Abstract

The motive of the current article is to study and characterize the geometrical and physical competency of the conharmonic curvature inheritance (Conh CI) symmetry in spacetime. We have established the condition for its relationship with both conformal motion and conharmonic motion in general and Einstein spacetime. From the investigation of the kinematical and dynamical properties of the conformal Killing vector (CKV) with the Conh CI vector admitted by spacetime, it is found that they are quite physically applicable in the theory of general relativity. We obtain results on the symmetry inheritance for physical quantities (μ,p,ui,σij,η,qi ) of the stress-energy tensor in imperfect fluid, perfect fluid and anisotropic fluid spacetimes. Finally, we prove that the conharmonic curvature tensor of a perfect fluid spacetime will be divergence-free when a Conh CI vector is also a CKV.
universe
Article
Conharmonic Curvature Inheritance in Spacetime of
General Relativity †
Musavvir Ali 1,*, Mohammad Salman 1and Mohd Bilal 2


Citation: Ali, M.; Salman, M.; Bilal,
M. Conharmonic Curvature
Inheritance in Spacetime of General
Relativity. Universe 2021,7, 505.
https://doi.org/10.3390/
universe7120505
Academic Editor: Kazuharu Bamba
Received: 17 November 2021
Accepted: 12 December 2021
Published: 17 December 2021
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Attribution (CC BY) license (https://
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4.0/).
1Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India; salman199114@gmail.com
2Department of Mathematical Sciences, Faculty of Applied Sciences, Umm Al Qura University,
Makkah P.O. Box 56199, Saudi Arabia; mohd7bilal@gmail.com
*Correspondence: musavvirali.maths@amu.ac.in
2010 Mathematics Subject Classification: 53B20; 83C45; 53A45; 83C20.
Abstract:
The motive of the current article is to study and characterize the geometrical and physical
competency of the conharmonic curvature inheritance (Conh CI) symmetry in spacetime. We have
established the condition for its relationship with both conformal motion and conharmonic motion in
general and Einstein spacetime. From the investigation of the kinematical and dynamical properties
of the conformal Killing vector (CKV) with the Conh CI vector admitted by spacetime, it is found
that they are quite physically applicable in the theory of general relativity. We obtain results on the
symmetry inheritance for physical quantities (
µ
,
p
,
ui
,
σij
,
η
,
qi
) of the stress-energy tensor in imperfect
fluid, perfect fluid and anisotropic fluid spacetimes. Finally, we prove that the conharmonic curvature
tensor of a perfect fluid spacetime will be divergence-free when a Conh CI vector is also a CKV.
Keywords: curvature; symmetry; inheritance; Einstein spacetime
1. Introduction
Let
(V4
,
g)
be a spacetime, where
V4
is a four-dimensional connected smooth Hausdorff
manifold and g is a smooth Lorentz metric of signature
(
,
+
,
+
,
+)
. Let
be the Levi–
Civita connection associated with g and
R
be the corresponding type (1, 3) Riemannian
curvature tensor. The type (1, 3) Weyl conformal curvature tensor of
(V4
,
g)
is denoted by
C
. The components of
R
and
C
are written as
Rh
ijk
and
Ch
ijk
and the Ricci tensor
Rij
and
Ricci scalar Rare defined, in components, by Ri j =Rh
ihj and R=Ri j gij , respectively.
In mathematics and theoretical physics, the study of spacetime symmetries is of great
interest for contemporary researchers. In addition, the spacetime symmetries are very
useful for finding the solutions to Einstein’s field equation (EFE) if its existence occurs, and
provide further intuition toward conservation laws of generators in dynamical systems [
1
].
Much interest has been shown in the various symmetries of the geometrical structures on
(V4
,
g)
, and details are available in ([
1
3
]). Gravitational classification can be carried out
through the help of geometrical symmetries of spacetime in general relativity. Moreover,
motion/isometry or Killing symmetry is one of the most primary symmetries of a spacetime.
This is defined along a vector field under the condition that the Lie derivative of metric
tensor vanishes.
An elegant restructuring form of classical mechanics is finalized by the general theory
of relativity. This theory wraps the time and the space co-ordinates into a single continuum,
called as spacetime. This theory is also called as the theory of gravitation in spacetime,
which is described by the Einstein’s field equation and these equations describe a system
of ten coupled highly nonlinear PDEs, given as the following
Rij 1
2Rgi j =κTij, (1)
Universe 2021,7, 505. https://doi.org/10.3390/universe7120505 https://www.mdpi.com/journal/universe
Universe 2021,7, 505 2 of 21
where
Tij
denotes the components of the stress-energy tensor and
κ
is the gravitational
constant. In the field Equation
(1)
, the left part depicts the geometrical meaning of
spacetime, whereas the right part describes the physical significance of the spacetime
of general relativity.
The study of spacetime symmetries is an important tool in finding the exact solution
of the system
(1)
. The spacetime symmetries play a pivotal role in understanding the
relationship between matter and geometry by EFE. The different classes of spacetime
symmetries, such as the isometries, homothetic motion, conformal motion, curvature sym-
metry, curvature inheritance symmetry, Ricci symmetry, Ricci inheritance symmetry, matter
collineations, matter inheritance collineations, conharmonic symmetries, semi-conformal
symmetry, etc., are well known in the literature ([
1
6
]). The spacetime symmetries are
important not only in finding the exact solutions of EFE, but also in providing spacetime
classifications along with an invariant basis (preferably, the basis of null tetrad can be cho-
sen). The spacetime symmetries are also a popular tool in investigating many conservation
laws in the theory of general relativity [
5
]. Moreover, certain geometrical and physical
notions are also described by spacetime symmetries, such as the conservation of linear
momentum, angular momentum and energy [
7
]. The symmetries regarding spacetime
(V4,g)are determine by the following mathematical equation [8]:
£ξ=2αΩ, (2)
where £
ξ
stands for the Lie derivative operator, with respect to the vector field
ξi
,
α
is
some smooth scalar function on the spacetime and
is any of the physical quantities
(
µ
,
p
,
ui
,
σij
,
η
,
qi
), where
µ
,
p
,
ui
,
σij
,
η
,
qi
are the energy density, the isotropic pressure, the
velocity vector, the shear tensor, the shear viscosity coefficient and the energy flux vector,
respectively, and geometrical quantities, such as the components of the metric tensor
(gij )
,
Riemannian curvature tensor
(Rh
ijk )
, Ricci tensor
(Rij )
, conharmonic curvature tensor
(Zh
ijk )
,
contracted conharmonic curvature tensor
(Zij )
, energy momentum tensor
(Tij)
, etc. The
most primary symmetry on
(V4
,
g)
is motion (M), which is obtained by setting
=gij
and
α=
0 in Equation
(2)
. Then, Equation
(2)
will be called the Killing equation, and the vector
satisfying it is known as the Killing vector. Equation (2) can also be explicitly written as the
following:
ξkgij,k+gikξk
,j+gjkξk
,i=0, (3)
where the subscript comma
(
,
)
stands for the partial differentiation, with respect to the
coordinates (xi)in the spacetime.
