On Axially Symmetric Space-Times Admitting Homothetic Vector Fields in Lyra's Geometry

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by  Ragab M. Gad

On Axially Symmetric Space-Times
Admitting Homothetic Vector Fields in
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0 Lyra’s Geometry
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a Ragab M. Gad1,2 ∗ and A. E. Al Mazrooei1†
J 1 Mathematics Department, Faculty of Science, University of Jeddah,

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21589 Jeddah, KSA
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2 Mathematics Department, Faculty of Science, Minia University,

] 61915 El-Minia, Egypt
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q
-
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g
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Abstract

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v Thispaperinvestigatesaxiallysymmetricspace-timeswhichadmit
1 a homothetic vector field based on Lyra’s geometry. The cases when
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the displacement vector is function of t and when it is constant are
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studied. In the context of this geometry, we find and classify the
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0 solutions of the Einstein’s field equations (EFE) for the space-time
. under consideration which display a homothetic symmetry.
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Keywords: Axially symmetric space-times; homothetic vector field; Lyra’s
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1 geometry; Einstein’s field equations
v: PACS Nos.: 04.20.Cv, 04.20.jb:
i
X
r 1 Introduction
a
Space-time symmetries play an important role in the features of space-time
that can be described as exhibiting some form of symmetry. The most im-
portant of symmetries is in simplifying Einstein field equations and provide
a classification of the space-times according to the structure of the corre-
sponding Lie algebra.
∗E-mail: [email protected]
†E-mail:[email protected]
1
Symmetries have been studied in the theory of General Relativity (GR)
based on Riemannian geometry and in the theory of teleparallel gravity
(TPG) based on the Weitzenbock geometry. In GR different kind of sym-
metries like isometry, homothetic, conformal, Ricci collineations and matter
collineations have been extensively studied [1]-[18]. Also some of them have
been studied in TPG, over the past few years [19]-[26].
In a series of paper, Gad and his collaborator [27]-[29] studied the homo-
thetic symmetry based on Lyra’s geometry. They classified the space-times
according to admitting such symmetry. For zero displacement vector field,
they obtained results agree with those obtained previously in GR, based
on Riemannian geometry. They shown also that in the case when the dis-
placement vector field is constant, it is not possible to compare the results
obtained in the context of Lyra’s geometry with that obtained in GR, using
Riemannian geometry. This means that in Lyra’s geometry if the displace-
ment vector field is considered as a constant this does not give meaningful
results.
In the framework of Lyra’s geometry, many authors have made attempts to
find solutions of EFE [30]- [36]. These attempts were by imposing certain
conditions upon the scale factors of the space-time in addition to the condi-
tions on the matter which represents such space-time and some restrictions
onitsphysicalproperties. Inthesamecontext,wewillobtainingsolutionsto
EFE for an axially symmetric space-time by assuming only that such space-
time displays a homothetic symmetry, that is, exhibits a self-similarity of
the first kind (cf. [6, 37, 38]).
The paper is organized as follows: In the next Section, we will study
the homothetic symmetry of axially symmetric space-times based on Lyra’s
geometry. Section 3 deals with the solutions of EFE and their classifications
of the space-times under considerations. Finally, in Section 4, concluding
remarks are given.
2 The version of model and homothetic vector field
in Lyra’s geometry
In Lyra’s geometry the metric or the measure of length of displacement
vector ζµ = xodxµ between two points p(xµ)n and q(xµ + dxµ)n is
µ=1 µ=1
given by absolute invariant underboth gauge function, xo = xo(xµ)n , and
µ=1
2
coordinate system, xµ n , as follows
{ }µ=1
ds2 =g xodxµxodxν,
µν
where g is a metric tensor as in Riemannian geometry.
µν
In Lyra’s geometry, a generalized affine connection characterized not only
bytheRiemannianconnection, α , butalsoby afunctionφ , whicharises
µν µ
{ }
through gauge transformation, and it is given by
1
Γα = (xo)−1 α + (δαφ +δαφ g φα), (2.1)
µν {µν} 2 µ ν ν µ− µν
where φ is called a displacement vector field and satisfies φα = gαβφ . We
β
consider φ to be a timelike vector, where
φ = (β(t),0,0,0). (2.2)
µ
Throughout the paper M will denoted a 4-dimensional Lyra manifold
with Lorentz metric g which is a generalization to the 4-dimensional Rie-
mannianmanifold [39,40]. As inRiemannian geometry, aglobal vector field
ζ = ζµ(t,x,y,z)4 on M is called homothetic vector field if the following
µ=1
condition
£ g = g ζρ+g ζρ = 2ψg , (2.3)
ζ µν ρν µ µρ ν µν
∇ ∇
holds where ψ is a constant (the homothetic constant) on M, £ denotes a
Lie derivative and is the covariant derivative, such that
∇
ζν = 1 ∂ ζν +Γν ζα,
∇µ xo µ µα (2.4)
ζ = 1 ∂ ζ Γα ζ .
