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.
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-
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://
1Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India; email@example.com
2Department of Mathematical Sciences, Faculty of Applied Sciences, Umm Al Qura University,
Makkah P.O. Box 56199, Saudi Arabia; firstname.lastname@example.org
† 2010 Mathematics Subject Classiﬁcation: 53B20; 83C45; 53A45; 83C20.
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 (
) 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
be a spacetime, where
is a four-dimensional connected smooth Hausdorff
manifold and g is a smooth Lorentz metric of signature
be the Levi–
Civita connection associated with g and
be the corresponding type (1, 3) Riemannian
curvature tensor. The type (1, 3) Weyl conformal curvature tensor of
is denoted by
. The components of
are written as
and the Ricci tensor
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 [
Much interest has been shown in the various symmetries of the geometrical structures on
, and details are available in ([
]). 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
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
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
denotes the components of the stress-energy tensor and
is the gravitational
constant. In the ﬁeld Equation
, 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
. 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 ([
]). 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 [
]. Moreover, certain geometrical and physical
notions are also described by spacetime symmetries, such as the conservation of linear
momentum, angular momentum and energy [
]. The symmetries regarding spacetime
(V4,g)are determine by the following mathematical equation :
stands for the Lie derivative operator, with respect to the vector ﬁeld
some smooth scalar function on the spacetime and
is any of the physical quantities
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
Riemannian curvature tensor
, Ricci tensor
, conharmonic curvature tensor
contracted conharmonic curvature tensor
, energy momentum tensor
, etc. The
most primary symmetry on
is motion (M), which is obtained by setting
0 in Equation
. Then, Equation
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
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
]. 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
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 [
] for the special conformal Killing vector (SCKV), and was
then further studied in 1990 for CKV ( [
]). 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
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.
If the Lie derivative of the Riemann curvature tensor, along a vector ﬁeld
0, then it is called a curvature collineation (CC), which was introduced by
Katzin et al. [
] in 1969. The Ricci collineation (RC) is obtained by the contraction of the
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)
is the conformal function on
is called the conformal Killing vector
(CKV). If αsatisﬁes the condition
α;ij =0and α;i6=0, (5)
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
and motion (M), if α=0.
The projective collineation (PC) satisﬁes £
denotes the Weyl
projective curvature tensor in (V4,g)and is deﬁned as follows:
kRij ]. (6)
The projective collineation is deﬁned in another way by a vector ﬁeld ξsatisfying
for a scalar ﬁeld
are the components of the Christoffel symbol of the
Riemannian metric g and δi
jstands for the Kronecker delta.
The curvature inheritance (CI) ([
]) along a vector ﬁeld
is deﬁned on the Rieman-
nian space as:
ijk , (8)
is an inheritance function of spacetime coordinates and vector ﬁeld
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
is called the Ricci inheritance vector (RIV). As we know that every
CIV is a RIV, and from , we have
Universe 2021,7, 505 4 of 21
£ξR=2αR−Rij ¯hi j, (11)
¯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  and deﬁned the following transformation,
gij =gi je2σ, (13)
where σstands for the scalar function and also the following condition holds:
, a quadratic Killing tensor is a generalization of a Killing vector
and is deﬁned as a second-order symmetric tensor Aij  satisfying the condition:
A vector ﬁeld
in a semi-Riemannian space is said to generate a one-parameter group
of curvature collineations  if it satisﬁes:
A Riemannian space is conformally ﬂat  if
ijk =0, (n>3). (17)
A Riemannian space is conharmonically ﬂat  if
ijk =0, (n>3). (18)
3. Conharmonic Curvature Inheritance Symmetry
A (1, 3)-type conharmonic curvature tensor
, which is unaltered under the conhar-
, can be explicitly expressed as the following
kRij +gik Rh
We introduce the notion of conharmonic curvature inheritance symmetry as follows.
with Lorentzian metric
, a smooth vector ﬁeld
is said to generate
a conharmonic curvature inheritance symmetry if it satisﬁes the following equation:
ijk , (20)
where α=α(xi)is an inheritance function.
Proposition 1. If a spacetime (V4,g)admits the following symmetry inheritance equations:
(b)£ξgij =2αgi j,
then that spacetime necessarily admits Conh CI along a vector ﬁeld ξ.
Universe 2021,7, 505 5 of 21
The proof is obtained directly by taking the Lie derivative of the Equation
andusing above symmetry inheritance equations we have £
. Thus, spacetime
admits Conh CI along a vector ﬁeld ξ.