The gravitational field consists of two parts viz., the free gravitational part and the
matter part, which is described by the Riemannian curvature tensor in the general theory
of relativity. The connection between these two parts is explained through Bianchi’s
identities [
9
]. The principal aim of all investigations in gravitational physics is focused on
constructing the gravitational potential (metric) satisfying the Einstein field equations.
In the present research paper, we raise the following fundamental problem:
how are the geometrical symmetries of the spacetime
(V4
,
g)
associated with the conhar-
monic curvature symmetry vector field, under the condition that this vector is inherited by
some of the source terms of the energy-momentum tensor in the field equations? In this
paper, we discuss the conharmonic curvature inheritance symmetry with respect to confor-
mal motion, conharmonic motion and source terms of perfect, imperfect and anisotropic
fluid spacetime. Our present work is mainly influenced by the work carried out towards
the symmetries, such as the curvature inheritance, Ricci inheritance, and matter inheritance
on the semi-Riemannian manifold. This concept of symmetry inheritance was initiated in
1989 by Coley and Tupper [
10
] for the special conformal Killing vector (SCKV), and was
then further studied in 1990 for CKV ( [
11
,
12
]). In 1992 and 1993, K. L. Duggal introduced
the concept of inheritance symmetry for the curvature tensor of Riemannian spaces with
physical applications to the fluid spacetime of general relativity ([2,13]).
Universe 2021,7, 505 3 of 21
The above abundant work motivated us to inquire about the inheritance symmetry of
the conharmonic curvature tensor in spacetime. The conharmonic curvature inheritance
symmetry is defined through Equation
(2)
, where
is replaced by the conharmonic
curvature tensor. The structure of our manuscript is as follows: the preliminaries are
given in Section 2. In Section 3, we elaborate on the concept of curvature inheritance
symmetry with some of the related results. In Section 4, we derive the relationship of
symmetry inheritance with other known symmetries, such as both conformal motion
and conharmonic motion in general and Einstein spacetime. We have established some
important results as a witness to the physical application of the Conh CI symmetry in
spacetime for perfect, imperfect and anisotropic fluid in Section 5. Finally, Section 6is a
brief conclusion. Furthermore, in an attempt to support our study, which is related to the
solution of EFE and conservation law of generators, we have constructed some non-trivial
examples that are embedded in the Appendix Aafter the conclusion.
2. Preliminaries
If the Lie derivative of the Riemann curvature tensor, along a vector field
ξ
, vanishes
i
.
e
., £
ξRh
ijk =
0, then it is called a curvature collineation (CC), which was introduced by
Katzin et al. [
5
] in 1969. The Ricci collineation (RC) is obtained by the contraction of the
expression £ξRh
ijk =0 and is given by £ξRi j =0.
Conformal motion (Conf M) along a vector ξis defined in the following manner:
¯hij =£ξgi j =2αgij ,α=α(xi), (4)
where
α
is the conformal function on
(V4
,
g)
and
ξ
is called the conformal Killing vector
(CKV). If αsatisfies the condition
α;ij =0and α;i6=0, (5)
then
ξ
is the special conformal Kiling vector field (SCKV), where the semi-colon (;) repre-
sents the covariant differentiation. The next subclass is homothetic motion (HM), if
α;i=
0
and motion (M), if α=0.
The projective collineation (PC) satisfies £
ξWh
ijk =
0, where
Wh
ijk
denotes the Weyl
projective curvature tensor in (V4,g)and is defined as follows:
Wh
ijk =Rh
ijk +1
3[δh
jRik δh
kRij ]. (6)
The projective collineation is defined in another way by a vector field ξsatisfying
£ξΓi
jk =δi
jρk+δi
kρj, (7)
where
ρi=iρ
for a scalar field
ρ
,
Γi
jk
are the components of the Christoffel symbol of the
Riemannian metric g and δi
jstands for the Kronecker delta.
The curvature inheritance (CI) ([
2
,
3
]) along a vector field
ξ
is defined on the Rieman-
nian space as:
£ξRh
ijk =2αRh
ijk , (8)
where
α
is an inheritance function of spacetime coordinates and vector field
ξ
is called the
curvature inheritance vector and is abbreviated as (CIV). Similarly, the Ricci inheritance
(RI) is defined as
£ξRij =2αRi j, (9)
The vector field
ξ
is called the Ricci inheritance vector (RIV). As we know that every
CIV is a RIV, and from [2], we have
£ξRi
j=2αRi
jRi
l¯hl
j(10)
Universe 2021,7, 505 4 of 21
and
£ξR=2αRRij ¯hi j, (11)
where
¯hij =£ξgi j =ξi;j+ξj;i. (12)
The study of the exact solutions of the Einstein field equation and related conservation
laws is carried out with symmetry assumptions on spacetime. In addition, such a study is
carried out by numerous authors by adopting various methods (cf., [1,5,14]).
The introduction of the conharmonic transformation as a subgroup of the conformal
transformation was given by Ishii [15] and defined the following transformation,
gij =gi je2σ, (13)
where σstands for the scalar function and also the following condition holds:
σi
;i+σ;iσi=0. (14)
On spacetime
(V4
,
g)
, a quadratic Killing tensor is a generalization of a Killing vector
and is defined as a second-order symmetric tensor Aij [16] satisfying the condition:
Aij;k+Ajk;i+Aki;j=0. (15)
A vector field
ξ
in a semi-Riemannian space is said to generate a one-parameter group
of curvature collineations [17] if it satisfies:
£ξR=0. (16)
A Riemannian space is conformally flat [18] if
Ch
ijk =0, (n>3). (17)
A Riemannian space is conharmonically flat [16] if
Zh
ijk =0, (n>3). (18)
3. Conharmonic Curvature Inheritance Symmetry
A (1, 3)-type conharmonic curvature tensor
Zh
ijk
, which is unaltered under the conhar-
monic transformation
(13)
and
(14)
, can be explicitly expressed as the following
equation [19]:
Zh
ijk =Rh
ijk +1
2(δh
jRik δh
kRij +gik Rh
jgij Rh
k). (19)
We introduce the notion of conharmonic curvature inheritance symmetry as follows.
Definition 1.
On spacetime
V4
with Lorentzian metric
g
, a smooth vector field
ξ
is said to generate
a conharmonic curvature inheritance symmetry if it satisfies the following equation:
£ξZh
ijk =2αZh
ijk , (20)
where α=α(xi)is an inheritance function.
Proposition 1. If a spacetime (V4,g)admits the following symmetry inheritance equations:
(a)£ξRh
ijk =2αRh
ijk ,
(b)£ξgij =2αgi j,
then that spacetime necessarily admits Conh CI along a vector field ξ.
Universe 2021,7, 505 5 of 21
Proof.