∇µ ν xo µ ν − µν α
Here Γσ is a Lyra connection given by equation (2.1).
µν
In equation (2.3), if ψ = 0, ζ is called proper homothetic vector field and
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the this equation is called homothetic equation. It is worth mention here if
ψ = 0, then equation (2.3) is called Killing equation and ζ is called a Killing
vector field on M.
Consider the axially symmetric metric in the form [41]
ds2 = dt2 A2(t)(dχ2+f2(χ)dφ2) B2(t)dz2, (2.5)
− −
with the convention x0 = t, x1 = χ, x2 = φ, x3 = z and A and B are
functions of t only while f is a function of the coordinate χ only.
3
The study of homothetic vector fields, ζ = ζµ(t,x,y,z)4 , on axially
µ=1
symmetricspace-times (2.5) is basedonan examination of thetenequations
obtained from (2.3). For the model (2.5), using (2.1) and apart from the
factor 1 , i.e., we choose the normal gauge xo = 1, equations (2.3) are
xo
reduced to the following system of equations:
A 1
ζ1 +( ,t + β)ζ0 = ψ, (2.6)
,χ A 2
ζ1 +f2ζ2 = 0, (2.7)
,y ,χ
A2ζ1 +B2ζ3 = 0, (2.8)
,z ,χ
ζ0 A2ζ1 = 0, (2.9)
,χ ,t
−
f A 1
ζ2 + ,χζ1+( ,t + β)ζ0 = ψ, (2.10)
,y f A 2
A2f2ζ2 +B2ζ3 = 0, (2.11)
,z ,y
ζ0 A2f2ζ2 = 0, (2.12)
,y ,t
−
B 1
ζ3 +( ,t + β)ζ0 = ψ, (2.13)
,z B 2
ζ0 B2ζ3 = 0, (2.14)
,z ,t
−
1
ζ0 + βζ0 = ψ. (2.15)
,t 2
Solving equation (2.15) and using the result back into equations (2.9),
(2.12) and (2.14), we get
ζ0 = [ψ e12 βdtdt+c ]e−12 βdt, (2.16)
Z R 0 R
ζ1 = F (x,y,z),
1
ζ2 = F (x,y,z),
2
ζ3 = F (x,y,z),
3
wherec isaconstantofintegrationandF (x,y,z),F (x,y,z)andF (x,y,z)
0 1 2 3
are arbitrary functions which are to be determined.
Differentiating equations (2.6) and (2.13) with respect to t and using (2.15)
and (2.16), we get respectively
A 1 a
t
+ β = , (2.17)
A 2 [ψ e12 βdtdt+c ]e−12 βdt
0
R R
R
4
B 1 c
t
+ β = , (2.18)
B 2 [ψ e12 βdtdt+c ]e−12 βdt
0
R R
R
where a and c are constants of integration. Without loss of generality,
we assume that a = c. Substituting these results back into (2.6) and (2.13),
using the obtained results in (2.7), (2.8), and (2.11), we get
ζ1 = (ψ a)x+c , (2.19)
1
−
ζ3 = (ψ a)z+c , (2.20)
3
−
where c and c are constants of integration. Consequently, ζ2 depends on
1 3
y only.
Integrating equations (2.17) and (2.18), using a = c, we get the following
relation
B(t)= nA(t), (2.21)
where n is a constant of integration
From equation (2.10), we have the following two cases
1. Case i: If we assume that
f
,χζ1 = c = 0. (2.22)
4
f 6
Integrating this equation, we get
c4
f(χ)= c5χψ−a, (2.23)
where c is a constant of integration.
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Inserting the above results into (2.10), we get
ζ2 = (ψ b)y+c , (2.24)
2
−
where c is a constant of integration and b = a+c .