Example 1. Consider the following line element of a de Sitter spacetime:
is a constant. This line element admits a proper CKV,
, 0, 0, 0), for which
A straightforward computation of the components
, and then taking the Lie derivative with
, indicates that
is a CIV and, therefore, an RIV. Thus, this example of
the above metric is compatible with Proposition 1, i.e., de Sitter spacetime satisfying the Conh
In this research article, we are considering the inheritance function as being the
same as the conformal function. If
= 0, then
reduces to £
0, which is called
conharmonic curvature collineation (Conh CC) . Contracting (20), we obtain
£ξZij =2αZij, (22)
denotes the contracted conharmonic curvature tensor on a spacetime
and it is invariant under the transformation (13).
, a smooth vector ﬁeld
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)
A vector ﬁeld
is called a contracted conharmonic curvature
collineation vector ﬁeld.
0, then a vector ﬁeld
is called a proper contracted Conh CI
vector. Contracting Equation (19), we obtain
2gij R. (24)
If a spacetime
admits the contracted conharmonic curvature tensor, then the
scalar curvature of the spacetime (V4,g)will be constant.
Recently, ref. [
] U. C. De, L. Velimirovic and S. Mallick studied the characteristics
of the contracted conharmonic curvature tensor (
) as follows: “In a spacetime, the
contracted conharmonic curvature tensor is a quadratic Killing tensor”, or it can be written
0 with the use of Equation
. 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
This completes the proof.
Universe 2021,7, 505 6 of 21
On the Lie derivative of
along a proper conformal Killing vector ﬁeld
using (25), we can easily show that Equation (22)is well deﬁned on spacetime (V4,g).
If a spacetime
admits Conh CI along a vector ﬁeld
, then the following
identities hold: (a)£ξZij =2αZij,
, we obtain £
, which proves (a) and implies
that every Conh CI is a contracted Conh CI. The proof of (b) follows by £
and the use of Equation
, which leads to £
ξZij =gjl (
. Now, comparing
with part (a) and the rearrangement, we obtain the required result (b). Since spacetime
also admits the conharmonic curvature tensor, and, in general, every conharmonic
is a contracted conharmonic curvature tensor
., 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).
Clearly, under the hypothesis of Theorem 1, spacetime
generates a one-parameter
group of curvature collineation .
In the empty spacetime
, the tensors
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.
If a spacetime (
) 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
kgij ) + R
and this expression is also written in terms of Zh
ijk and Zi j as
Taking the Lie derivative of (27) and using (20) and (22), we obtain
ijk . (28)
This completes the proof.
Now, we state Theorem 3(e) from [
], i.e., “If a spacetime (
) admits a CI, then the
following identity holds:
ijk , (29)
lgik ] + 1
k(R¯hij −R0gi j)−δh
j(R¯hik −R0gik )]
The above result raises the following open problem [
]: “Find condition on (
with a proper CI symmetry such that
vanishes”. From Theorem 2, we solve the above
open problem for the spacetime (
) to admit proper Conh CI. If a spacetime admits
Conh CI symmetry, then
is singled out as free from the term
. 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.
The Theorem 2gives us a motivation of the Conh CI symmetry of spacetime, since it
implies the conformal curvature inheritance symmetry
. 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
Under the hypothesis of Proposition 1, if a spacetime (
) admits the Weyl projective
tensor with Conh CI along a vector ﬁeld
, then the Weyl projective tensor also holds the symmetry
Let a spacetime (
) admit the Weyl projective tensor with a Conh CI along a
vector ﬁeld ξ; this tensor is expressed as
kRij ]. (30)
Taking the Lie derivative of (30), we have
k(£ξRik )]. (31)
Further, from Proposition 1, (V4,g) also admits a CIV and RIV, so we have
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
klm gi j +Zi
jlm gik =0. (35)
Taking the Lie derivative of (35), using Equations (20) and (4), we obtain
klm ¯hi j +Zi
jlm ¯hik =0. (36)
Now, using the expression of Zh
ijk and Equation (4) in Equation (36), we obtain
klm ¯hi j +Ri
jlm ¯hik =0. (37)
Applying the Ricci identity [
, we obtain
, which completes the proof.
Remark 4. If we multiply by √ggi l gjk in (33), we obtain the Komar’s identity 
[√g(ξi;j−ξj;i)];ji =0= [[√g(ξi;j−ξj;i)];j];i, (38)
) and Equation
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 [
) admits curvature symmetry properties. As Komar’s identity holds for all vector ﬁelds
¯hij;kl −¯hij;lk =0, (39)
for a CC, CI plays no restriction on this symmetry vector
. Hence, Conh CI are the necessary
symmetry properties of spacetime (
) that are embraced by the group of general curvilinear
co-ordinate transformations in V4.