The proof is obtained directly by taking the Lie derivative of the Equation
(19)
,
andusing above symmetry inheritance equations we have £
ξZh
ijk =
2
αZh
ijk
. Thus, spacetime
admits Conh CI along a vector field ξ.
Example 1. Consider the following line element of a de Sitter spacetime:
ds2=dt2+e2λt(dx2+dy2+dz2), (21)
where
λ
is a constant. This line element admits a proper CKV,
ξi
= (
eλt
, 0, 0, 0), for which
α=λeλt
.
A straightforward computation of the components
Rh
ijk
, and then taking the Lie derivative with
respect to
ξ
, indicates that
ξ
is a CIV and, therefore, an RIV. Thus, this example of
(V4
,
g)
with
the above metric is compatible with Proposition 1, i.e., de Sitter spacetime satisfying the Conh
CI symmetry.
In this research article, we are considering the inheritance function as being the
same as the conformal function. If
α
= 0, then
(20)
reduces to £
ξZh
ijk =
0, which is called
conharmonic curvature collineation (Conh CC) [4]. Contracting (20), we obtain
£ξZij =2αZij, (22)
where
Zij
denotes the contracted conharmonic curvature tensor on a spacetime
(V4
,
g)
[
16
],
and it is invariant under the transformation (13).
Definition 2.
On spacetime
(V4
,
g)
, a smooth vector field
ξ
is said to generate a contracted
conharmonic curvature inheritance symmetry if it satisfies the Equation (22).
Thus, in general, every Conh CI vector is a contracted Conh CI vector, but its converse
may not hold. In particular, if α= 0, (22) reduces to
£ξZij =0. (23)
Definition 3.
A vector field
ξ
satisfying
(23)
is called a contracted conharmonic curvature
collineation vector field.
If
α6=
0, then a vector field
ξ
satisfying
(22)
is called a proper contracted Conh CI
vector. Contracting Equation (19), we obtain
Zij =1
2gij R. (24)
Lemma 1.
If a spacetime
(V4
,
g)
admits the contracted conharmonic curvature tensor, then the
scalar curvature of the spacetime (V4,g)will be constant.
Proof.
Recently, ref. [
16
] U. C. De, L. Velimirovic and S. Mallick studied the characteristics
of the contracted conharmonic curvature tensor (
Zij
) as follows: “In a spacetime, the
contracted conharmonic curvature tensor is a quadratic Killing tensor”, or it can be written
as
Zij;k+Zjk;i+Zki;j=
0 with the use of Equation
(15)
. They also stated that “a necessary
and sufficient condition for contracted conharmonic curvature tensor [to] be a quadratic
Killing tensor is that the scalar curvature of the spacetime be constant”. Now, using
Equation (24) in Zij;k+Zjk;i+Zki;j=0, we obtain
R=constant. (25)
This completes the proof.
Universe 2021,7, 505 6 of 21
Remark 1.
On the Lie derivative of
(24)
along a proper conformal Killing vector field
ξ(4)
, and
using (25), we can easily show that Equation (22)is well defined on spacetime (V4,g).
Theorem 1.
If a spacetime
(V4
,
g)
admits Conh CI along a vector field
ξ
, then the following
identities hold: (a)£ξZij =2αZij,
(b)£ξZi
j=2αZi
jZk
j¯hi
k,
(c)£ξR=0.
Proof.
Contracting Equation
(20)
, we obtain £
ξZij =
2
αZij
, which proves (a) and implies
that every Conh CI is a contracted Conh CI. The proof of (b) follows by £
ξZij =
£
ξ(gjk Zk
i)
and the use of Equation
(4)
, which leads to £
ξZij =gjl (
£
ξZl
i+¯hl
kZk
i)
. Now, comparing
with part (a) and the rearrangement, we obtain the required result (b). Since spacetime
(V4
,
g)
also admits the conharmonic curvature tensor, and, in general, every conharmonic
curvature tensor
(Zh
ijk )
is a contracted conharmonic curvature tensor
(Zij )
., under the
hypothesis of Lemma 1, this implies that the scalar curvature is constant. Now, following
the Lie derivative of Equation (25) proves part (c).
Remark 2.
Clearly, under the hypothesis of Theorem 1, spacetime
(V4
,
g)
generates a one-parameter
group of curvature collineation [17].
In the empty spacetime
(Rij =
0
)
, the tensors
Rh
ijk
and
Zh
ijk
are identical. This implies
that, in empty spacetime, Conh CI reduces to curvature inheritance symmetry.
Now, here, we obtain the result on the symmetry inheritance for the spacetime admit-
ting the conformal curvature tensor under consideration of Conh CI.
Theorem 2.
If a spacetime (
V4
,
g
) admits the conharmonic curvature inheritance symmetry along
a vector field ξ, the conformal curvature tensor satisfies the symmetry inheritance property.
Proof. The conformal curvature tensor is
Ch
ijk =Rh
ijk +1
2(δh
jRik δh
kRij +Rh
jgik Rh
kgij ) + R
6(gij δh
kgik δh
j), (26)
and this expression is also written in terms of Zh
ijk and Zi j as
Ch
ijk =Zh
ijk +1
3(Zikδh
jZij δh
k). (27)
Taking the Lie derivative of (27) and using (20) and (22), we obtain
£ξCh
ijk =2αCh
ijk . (28)
This completes the proof.
Now, we state Theorem 3(e) from [
2
], i.e., “If a spacetime (
V4
,
g
) admits a CI, then the
following identity holds:
£ξCh
ijk =2αCh
ijk +Dh
ijk , (29)
where
Dh
ijk =1
2[Rh
j¯hik Rh
k¯hij +Rl
k¯hh
lgij Rl
j¯hh
lgik ] + 1
6[δh
k(R¯hij R0gi j)δh
j(R¯hik R0gik )]
and R0=2Ri
jξj
;i”.
The above result raises the following open problem [
13
]: “Find condition on (
V4
,
g
),
with a proper CI symmetry such that
Dh
ijk
vanishes”. From Theorem 2, we solve the above
open problem for the spacetime (
V4
,
g
) to admit proper Conh CI. If a spacetime admits
Conh CI symmetry, then
(29)
is singled out as free from the term
Dh
ijk
. At this point, we
Universe 2021,7, 505 7 of 21
mention that Conh CI is very important in the comparison of the CI symmetry; it restricts
(V4,g) to a very limited geometrical use, as well as physical use.
Remark 3.
The Theorem 2gives us a motivation of the Conh CI symmetry of spacetime, since it
implies the conformal curvature inheritance symmetry
(28)
. On the other hand, the CI does not
imply the conformal curvature inheritance symmetry.
Now, we shall investigate the role of such a symmetry inheritance for the spacetime
admitting the Weyl projective curvature tensor(Wh
ijk ).
Theorem 3.
Under the hypothesis of Proposition 1, if a spacetime (
V4
,
g
) admits the Weyl projective
tensor with Conh CI along a vector field
ξ
, then the Weyl projective tensor also holds the symmetry
inheritance property.
