Content uploaded by Mohammad Salman

Author content

All content in this area was uploaded by Mohammad Salman on Dec 19, 2021

Content may be subject to copyright.

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

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional afﬁl-

iations.

Copyright: © 2021 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

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]:

£ξΩ=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

Ω

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

Ω=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

Ω

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

j−Ri

l¯hl

j(10)

Universe 2021,7, 505 4 of 21

and

£ξR=2αR−Rij ¯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

j−gij 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

j−Zk

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

k−gik 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

j−Zij 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;j−ξj;i)];ji =0= [[√g(ξi;j−ξj;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;j−gjk 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).

Universe 2021,7, 505 10 of 21

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

j−Ri

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.

Universe 2021,7, 505 11 of 21

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:

ds∗2=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 6–8and

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

;k−2αRj

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.

Universe 2021,7, 505 12 of 21

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

j−gij 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).

Universe 2021,7, 505 13 of 21

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+p⊥Pij +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)( pk−p⊥)

, then the form of the energy

momentum tensor in anisotropic ﬂuid is identical to imperfect ﬂuid with qi=0.

Universe 2021,7, 505 14 of 21

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˙

ui−hj

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=−2αR−6α, (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)

Universe 2021,7, 505 15 of 21

Similarly, from Equation (79),

R=µ−3p. (93)

Now, using the dynamic result for

Tij

of imperfect ﬂuid by Equation

(79)

, it leads to

(cf., [2]), i.e.,

£ξµ=−2α−24−2αµ −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=−2α−2αµ −2α;ijuiuj. (95)

For imperfect ﬂuid, when using (EFE)

(89)

with conditions

uiui=−

1,

σijui=

0 and

qiui=0, we obtain

Rij uiuj= (µ−R

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

3−µ) = −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

3α−2αp−2

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(4µ−R)

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(Q−1£ξQ) = −wiwhere Q=qiqiand wiqi=0. (103)

Since the Tij of imperfect ﬂuid is represented by Equation (79), we have [10]

£ξqi= (Q−1£ξQ+α)qi+wi, (104)

Universe 2021,7, 505 16 of 21

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=−(µ−R

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=µ−3p. (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=−2α−2αµ −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= (µ−R

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

3−µ) = −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

3α−2αp−2

3α;ij uiuj. (113)

Universe 2021,7, 505 17 of 21

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=0⇒vi=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= (µ−R

2)(118)

and

Rij ninj= ( R

2+pk)(119)

respectively. Moreover, from Equations (97) and (118), we have

α;ij uiuj=α(R

3−µ). (120)

Since , for anisotropic ﬂuid, µmust satisfy the following [2],

£ξµ=−2α−2αµ −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=2α−2αpk−2α;ij ninj. (123)

Universe 2021,7, 505 18 of 21

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⊥=2α−2αp⊥−2α;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 ξ⊥uor 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

Universe 2021,7, 505 19 of 21

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

j−gij 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).

References

1.

Hall, G.S. Symmetries and Curvature Structure in General Relativity; World Scientiﬁc Publishing Co. Ltd.: Singapore, 2004;

https://doi.org/10.1142/1729.

2.

Duggal, K.L. Curvature inheritance symmetry in Riemannian spaces with applications to ﬂuid space times. J. Math. Phys.

1992

,

33, 2989–2997. https://doi.org/10.1063/1.529569.

3.

Duggal, K.L. Symmetry inheritance in Riemannian manifold with physical applications. Acta Appl. Math.

1993

,31, 225–247.

https://doi.org/10.1007/BF00997119.

4.

Abdussatar; Dwivedi B. Fluid space—Times and conharmonic symmetries. J. Math. Phys.

1998

,39, 3280–3295. https://doi.org/

10.1063/1.532441.

5.

Katzin, G.H.; Levine, J.; Davis, W.R. Curvature collineation: A fundamental symmetry property of the spacetimes of gen-

eral relativity deﬁned by the vanishing Lie derivative of the Riemann curvature tensor. J. Math. Phys.

1969

,10, 617–629.

https://doi.org/10.1063/1.1664886.

6.

Pundeer, N.A.; Ali, M.; Ahmad, N.; Ahsan, Z. Semiconformal symmetry-A new symmetry of the space-time manifold of the

general relativity. J. Math. Comput. Sci. 2020,20, 241–254. http://dx.doi.org/10.22436/jmcs.020.03.07.

Universe 2021,7, 505 21 of 21

7.

Bertschinger, T.H.; Flowers, N.A.; Moseley, S.; Pfeifer, C.R.; Tasson, J.D.; Yang, S. Spacetime symmetries and classical mechanics.

Symmetry 2019,11, 22. https://doi.org/10.3390/sym11010022.

8.

Yano, K. Integral Formulas in Riemannian Geometry; Marcel Dekker: New York, NY, USA, 1970; https://doi.org/10.1017/

S0008439500031520.

9.

Yano, K. Theory of Lie Derivatives and Its Applications; North-Holland Publishing Company: Amsterdam, The Netherlands, 1957.

Available online: https://archive.org/details/theoryoﬂiederiv029601mbp/page/n73/mode/2up (accessed on 11 December

2021).

10.

Coley, A.A.; Tupper, B.O.J. Special conformal Killing vector spacetimes and symmetry inheritance. J. Math. Phys.