2 4
Without loss of generality, we assume that c = c = c = c = 0,
0 1 2 3
therefore, from equations (2.16), (2.19), (2.20) and (2.24) we obtain
the following homothetic vector field
ζ = ([ψZ e12R βdtdt]e−12R βdt)∂t+(ψ−a)χ∂χ+(ψ−b)y∂y+(ψ−a)z∂z.
(2.25)
It is of interested to note that the function (2.23) does not satisfy
that f,χχ equals constant or zero. Consequently, the metric (2.5) with
f
5
c4
g22 = c5χψ−a doesnotasolutiontoEFE,asweseeinthenextsection.
Therefore, this function will withdraw from consideration inspect the
special case, when c = ψ a. The latter case gives
4
−
ζ2 = c
2
and
f(χ)= c χ. (2.26)
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In this case, the homothetic vector becomes
ζ = ([ψZ e12R βdtdt]e−12R βdt)∂t+(ψ−a)χ∂χ +c2y∂y +(ψ−a)z∂z.
(2.27)
2. Case ii: If
f
,χζ1 = 0. (2.28)
f
- When ζ1 =0, this implies
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f(χ)= c , (2.29)
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where c is a constant of integration.
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In this case the homothetic vector is the same as given by equation
(2.25).
-When ζ1 = 0, this implies
f(χ)= c ec7χ, (2.30)
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where c and c are constants of integration.
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In this case the homothetic vector is given by
ζ = ([ψZ e12R βdtdt]e−21R βdt)∂t+(ψ−b)y∂y +(ψ−c)z∂z. (2.31)
All the above considerations lead to the following theorem:
Theorem 2.1 In Lyra’s geometry, if a displacement vector is function of
t, that is, β = β(t), axially symmetric space-time described by metric (2.5)
admits the homothetic vector field if
B(t)= nA(t).
Such HVF is given by equation (2.25) if f(χ) is given by equation (2.29) and
it is given by equation (2.27) or (2.31), if f(χ) is given by equation (2.26)
or (2.30), respectively.
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Now we will investigate the situation when the displacement vector is
constant, that is, β = constant.
Inthiscase,fromequation(2.16), thetime-likecomponentofthehomothetic
vector ζ0, the only component depends on β, becomes
ζ0 = (2ψe12βt+c0)e−12βt.
β
While the space-like components are given as the case when β = β(t) ac-
cording to the values of f(χ).
Now we conclude that the homothetic vector field given by (2.25) and
with the above constrains on the scale factors is similar to the vector as
obtained in the theory of General Relativity (see for instance, [42, 43]).
It is worthy to note that, in the case β = const., we can not compare the
results with that obtained in the theory of General Relativity, using Rie-
mannian geometry, because in this case the component ζ0 tends to infinity
when β = 0. This means that in Lyra’s geometry if the displacement vector
field is considered as a constant this does not give meaningful results.
3 Field equations and their solutions
In this section, we shall determine the exact solutions of EFE by assuming
that the space-time under consideration admits a homothetic vector field
(self-similarity).
The field equations in normal gauge for Lyra’s geometry as obtained by
Sen [44] (in gravitational units c= 8πG = 1) read as
1 3 3
R Rg = T φ φ + g φ φα, (3.1)
µν µν µν µ ν µν α
− 2 − − 2 4
the left hand side is the usual Einstein tensor as in Riemannian geometry,
whereas φ is a time-like displacement field vector defined by (2.2) and T
µ µν
is the energy momentum tensor correspondingto perfect fluid. It is of inter-
esting to assume that the matter of field in space-time under consideration
is represented by a perfect fluid, that is, the energy-momentum tensor is
defined by
T = (ρ+p)u u pg , (3.2)
µν µ ν µν
−
Herepisthepressure,ρtheenergydensityandu thefourvelocity vector, it
µ
mustverify £ u = 0. For thespace-time (2.5) the4-velocity vector can be
ζµ µ
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definedbyuµ = u = (1,0,0,0) anditisverifiedg uµuν = 1. Inviewofthe
µ µν
metric (2.5), the volume element, the four-acceleration vector, the rotation,
the expansion scalar and the shear scalar can be written, respectively, in a
comoving coordinates system as (see [45])
V = √ g = A2fB, (3.3)
−
u˙ = 0,
i
ω = 0,
ij
(3.4)
Θ = 2A,t + B,t,
A B
σ2 = 1(11(At)2+5(Bt)2+ 2AtBt).