Proof.
Let a spacetime (
V4
,
g
) admit the Weyl projective tensor with a Conh CI along a
vector field ξ; this tensor is expressed as
Wh
ijk =Rh
ijk +1
3[δh
jRik δh
kRij ]. (30)
Taking the Lie derivative of (30), we have
£ξWh
ijk =£ξRh
ijk +1
3[δh
j(£ξRik )δh
k(£ξRik )]. (31)
Further, from Proposition 1, (V4,g) also admits a CIV and RIV, so we have
£ξWh
ijk =2αWh
ijk . (32)
This completes the proof.
Theorem 4. If a spacetime admits a Conh CI along vector ξ, then it satisfies the condition
¯hij;kl ¯hij;lk =0. (33)
Proof. As we know that the conharmonic curvature tensor satisfies the identity
Zjklm +Zkjlm =0. (34)
we can also write
Zi
klm gi j +Zi
jlm gik =0. (35)
Taking the Lie derivative of (35), using Equations (20) and (4), we obtain
Zi
klm ¯hi j +Zi
jlm ¯hik =0. (36)
Now, using the expression of Zh
ijk and Equation (4) in Equation (36), we obtain
Ri
klm ¯hi j +Ri
jlm ¯hik =0. (37)
Applying the Ricci identity [
9
] on
(37)
, we obtain
(33)
, which completes the proof.
Remark 4. If we multiply by ggi l gjk in (33), we obtain the Komar’s identity [20]
[g(ξi;jξj;i)];ji =0= [[g(ξi;jξj;i)];j];i, (38)
where
g
= det(
gij
) and Equation
(33)
is a necessary condition for a Conh CI and is also independent
of the inheritance function αof (20), and is the same as for CC and CI.
Universe 2021,7, 505 8 of 21
Komar’s identity directly interplays in the conservation law generator in general relativity [
20
],
where (
V4
,
g
) admits curvature symmetry properties. As Komar’s identity holds for all vector fields
ξon V4
¯hij;kl ¯hij;lk =0, (39)
for a CC, CI plays no restriction on this symmetry vector
ξ
. Hence, Conh CI are the necessary
symmetry properties of spacetime (
V4
,
g
) that are embraced by the group of general curvilinear
co-ordinate transformations in V4.
Furthermore, following the condition that Equation
(39)
is independent of the scalar
function
α
in a
(20)
, we observe that Conh CI retains this conharmonic transform character-
istics of the Conh CC of the spacetime geometry.
4. Relationship of Conh CI with Other Symmetries of Spacetime
In this section, we describe a relationship of Conh CI with other well-known symme-
tries of spacetime, such as conformal motion (Conf M) and conharmonic motion (Conh M).
We also obtain many results on the relationship between these symmetries. First, we give
the introduction and its characteristics’ results of those symmetries of spacetime, which are
required for the development of the present research work.
4.1. Conformal Motion
A spacetime
(V4
,
g)
admits Conf M [
5
] along a (CKV)
ξ
if the following equation
is satisfied,
¯hij =£ξgi j =2αgij , (40)
where
α=1
4ξk
;k
. If
α;ij =
0,
α;i6=
0, then
ξ
is called a (SCKV). Other CKVs are the homothetic
motion (HM) if α;i=0, α6=0 and the motion (M) if α=0.
A
(V4
,
g)
is said to admit a conformal collineation (Conf C) if a vector exists
ξsuch that
£ξΓi
jk =δi
jα;k+δi
kα;jgjk gil α;l, (41)
and along the vector field
ξ
, a Weyl conformal collineation (W Conf C) is said to be admitted
by a spacetime if
£ξCh
ijk =0. (42)
Every Conf M implies Conf C and W Conf C, but the converse is not necessarily true.
Further, we have the condition [2],
£ξRij =αgi j 2α;ij , (43)
where is the Laplacian–Beltrami operator defined by α=gij α;ij .
Theorem 5.
If a Conh CI vector
ξ
is also a conformal Killing vector (CKV) on a spacetime
(V4,g), then
(a)α;ij =1
3αRij ,
(b)α+1
3αR=0, (44)
(c)α;ij =1
3α(3Rij +Zij).
Proof. For a Conh CI, Proposition 1implies the following equation:
£ξRij =2αRi j. (45)
Universe 2021,7, 505 9 of 21
Since
ξ
is also a CKV, it must satisfy
(43)
. Thus, comparing Equations
(43)
and
(45)
,
we obtain
2αRij =αgi j 2α;ij , (46)
after simplification,
(46)
reduces to the
(44)
(a). Multiplying both side of
(44)
(a) by
gij
, we
obtain (44)(b). The proof of (44)(c) follows from Equations (46), (44)(b) and (24).
Remark 5.
All of the results of the Theorem 5are very useful in the further study of conharmonic
motion (Conh M) and Conh CI symmetry in the context of the space time of general relativity. They
have a direct role as applications in the anisotropic, perfect and imperfect fluid spacetimes.
Corollary 1.
If a spacetime
(V4
,
g)
admits the Conh CI with
ξ
as a conformal Killing vector, then
the conformal curvature tensor vanishes.
Proof.
The proof follows from Equations
(28)
,
(42)
and
(17)
. It may be noted that, for a CKV,
the conformal curvature tensor vanishes in (V4,g), i.e., spacetime is conformally flat.
Example 2.
It is well known that the Weyl conformal curvature tensor
Ch
ijk
= 0 if the spacetime is
conformally flat. By definition, the line element of a conformally flat spacetime can be written as
ds2=f2(t,x,y,z)(dt2+dx2+dy2+dz2).
All conformally flat solutions with a perfect fluid, an electromagnetic field or a pure radiation
field are known.
Corollary 2. If spacetime admits M, HM or SCKV, then Conh CI must be a Conh CC.
Proof.
In particular, a relationship of Conh CI with curvature collineation (CC) is described
by the hypothesis of Theorem 5. Since, in a
(V4
,
g)
, every motion (M) is a CC; therefore,
every HM and SCKV is also a CC. Thus, Conh CI must be a Conh CC when taking the Lie
derivative of Equation (19).
Now, we discuss Conh CI in Einstein spaces.
Theorem 6.
Every proper Conh CI in an Einstein space with a non-zero scalar curvature is a
proper Ricci inheritance.
Proof. Let (V4,g)be an Einstein spacetime with a non-zero scalar curvature,
Rij =R
4gij ,R=constant. (47)
Comparing Equation (47) with (24), we obtain
Zij =2Rij. (48)
Taking the Lie derivative of (48) and using (22), we obtain
£ξRij =2αRi j. (49)
Thus, the Einstein spaces admit the Ricci inheritance symmetry.
Corollary 3.
Under the hypothesis of Theorem 6, if
ξ
is a Ricci inheritance vector (RIV), then the
associated Conh CI must be a proper Conf M with the conformal function α.
Theorem 7.