1989

,30,

2616–2625. https://doi.org/10.1063/1.528492.

11.

Coley, A.A.; Tupper, B.O.J. Spacetimes admitting inheriting conformal Killing vector ﬁelds. Class. Quant. Grav.

1990

,7, 1961–1981.

https://doi.org/10.1088/0264-9381/7/11/009.

12.

Coley, A.A.; Tupper, B.O.J. Spherically symmetric spacetimes admitting inheriting conformal Killing vector ﬁelds. Class. Quant.

Grav. 1990,7, 2195–2214. https://doi.org/10.1088/0264-9381/7/12/005.

13.

Duggal, K.L.; Sharma, R. Symmetries of Spacetimes and Riemannian Manifolds; Kluwer Academic Press: Boston, MA, USA; London,

UK, 1999; https://doi.org/10.1007/978-1-4615-5315-1.

14.

Fatibene, L.; Francaviglia, M.; Mercadante, S. Noether symmetries and covariant conservation laws in classical relativistic and

quantum Physics. Symmetry 2010,2, 970–998. https://doi.org/10.3390/sym2020970.

15. Ishii Y. On conharmonic transformations. Tensor 1957,11, 73–80. https://ci.nii.ac.jp/naid/10025303733/.

16.

De, U.C.; Velimirovic, L.; Mallick, S. On a type of spacetime. Int. J. Geom. Methods Mod. Phys.

2017

,14, 1750003.

https://doi.org/10.1142/S0219887817500037.

17.

Duaggal, K.L.; Sharma, R. Hypersurfaces in a conformally ﬂat space with curvature collineation. Internat. J. Math. Math. Sci.

1991

,14, 595–604. Available online: https://www.researchgate.net/publication/26535459_Hypersurfaces_in_a_Conformally_

Flat_Space_With_Curvature_Collineation (accessed on 11 December 2021).

18. Weyl, H. Reine Inﬁnitesimalgeometrie. Math. Z. 1918,2, 384–411. https://doi.org/10.1007/BF01199420.

19.

Mishra, R.S. Structures on a Differentiable Manifold and Their Applications; Chandrama Prakashan: Allahabad, India, 1984. Avail-

able online: https://www.worldcat.org/title/structures-on-a-differentiable-manifold-and-their-applications/oclc/16997067

(accessed on 11 December 2021).

20.

Komar, A. Covariant conservation laws in general relativity. Phys. Rev.

1959

,113, 934. https://doi.org/10.1103/PhysRev.113.934.

21.

Petrov, A.Z. Einstein Spaces; Peragamon Press: Oxford, UK, 1969. Available online: https://www.sciencedirect.com/book/978008

0123158/einstein-spaces (accessed on 11 December 2021).

22.

Sinyukov, N.S. Scientiﬁc Annual; Odessa University: Odessa, Ukraine, 1957. Available online: https://scholar.google.com/

scholar?hl=en&q=N.+S.+Sinyukov%2C+Scientiﬁc+Annual%2C+Odessa+University+%281957%29 (accessed on 11 December

2021).

23.

Sharma, R.; Duggal, K.L. Differential geometry and mathematical physics. Am. Math. Soc. Contemp. Math. Ser.

1994

,170, 215–224.

http://dx.doi.org/10.1090/conm/170/01756.

24.

Singh, K.P.; Singh, D.N. A plane symmetric cosmological model. Mon. Not. R. Astron. Soc.

1968

,140, 453–460.

https://doi.org/10.1093/mnras/140.4.453.

25.

Saridakis, E.; Tsamparlis, M. Symmetry inheritance of conformal Killing vectors. J. Math. Phys.

1991

,32, 1541–1551.

https://doi.org/10.1063/1.529263.

26.

Ahsan, Z. On a geometrical symmetry of the spacetime of general relativity. Bull. Cal. Math. Soc.

2005

,97, 191–200. Available

online: https://www.researchgate.net/publication/353348171_ON_A_GEOMETRICAL_SYMMETRY_OF_THE_SPACE-TIME_

OF_GENERAL_RELATIVITY (accessed on 11 December 2021).

27.

Maartens, R.; Mason, D.P.; Tsamparlis, M. Kinematic and dynamic properties of conformal Killing vectors in anisotropic ﬂuids.

J. Math. Phys. 1986,27, 2987–2994. https://doi.org/10.1063/1.527225.

28.

Siddiqui, S.A.; Ahsan, Z. Conharmonic curvature tensor and the spacetime of general relativity. Diff. Geom. Dyn. Syst.

2010

,12,

213–220. Available online: http://www.mathem.pub.ro/dgds/v12/D12-SI.pdf (accessed on 11 December 2021).

29.

Narlikar, J.V. An Introduction to Relativity; Cambridge University Press: Cambridge, UK, 2010; https://doi.org/10.1017/CBO

9780511801341.

30.

Kumar, R.; Srivastava, S.K. FRW-cosmological model for conharmonically ﬂat spacetime. Int. J. Theor. Phys.

2013

,52, 589–596.

Available online: https://ui.adsabs.harvard.edu/link_gateway/2013IJTP...52..589K/doi:10.1007/s10773-012- 1364-7 (accessed on

11 December 2021).