9 A B AB
The non vanishing components of the shear tensorσ are
ij
σ = A(1ΘA A ),
11 3 − ,t
σ = Af2(1ΘA A ),
22 3 − ,t (3.5)
σ = B(1ΘB B ),
33 3 − t
σ = 2Θ.
44 −3
For the line element (2.5), the field equations (3.1) with equation (??)
lead to the following system of equations (see for instance Gad [45])
A B A B 3
,tt + ,tt + ,t t + β2 = p, (3.6)
A B AB 4 −
2A A f 3
tt +( ,t)2 ,χχ + β2 = p, (3.7)
A A − fA2 4 −
A 2A B f 3
( ,t)2+ t t ,χχ β2 = ρ, (3.8)
A AB − fA2 − 4
2A B
t t
ρ +(ρ+p)( + ) = 0, (3.9)
,t
A B
InGR,CahillandTaub[6]andBicknellandHenriksen[15](seealso[16])
pointed out, if the matter field is a perfect fluid, then the only barotropic
equation of state which is compatible with self-similarity (characterized by
the existence of homothetic vector field) is of the form
p = kρ, (3.10)
whereρisthetotalenergydensity,pisthepressureandk isaconstantinthe
range 0 k 1. This equation of state is nevertheless physically consistent
≤ ≤
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in the whole range of k. When k = 0, the above equation describes dust,
k = 1/3 gives the equation of state for radiation and k = 1 considers the
effective ”stiff fluid” distribution. The latter was apparently first proposed
by Zeldovich [46]. It should have applied in the early Universe, with this
case, the velocity of sound equals the velocity of light, so no material in this
Universe could be more stiff.
Exact solutions for the Einstein field equations (3.6)-(3.9) can be found
under the assumption that the space-time (2.5) admits homothetic vector
field. The obtained solutions can be classified according to the value of the
scale factor f(χ) and the constant k.
Using (2.21) and (3.10), the Einstein field equations (3.6)-(3.9) reduce to
the following equations
2A A 3
,tt +( ,t)2+ β2 = kρ, (3.11)
A A 4 −
A 3
3( ,t)2 β2 = ρ, (3.12)
A − 4
m
ρ = , (3.13)
A3(1+k)
where m is a constant of integration.
From equations (3.11)-(3.13), we get
2A A m(1 k)
tt t 2
+4( ) = − (3.14)
A A A3(1+k)
In the following we discuss two cases, when k = 1 and k = 0:
Class I: k=1
In this case
1
A(t) = (αt+α1)3, (3.15)
where α and α are constants of integration.
1
Accordingtothevaluesoff(χ),givenbyequations(2.26), (2.29) and(2.30),
we have, respectively, the following solutions
2
ds2 = dt2 (αt+α1)3(dχ2+c5χdφ2+ndz2), (3.16)
−
2
ds2 =dt2 (αt+α1)3(dχ2 +c6dφ2+ndz2), (3.17)
−
ds2 = dt2 (αt+α1)32(dχ2+c8ec7χdφ2+ndz2), (3.18)
−
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Class II: k = 0
From equation (3.14), we have the following equation
m
A2A +2AA2 = . (3.19)
,tt ,t 2
Solving the above equation, we get
1 1
A(t) = (6mt2 α2t+α3)3. (3.20)
2 −
where α and α are constants of integration.
2 3
For the values of f(χ), as given in theorem (2.1), we have the following class
of solutions
1 2
ds2 = dt2 (6mt2 α2t+α3)3(dχ2 +c5χdφ2+ndz2), (3.21)
− 4 −
1 2
ds2 = dt2 (6mt2 α2t+α3)3(dχ2+c6dφ2+ndz2), (3.22)
− 4 −
ds2 = dt2 1(6mt2 α2t+α3)23(dχ2+c8ec7χdφ2 +ndz2), (3.23)
− 4 −
For the class I of solutions the expressions for density ρ, pressure p and
displacement field β are given by
m
p = ρ = ,
(αt+α )2
1
which shows that ρ and p are not singular,
4 α2 3m
β2 = ( − ).
9 (αt+α )2
1
It is observed, from equations (3.15) and (2.21) that A(t) and B(t) can be
singular only for t . Thus the line element (2.5) is singular free even at
→ ∞
t = 0.
For the same class of solutions, usingequations (3.3) - (3.5), we have the
following physical properties:
The volume element is
V = nf(χ)(αt+α ).
1
Here f(χ) takes one of the values given by equations (2.26), (2.29) and
(2.30). For all values of f(χ), we see that the volume element increases as
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