An Einstein spacetime admits Conh CI along a vector field
ξ
if
ξ
is a curvature
inheritance vector (CIV).
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Proof. Let (V4,g)be an Einstein spacetime that admits a Conh CI vector ξ, i.e.,
£ξZh
ijk =2αZh
ijk . (50)
Now, using Theorem 6, we obtain
£ξRij =2αRi j. (51)
Again, by virtue of Corollary 3,
£ξgij =2αgi j. (52)
Now, Rh
ijk can be expressed as follows:
Rh
ijk =Zh
ijk 1
2(δh
jRik δh
kRij +Rh
jgik Rh
kgij ). (53)
Taking the Lie derivative of (53) and using Equations (50)–(52), we obtain
£ξRh
ijk =2αRh
ijk . (54)
Thus, Conh CI reduces to a CI.
The converse part is also obvious: if an Einstein spacetime admits CI symmetry along
a vector field ξ, i.e.,
£ξRij =2αRi j. (55)
From Duggal (cf., [
2
], p. 2992), if an Einstein spacetime admits proper CI, then the
spacetime also admits proper Conf M, i.e.,
£ξgij =2αgi j. (56)
taking the Lie derivative of Equation
(19)
and using Equations
(8)
(10)
and
(56)
. Thus, we
conclude that CI reduces to a Conh CI.
Now, we derive a necessary condition for Conh CI symmetry, admitted by a spacetime
that is not an Einstein spacetime.
Theorem 8.
A necessary condition for spacetime
(V4
,
g)
admitting Conh CI is that the spacetime
admits both CI and Conf M together.
Proof.
We let spacetime
(V4
,
g)
admit
ξ
as the CIV on it; this implies that
ξ
satisfies
Equations
(8)
and
(9)
. Moreover, it is known (cf., [
2
], p. 2991) that if the spacetime admits
CI, then the following identity holds:
£ξRi
j=2αRi
jRi
l¯hl
j. (57)
Taking the Lie derivative of
(19)
and using Equations
(8)
,
(9)
,
(57)
and the equation of
Conf M, we see that the spacetime admits Conh CI symmetry along the vector field
ξ
, i.e.,
£ξZh
ijk =2αZh
ijk . (58)
This completes the proof.
Remark 6.
In general, the converse of Theorem 8is not true, while the converse holds if
(V4
,
g)
is
an Einstein spacetime. We conclude that the advantage of the selection of the Einstein spacetime is
the relaxation of the condition for Conf M.
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Example 3.
Let
(V4
,
g
) be an Einstein spacetime admitting a Conh CI, which implies admitting
the CI, as well as Conf M. Then, following on from Corollary 3using
(47)
in the first result of
Theorem 5, we obtain
α;ij = (αR
12 )gij , (59)
where
α
and
R
are both scalar functions of spacetime co-ordinates. We consider the single scalar
function φinstead of (αR
12 )in (59), and then we obtain
α;ij =φgi j. (60)
Petrov [
21
] referred to a finding of Sinyukov [
22
] that explains that, if a spacetime
(V4
,
g)
admits a vector field
φ;i
satisfying
(60)
for
φ6=
0, then a system of co-ordinates exists where the
metric has the form:
ds2=g11 dx1dx1+ ( 1
g11
)Γab(x2,x3,x4)dxadxb, (61)
where a, b 6=1and g11 = [2Rφ(x1)dx1+C]1, and the arbitrary function φ=φ(x1).
The above example of
(V4
,
g)
with metric
(61)
is well suited for Theorems 68and
Corollary 3.
4.2. Conharmonic Motion
Abdussatar and Babita Dwivedi [
4
] introduced a new type of conformal symmetry
called conharmonic symmetry. Conharmonic motion (Conh M) is defined through a
definition of Conf M (40) as follows:
α=gij α;ij =0, α;ij 6=0. (62)
Similarly, a Conh CC is defined through Conf C if Equation
(41)
holds with the
condition (62)
.
If a vector field ξsatisfies
£ξZh
ijk =0, (63)
then
(V4
,
g)
admits a conharmonic curvature collineation (Conh CC), and
ξ
is also known
as a conharmonic Killing vector (Conh KV). Clearly, every conharmonic motion is a Conh
CC, but the converse is not true in general. From Equation
(28)
, it is evident that every
Conh CC is a W Conf C. Every Conh M satisfies
£ξRh
ijk =δh
jα;ik δh
kα;ij +αh
;jgik αh
;kgij , (64)
£ξRij =2α;i j, (65)
£ξRj
k=2αj
;k2αRj
k, (66)
£ξR=2αR. (67)
Multiplying by gij in Equation (65) and in view of (62), we observe that
gij £ξRij =0. (68)
Thus, we can say that every Conh M reduces to a contracted Ricci collineation, but that the
converse is not true.
We also have the following:
Theorem 9.
If the spacetime
(V4
,
g)
admits a Conh CI as well as a proper Conh M, then the scalar
curvature of spacetime vanishes.
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Proof.
Comparing
(67)
with Equation (c) in Theorem 1, we obtain
2
αR=
0; that is, the
scalar curvature Rof the spacetime vanishes.
Remark 7.
Now, here, we will be discussing the motivation of Theorem 9. From [
2
], every CIV
is a RIV , but the converse is not true in general (for further details, see Theorem 3.2 in [
4
] and
Propositions 1 and 2 in [
23
]). If spacetime admits a Conh CIV that is also a RIV, then every RIV is
a CIV. This information was not available to Sharma, R. and Duggal, K.L. et al. [
23
] in 1994, when
they introduced CI. This is certainly an improvement over the use of Conh curvature symmetries
because the proper CIV exists together with the proper CKV, which has greater physical significance.
Moreover, we have the following result:
Theorem 10. If ξis a Conh CIV as well as a RIV, then
£ξRh
ijk =2αRh
ijk . (69)
Proof.
Using the Lie derivation of Equation
(19)
with respect to
ξ
, and then using the
inherited symmetry properties of Rij ,Zh
ijk and gi j, we obtain
£ξRh
ijk =2αRh
ijk , (70)
i.e., the Riemann curvature tensor is inherited in spacetime.
Next, we also have
Theorem 11.
If a spacetime
(V4
,
g)
admits proper Conh CI along a conharmonic Killing vector
ξ
,
then that spacetime is conharmonically flat.
Proof. Zh
ijk is expressed as
Rh
ijk +1
2(δh
jRik δh
kRij +gik Rh
jgij Rh
k). (71)
Taking the Lie derivative of (71), along the vector field ξ,
£ξZh
ijk =£ξ(Rh
ijk ) + 1
2(δh
j£ξ(Rik )δh
k£ξ(Rij ) + £ξ(gik Rh
j)£ξ(gij Rh
k)). (72)
Since the spacetime admits Conh CC and Conh M, then, using Equations
(8)
(10)
and (40) in Equation (72), we obtain
£ξZh
ijk =0. (73)
Now, applying the Conh CI Equation (20), we obtain
Zh
ijk =0(since α6=0). (74)
Thus, spacetime is conharmonically flat.
