
−
which gives four independent MCs. This case is worth mentioning as we have
found finite number of MCs even for the degenerate energy-momentum tensor.
Incase(3bii), Eq.(11)becomesidentitywhileEqs.(7)and(8)giveξ0 =ξ0(θ,φ).
Also,Eqs.(12)and(13)respectively showthatξ2 andξ3 arefunctions oft,θ,φ.
From the remaining MC equations, we have the following solution
ξ =c ∂ +c (cosφ∂ cotθsinφ∂ )+c (sinφ∂ +cotθcosφ∂ )+c ∂
0 t 1 θ φ 2 θ φ 3 φ
−
1 ∂
[f (z)+f (z¯)]sinθ ∂ +[f (z)+f (z¯)]sinθ∂
−(cid:20)ln√T2∂θ{ + − }(cid:21) r + − θ
[f (z)+f (z¯)]∂ , (36)
+ φ
− −
7
wherez =ln cscθ cotθ +ιφandf (z)andf (z¯)arearbitraryfunctions of
+
| − | −
z and z¯(the complex conjugate of z) such that their sum is real and difference
is imaginary.
For the case (3biii), one obtains the following MCs
f˙(t)
ξ =f(t)∂ + − ∂ +c (cosφ∂ cotθsinφ∂ )+c (sinφ∂
t r 1 θ φ 2 θ
[ln√T ] −
0 ′
+cotθcosφ∂ )+c ∂ (37)
φ 3 φ
which gives an infinite number of MCs.
The case (3biv) yields the following MC vectors
ξ =c ∂ +f(xa)∂ +c (cosφ∂ cotθsinφ∂ )+c (sinφ∂ +cotθcosφ∂ )+c ∂ .
0 t r 1 θ φ 2 θ φ 3 φ
−
(38)
Thus one has an infinite number of independent MCs in all subcases of of the
case (3) except for (3bi) which has four independent MCs.
4 Matter Collineations in the Non-degenerate
Case
In this section we shallsolve MC equations when the energy-momentumtensor
is non-degenerate, i.e, T =0 since det(T )=0.
a a
6 6
IfwesolveMCEqs.(7)-(16),aftersometediousalgebra,wearriveatthefollow-
ing solution
T sinθ T
ξ0 = 2 [A˙ sinφ A˙ cosφ]+ 2A˙ cosθ+A , (39)
− T 1 − 2 T 3 4
0 1
T sinθ T
ξ1 = 2 [A sinφ A cosφ]+ 2A cosθ+A , (40)
− T ′1 − ′2 T ′3 5
1 1
ξ2 = cosθ[A sinφ A cosφ]+A sinθ+c sinφ c cosφ, (41)
1 2 3 1 2
− −
ξ3 = cosecθ[A cosφ+A sinφ]+[c cosφ+c sinφ]cotθ+c , (42)
1 2 1 2 0
where c ,c and c are arbitrary constants and A = A (t,r),µ = 1,2,3,4,5.
0 1 2 µ µ
These ξa are satisfied subject to the following differential constraints on A
µ
2T A¨ +T A =0, i=1,2,3, (43)
1 i 0,1 ′i
2T A˙ +T A =0, (44)
0 4 0,1 5
2T A˙ +T T2 ′A˙ =0, (45)
2 ′i 0 T i
(cid:18) 0(cid:19)
T A +T A˙ =0, (46)
0 ′4 11 5
T1,1TT2 +2T1 TT2 ′ A′i+2T2A′i′ =0, (47)
( 1 (cid:18) 1(cid:19))
T A +2T A =0, (48)
1,1 5 1 ′5
8
T A +2T A =0, c =0, (49)
2,1 ′i 1 i 0
T A =0. (50)
2,1 5
NowtheproblemofworkingoutMCsforallpossibilitiesofA ,A ,A isreduced
i 4 5
to solving the set of Eqs.(39)-(42) subject to the above constraints. We start
the classification of MCs by considering the constraint Eq.(50). This can be
satisfied for three different possible cases.