Furthermore, following the condition that Equation
is independent of the scalar
, 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
admits Conf M [
] along a (CKV)
if the following equation
¯hij =£ξgi j =2αgij , (40)
is called a (SCKV). Other CKVs are the homothetic
motion (HM) if α;i=0, α6=0 and the motion (M) if α=0.
is said to admit a conformal collineation (Conf C) if a vector exists
kα;j−gjk gil α;l, (41)
and along the vector ﬁeld
, a Weyl conformal collineation (W Conf C) is said to be admitted
by a spacetime if
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 ,
£ξRij =−αgi j −2α;ij , (43)
where is the Laplacian–Beltrami operator deﬁned by α=gij α;ij .
If a Conh CI vector
is also a conformal Killing vector (CKV) on a spacetime
Proof. For a Conh CI, Proposition 1implies the following equation:
£ξRij =2αRi j. (45)
Universe 2021,7, 505 9 of 21
is also a CKV, it must satisfy
. Thus, comparing Equations
2αRij =−αgi j −2α;ij , (46)
reduces to the
(a). Multiplying both side of
obtain (44)(b). The proof of (44)(c) follows from Equations (46), (44)(b) and (24).
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.
If a spacetime
admits the Conh CI with
as a conformal Killing vector, then
the conformal curvature tensor vanishes.
The proof follows from Equations
. It may be noted that, for a CKV,
the conformal curvature tensor vanishes in (V4,g), i.e., spacetime is conformally ﬂat.
It is well known that the Weyl conformal curvature tensor
= 0 if the spacetime is
conformally ﬂat. By deﬁnition, the line element of a conformally ﬂat spacetime can be written as
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.
In particular, a relationship of Conh CI with curvature collineation (CC) is described
by the hypothesis of Theorem 5. Since, in a
, 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.
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,
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.
Under the hypothesis of Theorem 6, if
is a Ricci inheritance vector (RIV), then the
associated Conh CI must be a proper Conf M with the conformal function α.
An Einstein spacetime admits Conh CI along a vector ﬁeld
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.,
ijk . (50)
Now, using Theorem 6, we obtain
£ξRij =2αRi j. (51)
Again, by virtue of Corollary 3,
£ξgij =2αgi j. (52)
ijk can be expressed as follows:
kgij ). (53)
Taking the Lie derivative of (53) and using Equations (50)–(52), we obtain
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., [
], 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
and using Equations
. 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.
A necessary condition for spacetime
admitting Conh CI is that the spacetime
admits both CI and Conf M together.
We let spacetime
as the CIV on it; this implies that
. Moreover, it is known (cf., [
], p. 2991) that if the spacetime admits
CI, then the following identity holds:
Taking the Lie derivative of
and using Equations
and the equation of
Conf M, we see that the spacetime admits Conh CI symmetry along the vector ﬁeld
ijk . (58)
This completes the proof.
In general, the converse of Theorem 8is not true, while the converse holds if
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
) be an Einstein spacetime admitting a Conh CI, which implies admitting
the CI, as well as Conf M. Then, following on from Corollary 3using
in the ﬁrst result of
Theorem 5, we obtain
α;ij = (−αR
12 )gij , (59)
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)
] referred to a ﬁnding of Sinyukov [
] that explains that, if a spacetime
admits a vector ﬁeld
0, then a system of co-ordinates exists where the
metric has the form:
ds∗2=g11 dx1dx1+ ( 1
where a, b 6=1and g11 = [2Rφ(x1)dx1+C]−1, and the arbitrary function φ=φ(x1).
The above example of
is well suited for Theorems 6–8and
4.2. Conharmonic Motion
Abdussatar and Babita Dwivedi [
] 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
holds with the
If a vector ﬁeld ξsatisﬁes
ijk =0, (63)
admits a conharmonic curvature collineation (Conh CC), and
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
, it is evident that every
Conh CC is a W Conf C. Every Conh M satisﬁes
;kgij , (64)
£ξRij =−2α;i j, (65)
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:
If the spacetime
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
with Equation (c) in Theorem 1, we obtain
0; that is, the
scalar curvature Rof the spacetime vanishes.