Corollary 4.
If a spacetime
(V4
,
g)
admits proper Conh CI and Conh CC, then the spacetime is
conharmonically flat.
Proof. The proof directly follows from Equation (63).
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Example 4.
We consider a plane symmetric perfect fluid cosmological model obtained by Singh and
Singh [
24
] that does not have a conformally flat spacetime. The geometry of this model is defined by
the line element
ds2= (1+at)2[dt2+dx2+dy2]+(1+bt)dz2
where a and b are non-zero arbitrary constants. The above line element is found to admit a CIV,
which is also a Conh CC
ξi= (A/a)δi
0
when a=b with
α=A
(1+at)
, where A is an arbitrary
constant. However, when
a=b
, the model becomes conharmonic to flat spacetime and reduces to a
special case
(k=
0
)
of the Friedmann–Robertson–Walker ( FRW) model, representing a universe
filled with disordered radiation.
Theorem 12. If a spacetime (V4,g)admits Conh M, then
£ξZij =0. (75)
Proof. Let a spacetime (V4,g)admit a conharmonic curvature tensor; then,
Zij =1
2gij R. (76)
Now, taking the Lie derivative of Equation (76),
£ξZij =1
2[(£ξgij)R+gi j(£ξ(R)], (77)
using Equation (67) with the condition of conharmonic motion, we obtain
£ξZij =0. (78)
This implies that spacetime
(V4
,
g)
admits contracted conharmonic curvature
collineation.
5. Physical Interpretation to Fluid Spacetimes of General Relativity
In this section, we consider different types of fluid spacetimes as applications of Conh
CI. If
(V4
,
g)
is a spacetime of the general theory of relativity with imperfect fluid (heat
conducting and viscous) and a stress-energy tensor of the form:
Tij =µuiuj+phij 2σijη+uiqj+ujqi, (79)
where projection tensor
hij =gi j +uiuj
and shear viscosity coefficient
η
is non-negative,
and the term (2
σijη+uiqj+ujqi
) in Equation
(79)
vanishes if
σij =
0 and
qi=
0 separately,
then Equation (79) represents the stress-energy tensor for perfect fluid spacetime, i.e.,
Tij = (µ+p)uiuj+pgi j. (80)
In anisotropic fluid spacetime, the stress-energy tensor is of the form:
Tij =µuiuj+pPij +pkninj, (81)
where
pk
and
p
are the parallel and perpendicular components of the isotropic pressure
to a unit vector
ni
orthogonal to
ui
, respectively.
Pij =hi j ninj
is the projection tensor
onto the two orthogonal planes of vectors uiand ni.
If
p=1
3(pk+
2
p)
and 2
σijη= ( 1
3hij ninj)( pkp)
, then the form of the energy
momentum tensor in anisotropic fluid is identical to imperfect fluid with qi=0.
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Since self similar imperfect fluid spacetime admits homothetic vector
ξi
, i.e., self
similarity is imposed on Equation (79), then the following equation holds [2]:
(a)£ξµ=2αµ,(b)£ξp=2αp,(c)£ξui=αui,
(d)£ξσij =ασij,(e)£ξη=αη,(f)£ξqi=αqi. (82)
From
(82)
, we conclude that all physical quantities (
µ
,
p
,
ui
,
σij
,
η
,
qi
) inherit the spacetime
symmetry defined by
ξi
. Tupper and Coley [
10
] have investigated the conditions for an
imperfect fluid to inherited symmetry
(82)
for a SCKV. Saridakis [
25
] et al. have solved the
problem of symmetry inheritance for a spacelike proper CKV and other types of symmetry.
Furthermore, Duggal [
2
] has also investigated the conditions for imperfect fluid, perfect
fluid and anisotropic fluid to inherited symmetry
(82)
for a CIV, and Z. Ahsan [
26
] has
investigated the necessary and sufficient conditions for perfect fluid spacetimes to admit
Ricci inheritance symmetry.
We shall now consider spacetimes that admit a CKV ξi, i.e.,
£ξgij =2αgi j, (83)
where
α(xi)
is the conformal function. As this is known for a CKV
ξ
in fluid spacetime,
then the following result holds [27]:
£ξui=αui+vi, (84)
where
vi
is the spacelike vector orthogonal to
ui
, i.e.,
uivi
=0. Maartens [
27
] et al. have
shown that vi6=0 generally, and is given by
vi=2ξjωij +β˙
uihj
iβ,j, (85)
where the vorticity tensor is denoted by
ωij
and
β=uiξi
. Fluid flow lines are mapped
onto fluid flow lines by the action of
ξi
if
vi=
0. They are also said to be “frozen in” curves
to the fluid.
For a CKV ξi[10], the following results hold :
£ξRij =αgi j 2α;ij , (86)
£ξR=2αR6α, (87)
£ξTij =2(αgij α;ij), (88)
and the Einstein field equations are in the form
Gij =Rij 1
2Rgi j =Tij. (89)
In this section, we shall prove some results for the perfect fluid, imperfect fluid and
anisotropic fluid on spacetime (V4,g)that admit the Conh CI vector ξi.
Theorem 13.
Let an imperfect fluid spacetime admit Conh CI symmetry along a vector field
ξ
,
where fluid flow lines are mapped conformally by ξ. Then, the following equations hold:
(a)£ξµ=0(b)£ξp=0(c)£ξui=αui, (90)
(a)£ξσij =ασij (b)£ξη=αη (c)£ξqi=αqi. (91)
Proof. The contraction of the Einstein field Equations (89) leads to
T=R or R =T, (92)
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Similarly, from Equation (79),
R=µ3p. (93)
Now, using the dynamic result for
Tij
of imperfect fluid by Equation
(79)
, it leads to
(cf., [2]), i.e.,
£ξµ=2α242αµ 2α;ij uiuj, (94)
where
4=qivi
. It is seen that the fluid flow lines are mapped conformally by
ξi
. This
implies that vi=0. Hence, 4=0 and Equation (94) reduces to
£ξµ=2α2αµ 2α;ijuiuj. (95)
For imperfect fluid, when using (EFE)
(89)
with conditions
uiui=
1,
σijui=
0 and
qiui=0, we obtain
Rij uiuj= (µR
2) = ( 3p+µ
2). (96)
If we set,
α;ij =α
2[R
3gij 2Ri j], (97)
then, from [
2
], every CIV is also a CKV. Theorem 8implies that spacetime admits Conh CI
symmetry. If we multiply Equation
(97)
by
uiuj
, and using Equation
(96)
and
uiui=
1 (
ui
is timelike), then we obtain
α;ij uiuj=α(R
3µ) = α
3(2µ+3p). (98)
In view of
(44)
(b), Equations
(95)
and
(98)
yield £
ξµ=
0; this implies that
µ
is constant
under Lie differentiation. The proof of Equation (90)(b) follows from
£ξp=2
34+4
3α2αp2
3α;ij uiuj, (99)
and 4=0, (44)(b), (93) and (98).