(1) T =0, A =0,
2,1 5
6
(2) T =0, A =0,
2,1 5
6
(3) T =0, A =0.
2,1 5
Case (1): In this case, all the constraints remain unchanged except (43), (47)
and (49). Thus we have
1T
A˙ 0,1A˙ =0, (51)
′i− 2 T i
0
T
A 1,1A =0, (52)
′i′− 2T ′i
1
T A =0. (53)
1 i
The last equation is satisfied only if A = 0. As a result, all the differential
i
constraintsinvolvingA anditsderivativesdisappearidenticallyandweareleft
i
with Eqs.(44), (46) and (48) only. Now integrating constraint Eq.(48) w.r.t. r
and replacing the value of A in constraint Eq.(44), we have
5
A(t)
T +2T A˙ =0,
0,1 0 4
√T
1
where A(t) is an integration function. This can be satisfied for the following
two possibilities:
(a) T =0, A˙ =0, (b) T =0, A˙ =0.
0,1 4 0,1 4
6 6
For the case (1a), after some algebra, we arrive at the following set
1 t 1
ξ =c ∂ +c [ − T dr∂ + ∂ ]+c ∂
0 t 4 { a 1 t √T } r 5√T r
Z 1 1
+c (cosφ∂ cotθsinφ∂ )+cp(sinφ∂ +cotθcosφ∂ )+c ∂ , (54)
1 θ φ 2 θ φ 3 φ
−
where a,c ,c ,c ,c ,c ,c are arbitraryconstants. This shows thatwe havesix
0 1 2 3 4 5
MCs.
In the case (1b), we have A˙ = 0 and T = 0. Solving Eqs.(44) and (46) and
4 0,1
6
rearrangingterms, we get
A¨ = 1 T0,1 ′ T0 =α, (55)
A 2 T √T √T
(cid:20) 0 1(cid:21) 1
9
whereαisaseparationconstantandthisgivesthefollowingthreepossiblecases:
(i) α<0, (ii) α=0, (iii) α>0.
First we take α<0 and assume that α= p, where p is positive, then Eq.(55)
−
yields
A=A cos√pt+A sin√pt
11 12
and also
1 T0,1 ′ = p√T1. (56)
2 T √T − T
(cid:20) 0 1(cid:21) 0
After some algebra, we can write the values of A and A given as
4 5
√T
A =√p 1dr(A sin√pt A cos√pt)+c ,
4 T 11 − 12 0
Z 0
A11cos√pt+A12sin√pt
A = .
5
√T
1
Substituting these values in Eqs.(39)-(42) and using the constraints A =0=
11
A , we obtain four independent MCs which are exactly similar to the case
12
(3bi).
The subcase (1bii) gives
A(t)=c t+c
3 4
and
T
0,1 =β, (57)
T √T
0 1
where β is an integration constant which yields the following two possibilities
( ) β =0, ( ) β =0.
∗ 6 ∗∗
The first possibility implies that
T =β eβ √T1dr,
0 0
R
where β is an integration constant. Now we solve Eqs.(48) and (50) by using
0
this constraint, we can get the following set
c √T c t+c
ξ0 = 3 1 dr+c , ξ1 = 3 4,
−β0 Z eβ √T1dr 0 √T1
ξ2 =c1sinφ cR2cosφ, ξ3 =c1cosφ+c2sinφ. (58)
−
This gives five independent MCs.
For the case (1bii**), T =constant. Using this fact Eq.(48) yields A =g(r).
0 4
Thus we have the solution
c c t+c
ξ0 = 3 T dr+c , ξ1 = 3 4,
− b 1 0 √T
Z 1
ξ2 =c sinφ c copsφ, ξ3 =c cosφ+c sinφ. (59)
1 2 1 2
−
10