Now, here, we will be discussing the motivation of Theorem 9. From [
], every CIV
is a RIV , but the converse is not true in general (for further details, see Theorem 3.2 in [
Propositions 1 and 2 in [
]). 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. [
] 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
ijk . (69)
Using the Lie derivation of Equation
with respect to
, and then using the
inherited symmetry properties of Rij ,Zh
ijk and gi j, we obtain
ijk , (70)
i.e., the Riemann curvature tensor is inherited in spacetime.
Next, we also have
If a spacetime
admits proper Conh CI along a conharmonic Killing vector
then that spacetime is conharmonically ﬂat.
ijk is expressed as
kRij +gik Rh
Taking the Lie derivative of (71), along the vector ﬁeld ξ,
ijk ) + 1
k£ξ(Rij ) + £ξ(gik Rh
Since the spacetime admits Conh CC and Conh M, then, using Equations
and (40) in Equation (72), we obtain
ijk =0. (73)
Now, applying the Conh CI Equation (20), we obtain
ijk =0(since α6=0). (74)
Thus, spacetime is conharmonically ﬂat.
If a spacetime
admits proper Conh CI and Conh CC, then the spacetime is
Proof. The proof directly follows from Equation (63).
Universe 2021,7, 505 13 of 21
We consider a plane symmetric perfect ﬂuid cosmological model obtained by Singh and
] that does not have a conformally ﬂat spacetime. The geometry of this model is deﬁned by
the line element
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
when a=b with
, where A is an arbitrary
constant. However, when
, the model becomes conharmonic to ﬂat spacetime and reduces to a
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,
2gij R. (76)
Now, taking the Lie derivative of Equation (76),
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
admits contracted conharmonic curvature
5. Physical Interpretation to Fluid Spacetimes of General Relativity
In this section, we consider different types of ﬂuid spacetimes as applications of Conh
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
and the term (2
) in Equation
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)
are the parallel and perpendicular components of the isotropic pressure
to a unit vector
Pij =hi j −ninj
is the projection tensor
onto the two orthogonal planes of vectors uiand ni.
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
, i.e., self
similarity is imposed on Equation (79), then the following equation holds :
(d)£ξσij =ασij,(e)£ξη=−αη,(f)£ξqi=−αqi. (82)
, we conclude that all physical quantities (
) inherit the spacetime
symmetry deﬁned by
. Tupper and Coley [
] have investigated the conditions for an
imperfect ﬂuid to inherited symmetry
for a SCKV. Saridakis [
] et al. have solved the
problem of symmetry inheritance for a spacelike proper CKV and other types of symmetry.
Furthermore, Duggal [
] has also investigated the conditions for imperfect ﬂuid, perfect
ﬂuid and anisotropic ﬂuid to inherited symmetry
for a CIV, and Z. Ahsan [
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)
is the conformal function. As this is known for a CKV
in ﬂuid spacetime,
then the following result holds :
is the spacelike vector orthogonal to
=0. Maartens [
] et al. have
shown that vi6=0 generally, and is given by
where the vorticity tensor is denoted by
. Fluid ﬂow lines are mapped
onto ﬂuid ﬂow lines by the action of
0. They are also said to be “frozen in” curves
to the ﬂuid.
For a CKV ξi, the following results hold :
£ξRij =−αgi j −2α;ij , (86)
£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.
Let an imperfect ﬂuid spacetime admit Conh CI symmetry along a vector ﬁeld
where ﬂuid ﬂow lines are mapped conformally by ξ. Then, the following equations hold:
(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),
Now, using the dynamic result for
of imperfect ﬂuid by Equation
, it leads to
(cf., ), i.e.,
£ξµ=−2α−24−2αµ −2α;ij uiuj, (94)
. It is seen that the ﬂuid ﬂow lines are mapped conformally by
implies that vi=0. Hence, 4=0 and Equation (94) reduces to
£ξµ=−2α−2αµ −2α;ijuiuj. (95)
For imperfect ﬂuid, when using (EFE)
qiui=0, we obtain
Rij uiuj= (µ−R
2) = ( 3p+µ
If we set,
3gij −2Ri j], (97)
then, from [
], every CIV is also a CKV. Theorem 8implies that spacetime admits Conh CI
symmetry. If we multiply Equation
, and using Equation
is timelike), then we obtain
3−µ) = −α
In view of
0; this implies that
under Lie differentiation. The proof of Equation (90)(b) follows from
3α;ij uiuj, (99)
and 4=0, (44)(b), (93) and (98).
Using vi=0 in Equation (84), we obtain (90)(c).