Using vi=0 in Equation (84), we obtain (90)(c).
Moreover, from [2], it follows that,
vi=0£ξσij =ασij, (100)
which proves Equation
(91)
(a). For imperfect fluid spacetime (with
Tij
of the form
(79)
), we
have [2]
£ξ(σij η) = (αη +£ξη)σij =α(2µ+R)
6gij αRi j +α(4µR)
3uiuj+α(qiuj+qjui). (101)
Contracting Equation
(101)
with
σij
and using
(79)
,
σijui=
0,
σijgij =
0 and Einstein
field Equation (89), we obtain
(£ξη+αη)(2σ2) = 4αησ2, where σijσi j =2σ2, (102)
which leads to £ξη=αη i.e., (91)(b) is proved. Finally, we prove (91)(c):
qi(Q1£ξQ) = wiwhere Q=qiqiand wiqi=0. (103)
Since the Tij of imperfect fluid is represented by Equation (79), we have [10]
£ξqi= (Q1£ξQ+α)qi+wi, (104)
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from (103), Equation (104) leads to £ξqi=αqi.
Theorem 14.
Let an imperfect fluid spacetime admit a Conh CIV
ξi
with
(p+µ)6=
0 and
qi=0. Then,
(a) An eigenvector of α;i j is ui;
(b) ξiis conformally mapped by fluid flow lines.
Proof.
For an imperfect fluid, using the Einstein field equation
(89)
with conditions
uiui=1, σijui=0 and qiui=0, we obtain
Rij uj=(µR
2)ui=(µ+3p
2)ui. (105)
Notice that, from Equation
(105)
,
ui
is a timelike eigenvector of
Rij
. After multiplying
ui
in
Equation (97), and from (105) and (93), we obtain
α;ij uj= ( α
3)[3p+2µ]ui, (106)
which shows that
ui
is an eigenvector of
α;ij
; this proves the first part of the theorem. Now,
using Equation
(90)
(c) in
(84)
, we obtain
vi=
0, i.e, the vector
ξ
is conformally mapped by
fluid flow lines, and, hence, the proof of part (b) is complete.
Theorem 15.
Let a perfect fluid spacetime
(V4
,
g)
admit a Conh CIV
ξ
with
(p+µ)6=
0; then,
the following equations hold:
(a)£ξµ=0(b)£ξp=0. (107)
Proof. First, contracting Equation (89), we obtain
T=R or R =T, (108)
and then, contracting Equation (80), we obtain
R=µ3p. (109)
Next, we use a dynamic result for perfect fluid with
Tij
of the form
(80)
along a CKV vector
field ξithat was derived by Duggal in [2]:
£ξµ=2α2αµ 2α;ijuiuj. (110)
In a perfect fluid spacetime, using the (EFE)
(89)
with conditions
uiui=
1,
σijui=
0
and qiui=0, we obtain
Rij uiuj= (µR
2) = ( 3p+µ
2). (111)
If we multiply both sides by
uiuj
in
(97)
and use Equation
(111)
and
uiui=
1 (
ui
is
timelike), then we obtain
α;ij uiuj=α(R
3µ) = α
3(2µ+3p). (112)
Now using Equations
(44)
(b) and
(112)
in
(110)
, we obtain £
ξµ=
0, i.e.,
Equation (107)(a)
holds. Equation (107)(b) follows from
£ξp=4
3α2αp2
3α;ij uiuj. (113)
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Moreover, the use of Equations
(44)
(b),
(112)
and
(109)
in Equation
(113)
establishes
the proof.
Theorem 16. Let a perfect fluid spacetime admit a Conh CIV ξiand (p+µ)6=0. Then,
(a)
An eigenvector of α;ij is ui;
(b)
Fluid flow lines are mapped conformally along the vector field ξi;
(c)
£ξui=αui.
Proof.
The proof of the first part (a) is the same as the proof of the first part of Theorem 14.
Now, we prove the second part of the theorem. By applying a dynamic result for a Conh
CI vector in perfect fluid spacetime, we have [2]
(p+µ)vi=2[(α;kl ukul)ui+α;ik uk]. (114)
Now, using Equations (106) and (112) in (114), we obtain
(p+µ)vi=0vi=0(as, µ+p6=0). (115)
Finally, using Equation (115) in Equation (84), we obtain £ξui=αui.
Now, we conclude that, by vector field
ξ
, the fluid flow lines are mapped conformally
to Conh CI admitted by perfect fluid spacetime; consequently, the four-velocity vector
(ui)
is also inherited.
Theorem 17.
Let anisotropic fluid spacetime
(V4
,
g)
admit a Conh CIV
ξ
with
(P+µ)6=
0and
(Pk+µ)6=0; then, the following equations hold:
(a)£ξµ=0, (b)£ξPk=0, (c)£ξP=0, (116)
(d)£ξui=αui,i.e., vi=0. (117)
Proof.
For anisotropic fluid spacetime, the stress energy tensor is given by Equation
(81)
.
Now multiplying both sides of Equation (89) by uiujand ninj, we obtain
Rij uiuj= (µR
2)(118)
and
Rij ninj= ( R
2+pk)(119)
respectively. Moreover, from Equations (97) and (118), we have
α;ij uiuj=α(R
3µ). (120)
Since , for anisotropic fluid, µmust satisfy the following [2],
£ξµ=2α2αµ 2α;ijuiuj. (121)
From Theorem 5(a), and Equation
(120)
, Equation
(121)
reduces to
(116)
(a). The proof
of the second part of (116) is followed by combining Equation (97) and (119); therefore,
α;ij ninj=α(pk+R
3). (122)
In anisotropic fluid, pkmust satisfy the following [2]:
£ξpk=2α2αpk2α;ij ninj. (123)
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Again, using Equations
(44)
(a) and
(122)
, Equation
(123)
reduces to
(116)
(b). The
proof of the third part is as follows:
Rij Pi j =2p+R, (124)
and using Equation (97), we obtain
α;ij Pi j =2α(p+R
3). (125)
We also have [2]
£ξp=2α2αp2α;ij Pi j. (126)
If we put the value of
α
and
α;ij Pi j
in Equation
(126)
, then
(116)
(c) holds, as
we know that
α;ij pik uj=0, α;i jnjui=0. (127)
For an anisotropic fluid, we have
(pk+µ)vjnj=2α;ij njui(128)
and
(p+µ)vjpjk =2α;ij pjkui. (129)
Now, by virtue of Equations (127) and (128), Equation (129) reduces to
nivs.i=0 and pij vj=0, where (µ+pk)6=0, (µ+p)6=0. (130)
We conclude that, from the above equations,
ui
and
vj
must be parallel. This result,
combined with viui=0, implies vi=0; thus, from (84), we have £ξui=αui.
Theorem 18.