Moreover, from , it follows that,
vi=0⇒£ξσij =ασij, (100)
which proves Equation
(a). For imperfect ﬂuid spacetime (with
of the form
£ξ(σij η) = (αη +£ξη)σij =α(2µ+R)
6gij −αRi j +α(4µ−R)
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 
£ξqi= (Q−1£ξQ+α)qi+wi, (104)
Universe 2021,7, 505 16 of 21
from (103), Equation (104) leads to £ξqi=αqi.
Let an imperfect ﬂuid spacetime admit a Conh CIV
(a) An eigenvector of α;i j is ui;
(b) ξiis conformally mapped by ﬂuid ﬂow lines.
For an imperfect ﬂuid, using the Einstein ﬁeld equation
uiui=−1, σijui=0 and qiui=0, we obtain
Notice that, from Equation
is a timelike eigenvector of
. After multiplying
Equation (97), and from (105) and (93), we obtain
α;ij uj= ( α
which shows that
is an eigenvector of
; this proves the ﬁrst part of the theorem. Now,
, we obtain
0, i.e, the vector
is conformally mapped by
ﬂuid ﬂow lines, and, hence, the proof of part (b) is complete.
Let a perfect ﬂuid spacetime
admit a Conh CIV
the following equations hold:
Proof. First, contracting Equation (89), we obtain
T=−R or R =−T, (108)
and then, contracting Equation (80), we obtain
Next, we use a dynamic result for perfect ﬂuid with
of the form
along a CKV vector
ﬁeld ξithat was derived by Duggal in :
£ξµ=−2α−2αµ −2α;ijuiuj. (110)
In a perfect ﬂuid spacetime, using the (EFE)
and qiui=0, we obtain
Rij uiuj= (µ−R
2) = ( 3p+µ
If we multiply both sides by
and use Equation
timelike), then we obtain
3−µ) = −α
Now using Equations
, we obtain £
holds. Equation (107)(b) follows from
3α;ij uiuj. (113)
Universe 2021,7, 505 17 of 21
Moreover, the use of Equations
Theorem 16. Let a perfect ﬂuid spacetime admit a Conh CIV ξiand (p+µ)6=0. Then,
An eigenvector of α;ij is ui;
Fluid ﬂow lines are mapped conformally along the vector ﬁeld ξi;
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 
(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
, the ﬂuid ﬂow lines are mapped conformally
to Conh CI admitted by perfect ﬂuid spacetime; consequently, the four-velocity vector
is also inherited.
Let anisotropic ﬂuid spacetime
admit a Conh CIV
(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)
For anisotropic ﬂuid spacetime, the stress energy tensor is given by Equation
Now multiplying both sides of Equation (89) by uiujand ninj, we obtain
Rij uiuj= (µ−R
Rij ninj= ( R
respectively. Moreover, from Equations (97) and (118), we have
Since , for anisotropic ﬂuid, µmust satisfy the following ,
£ξµ=−2α−2αµ −2α;ijuiuj. (121)
From Theorem 5(a), and Equation
(a). The proof
of the second part of (116) is followed by combining Equation (97) and (119); therefore,
In anisotropic ﬂuid, pkmust satisfy the following :
£ξpk=2α−2αpk−2α;ij ninj. (123)
Universe 2021,7, 505 18 of 21
Again, using Equations
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
We also have 
£ξp⊥=2α−2αp⊥−2α;ij Pi j. (126)
If we put the value of
α;ij Pi j
(c) holds, as
we know that
α;ij pik uj=0, α;i jnjui=0. (127)
For an anisotropic ﬂuid, we have
(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,
must be parallel. This result,
combined with viui=0, implies vi=0; thus, from (84), we have £ξui=−αui.
A perfect ﬂuid spacetime admits Conh CI along a conformal Killing vector ﬁeld
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)
explores a new equation of state for various matter. Perfect ﬂuid
spacetime satisﬁes (80) with ξ⊥uor ξku. Then,
Now, we use
in the above equation to obtain
ijk;h=0 (cf., Theorem (2.1) in ).
One can prove a similar result for an anisotropic ﬂuid and imperfect ﬂuid spacetime.
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.,
or expresses the equation of state for a vacuum-like case, which is not a perfect ﬂuid but is
instead an Einstein spacetime.
The authors contributed equally to this work. All authors have read and
agreed to the published version of the manuscript.
Institute of Scientiﬁc Research and Revival of Islamic Heritage at Umm Al-Qura University,
Saudi Arabia (Project # 43405050).