A perfect fluid spacetime admits Conh CI along a conformal Killing vector field
ξ
and also satisfies the EFE (1); then, the divergence of the conharmonic curvature tensor vanishes.
Proof. Let ξbe a Conh CI vector and also a CKV satisfying (20); then,
(Rij ξj);i=3α. (131)
With the Einstein field Equations (90) and (44)(b), we obtain
((Ti j +R
2gij )ξj);i=αR. (132)
Equation
(132)
explores a new equation of state for various matter. Perfect fluid
spacetime satisfies (80) with ξuor ξku. Then,
((p+R
2)ξi);i=αR. (133)
Now, we use
ξi
;i=
4
α
and Equation
(109)
in the above equation to obtain
p+µ=
0;
therefore, Zh
ijk;h=0 (cf., Theorem (2.1) in [4]).
One can prove a similar result for an anisotropic fluid and imperfect fluid spacetime.
6. Conclusions
The idea of symmetry inheritance for a conharmonic curvature tensor is explored,
and some related results are obtained on the Conh CI with both conformal motion and
conharmonic motion in general and Einstein spacetime. We have obtained the necessary
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conditions for CI and conformal motion to have conharmonic curvature inheritance sym-
metry. We have also derived a result as a physical application for imperfect fluid, perfect
fluid and anisotropic fluid in the spacetime of general relativity. In the last result, it is
concluded that the perfect fluid spacetime becomes either empty/Ricci flat, i.e.,
(p+µ=
0
)
,
or expresses the equation of state for a vacuum-like case, which is not a perfect fluid but is
instead an Einstein spacetime.
Author Contributions:
The authors contributed equally to this work. All authors have read and
agreed to the published version of the manuscript.
Funding:
Institute of Scientific Research and Revival of Islamic Heritage at Umm Al-Qura University,
Saudi Arabia (Project # 43405050).
Acknowledgments:
The authors are grateful to G. S. Hall, University of Aberdeen, Scotland for
helpful discussions and suggestions. We are thankful to Md Danish Iqbal, Department of English,
and S. S. Z. Ashraf, Department of Physics, AMU, for editing the language of the paper. The authors
also wish to thanks the reviewers for constructive comments, which have led to extensive revision
and improvement of the manuscript and acknowledge the finicial support by Institute of Scientific
Research and Revival of Islamic Heritage at Umm Al-Qura University, Saudi Arabia.
Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design
of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript, or
in the decision to publish the results.
Appendix A
Appendix A.1. Application to Cosmology
Siddiqui and Ahsan [
28
] have studied the relativistic significance of conharmoni-
cally flat spacetime. A conharmonically flat spacetime is an Einstein spacetime that is
consequently a space of constant curvature. The significance of the space of constant
curvature is of great interest in the study of the cosmology (for further details, see [
29
]).
For conharmonically flat spacetime, we have Equation (19):
Zh
ijk =Rh
ijk +1
2(δh
jRik δh
kRij +gik Rh
jgij Rh
k) = 0. (A1)
Contracting this, we obtain
Rij =1
4Rgi j. (A2)
Substituting this into the Einstein field Equations (1) with κ=1, we obtain
3Rij =Tij or 3
4Rgi j =Tij. (A3)
Many authors have found solutions to the modified field Equation
(A3)
. However, there is
a very important problem with these solutions.
We illustrate this by means of an example studied by Kumar and Srivastava [
30
]. For
the FRW model,
ds2=dt2+a(t)2[dr2
(1+kr2)+r2(dθ2+sin2θdφ2)], (A4)
the field Equations (A3) yield
¨
a(t) + [ µ
9]a(t) = 0, (A5)
¨
a
a+2˙
a2
a2+2k
a2=p
3. (A6)
In Equations
(A5)
and
(A6)
, p and
µ
denote the pressure and density, respectively, of
the perfect fluid
(80)
, and k is an arbitrary constant. In addition, we see that Equation
(A5)
Universe 2021,7, 505 20 of 21
is satisfied for
a(t) = A cos(µ
3t) + B sin(µ
3t)
. Assuming
k=
2, in the cases (i) A = 1 , B
= 0 (ii) A = 0, B = 1, Equations
(A5)
and
(A6)
have the common solutions when
p+µ=
0.
This implies that the condition of the equation of state occurred for the FRW metric (A4).
If we further contract Equation
(A2)
, we obtain the “vacuum”, i.e.,
Rij =
0. This is
a very strong imposition. Thus, the additional symmetry requirement of conharmonic
flatness reduces the space of solutions to “vacuum” solutions in general relativity.
Appendix A.2. Conh CI with Conservation Law Generator
Under the hypothesis of Theorem 9, spacetime
(V4
,
g)
possesses
R=
0 and a Ricci
tensor
Rij 6=
0 along a Conh Killing vector
ξ
(Conh M) with the condition that
ξ
satisfies
Equation (68). Thus, it follows that
£ξR=£ξ(Rij gi j) = (£ξgij )Rij =0, (A7)
where £ξgij =gik gjl£ξgi j; then, Equation (A7) reduces to
Rkl £ξgkl =0. (A8)
Now, using £ξgkl =ξk;l+ξl;kin (A8), we obtain
Rl
kξk
;l=0. (A9)
From the twice-contracted Bianchi identity [8], we find ( using R=0 )
Rl
k;l=0. (A10)
Combining Equations (A8) and (A10), we obtain
(Rl
kξk);l=0. (A11)
In a spacetime with R=0, the Einstein field Equations (1) take the form
Rl
k=κTl
k, (A12)
where κis a constant and Tl
kis an energy-momentum tensor with trace Tl
l=T=0.
Substituting (A12) in (A11) gives
(gTl
kξk);l= (gTl
kξk),l=0, (A13)
where
g=|det gij |
and
ξk
is defined by Conh CI. Thus, we conclude that, if a space-time
V4
with R = 0 and
Rij 6=
0 admits Conh CI along a Conh Killing vector
ξ
, then there exists a
covariant conservation law generator of the form (A13).
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... To date, more than 30 geometric symmetries have been found in the literature. For a detailed study of symmetry inheritance, see [7][8][9][10]. Many other studies were also conducted on curvature inheritance symmetry in various research subfields of mathematics and physics (for more details, see [11][12][13][14][15][16][17][18]). ...
... Recent studies [8][9][10][11][12][13][14][15][16][17][18][19] exhibited a deep interest in the study of the different symmetries (in particular, curvature, Ricci, projective, matter, semiconformal symmetry [23,24], and conharmonic curvature inheritance [9]). These geometrical symmetries appear strongly beneficial towards the exact solutions of Einstein field Equation (1). ...
... Recent studies [8][9][10][11][12][13][14][15][16][17][18][19] exhibited a deep interest in the study of the different symmetries (in particular, curvature, Ricci, projective, matter, semiconformal symmetry [23,24], and conharmonic curvature inheritance [9]). These geometrical symmetries appear strongly beneficial towards the exact solutions of Einstein field Equation (1). ...
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