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.1. Application to Cosmology
Siddiqui and Ahsan [
] 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 [
For conharmonically ﬂat spacetime, we have Equation (19):
kRij +gik Rh
k) = 0. (A1)
Contracting this, we obtain
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
. However, there is
a very important problem with these solutions.
We illustrate this by means of an example studied by Kumar and Srivastava [
the FRW model,
the ﬁeld Equations (A3) yield
a(t) + [ µ
9]a(t) = 0, (A5)
, p and
denote the pressure and density, respectively, of
the perfect ﬂuid
, and k is an arbitrary constant. In addition, we see that Equation
Universe 2021,7, 505 20 of 21
is satisﬁed for
a(t) = A cos(√µ
3t) + B sin(√µ
2, in the cases (i) A = 1 , B
= 0 (ii) A = 0, B = 1, Equations
have the common solutions when
This implies that the condition of the equation of state occurred for the FRW metric (A4).
If we further contract Equation
, we obtain the “vacuum”, i.e.,
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
0 and a Ricci
0 along a Conh Killing vector
(Conh M) with the condition that
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
From the twice-contracted Bianchi identity , we ﬁnd ( using R=0 )
Combining Equations (A8) and (A10), we obtain
In a spacetime with R=0, the Einstein ﬁeld Equations (1) take the form
where κis a constant and Tl
kis an energy-momentum tensor with trace Tl
Substituting (A12) in (A11) gives
g=|det gij |
is deﬁned by Conh CI. Thus, we conclude that, if a space-time
with R = 0 and
0 admits Conh CI along a Conh Killing vector
, then there exists a
covariant conservation law generator of the form (A13).
Hall, G.S. Symmetries and Curvature Structure in General Relativity; World Scientiﬁc Publishing Co. Ltd.: Singapore, 2004;
Duggal, K.L. Curvature inheritance symmetry in Riemannian spaces with applications to ﬂuid space times. J. Math. Phys.
33, 2989–2997. https://doi.org/10.1063/1.529569.
Duggal, K.L. Symmetry inheritance in Riemannian manifold with physical applications. Acta Appl. Math.
Abdussatar; Dwivedi B. Fluid space—Times and conharmonic symmetries. J. Math. Phys.
,39, 3280–3295. https://doi.org/
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.
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
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.
Yano, K. Integral Formulas in Riemannian Geometry; Marcel Dekker: New York, NY, USA, 1970; https://doi.org/10.1017/
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
Coley, A.A.; Tupper, B.O.J. Special conformal Killing vector spacetimes and symmetry inheritance. J. Math. Phys.
Coley, A.A.; Tupper, B.O.J. Spacetimes admitting inheriting conformal Killing vector ﬁelds. Class. Quant. Grav.
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.
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.
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/.
De, U.C.; Velimirovic, L.; Mallick, S. On a type of spacetime. Int. J. Geom. Methods Mod. Phys.
Duaggal, K.L.; Sharma, R. Hypersurfaces in a conformally ﬂat space with curvature collineation. Internat. J. Math. Math. Sci.
,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.
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).
Komar, A. Covariant conservation laws in general relativity. Phys. Rev.
,113, 934. https://doi.org/10.1103/PhysRev.113.934.
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).
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
Sharma, R.; Duggal, K.L. Differential geometry and mathematical physics. Am. Math. Soc. Contemp. Math. Ser.
Singh, K.P.; Singh, D.N. A plane symmetric cosmological model. Mon. Not. R. Astron. Soc.
Saridakis, E.; Tsamparlis, M. Symmetry inheritance of conformal Killing vectors. J. Math. Phys.
Ahsan, Z. On a geometrical symmetry of the spacetime of general relativity. Bull. Cal. Math. Soc.
,97, 191–200. Available
OF_GENERAL_RELATIVITY (accessed on 11 December 2021).
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.
Siddiqui, S.A.; Ahsan, Z. Conharmonic curvature tensor and the spacetime of general relativity. Diff. Geom. Dyn. Syst.
213–220. Available online: http://www.mathem.pub.ro/dgds/v12/D12-SI.pdf (accessed on 11 December 2021).
Narlikar, J.V. An Introduction to Relativity; Cambridge University Press: Cambridge, UK, 2010; https://doi.org/10.1017/CBO
Kumar, R.; Srivastava, S.K. FRW-cosmological model for conharmonically ﬂat spacetime. Int. J. Theor. Phys.
Available online: https://ui.adsabs.harvard.edu/link_gateway/2013IJTP...52..589K/doi:10.1007/s10773-012- 1364-7 (accessed on
11 December 2021).