Semiconformal symmetry- A new symmetry of the spacetime manifold of the
general relativity
MUSAVVIR ALI1***Corresponding author, NAEEM AHMAD PUNDEER2 AND ZAFAR AHSAN3

Abstract. In this paper we have introduced a new symmetry property of
spacetime which is named as semiconformal curvature collineation, and its
relationship with other known symmetry properties has been established. This
new symmetry property of the spacetime has also been studied for non-null and
null electromagnetic fields.

MSC 2010 Classification: 53A45, 53C25, 53C50, 83C20, 83C22, 83C50.
Keywords curvature tensors, symmetries, Spacetime Manifold, electromagnetic
fields.

##  1 Introduction

In recent years general relativists have been much interested in symmetries of
spacetime in general relativity. Such interest is due to the need to simplify
the Einstein’s field equations in the search for their exact solutions. These
geometrical symmetries of the spacetime are often defined by the vanishing of
the Lie derivative of certain tensors with respect to a vector ( this vector
may be time-like, space-like or null). The symmetries in general theory of
relativity have been introduced by Katzin, Levine and Davis in the papers
([24] and [25]). These symmetries which are also known as collineations, were
further studied by Ahsan ([3] \- [8]), Ahsan and Hussain ([13]), Ahsan and
Siddiqui ([14]), Ahsan and Ali ([9] \- [12] ) and Ali and Ahsan ([15]-[17])
among many others. However, in this paper our study is focused on these
symmetries which can be used as simplifying assumptions in the exact solution
of Einstein’s field equations but solving EFE by our findings will be the next
target. Main objective of this paper is to give new symmetry in mathematical
approach and analyse it on parameters of the well established literature on
symmetries of spacetime manifolds.
As a special subgroup of the conformal transformation group, Ishii([23])
defined a rank four tensor $L^{h}_{bcd}$ that remains invariant under
conharmonic transformation for a n-dimensional Riemannian differentiable
manifold ($M^{n},g$) of dimension $n\geq 4$, as follows:

$L^{h}_{bcd}=R^{h}_{bcd}+\dfrac{1}{n-2}(\delta^{h}_{c}R_{bd}-\delta^{h}_{d}R_{bc}+g_{bd}R^{h}_{c}-g_{bc}R^{h}_{d}),$
(1.1)

where $R^{h}_{bcd},~{}R_{bd}$ are Riemann curvature tensor and Ricci tensor
respectively. The geometric properties of conharmonic curvature tensor have
been discussed by Shaikh and Hui ( [33] ), while the relativistic significance
of this tensor has been investigated by Abdussattar and Dwivedi ([1]) and
Siddiqui and Ahsan ([31]). In 2016, J. Kim ([27]) introduced curvature like
tensor field which remain invariant under conharmonic transformation. He named
this new tensor as semiconformal curvature tensor and denoted it by
$P^{h}_{bcd}$. For a Riemannian manifold $M^{n}$ with metric $g$, this tensor
is defined as (see also [28])

$P^{h}_{bcd}=-(n-2)BC^{h}_{bcd}+[A+(n-2)B]L^{h}_{bcd},$ (1.2)

provided the constants A and B are not simultaneously zero, $C^{h}_{bcd}$ is
conformal curvature tensor defined as

$C^{h}_{bcd}=R^{h}_{bcd}+\dfrac{1}{n-2}(\delta^{h}_{c}R_{bd}-\delta^{h}_{d}R_{bc}+g_{bd}R^{h}_{c}-g_{bc}R^{h}_{d})+\dfrac{R}{(n-1)(n-2)}(\delta^{h}_{d}g_{bc}-\delta^{h}_{c}g_{bd}),$
(1.3)

where $R_{ab}$ is the Ricci tensor and $R$ is the scalar curvature.

For a special substitution $A=1$ and $B=\dfrac{-1}{(n-2)},$ the semiconformal
curvature tensor reduces to conformal curvature tensor, while for $A=1$ and
$B=0$, it reduces to conharmonic curvature tensor. The semiconformal curvature
$P_{hbcd}$ satisfy the following symmetry properties

$P_{hbcd}=-P_{bhcd}=-P_{hbdc}=P_{cdhb},$ (1.4)

and

$P_{hbcd}+P_{chbd}+P_{bchd}=0.$ (1.5)

In this paper we define a new symmetry in terms of semiconformal curvature
tensor and study its relationship with other symmetries of the spacetime. We
call this new symmery as semiconformal curvature collineation. Section 2
contains some known results that are required for our investigation. In
sections 3 and 4, the relationship between semiconformal cuvature collineation
and the other symmetry properties for a general Riemannian space and for a
Riemannian space with vanishing Ricci Tensor, respectively, have been
established. Finally in section 5, the semiconformal curvature collineation
has been studied for non-null and null electromagnetic fields.

##  2 Preliminaries

A geometrical symmetry of the spacetime is often defined in terms of the Lie
derivative of a tensor. These symmetries are also known as collineations. The
literature on such collineations is very large and still expanding with
results of elegance. Here we shall mention only those symmetry assumptions
that are necessary for our study and we have (cf. [20], [34], [36])

######  Definition 2.1.

Motion (M) A spacetime is said to admit motion if there exists a vector field
$\xi^{a}$ such that†††Indices takes the values from $1,2,3......,n$ and the
summation convention is used. Covariant differentiation is indicated by a
semicolon (;) and partial differentiation by a comma (,).

${\Huge\eta}_{ab}\equiv\pounds_{\xi}g_{ab}=\xi_{a;b}+\xi_{b;a}=0.$ (2.1)

Equation (2.1) is known as Killing equation and the vector $\xi^{a}$ is known
as a Killing vector field.

######  Definition 2.2.

Affine Collineation (AC) A spacetime is said to be an affine collineation if
there is a vector field $\xi^{a}$ such that

$\pounds_{\xi}\begin{Bmatrix}c\\\
ab\end{Bmatrix}\equiv\dfrac{1}{2}g^{cd}({\Huge\eta}_{da;b}+{\Huge\eta}_{db;a}-{\Huge\eta}_{ab;d})=0,$
(2.2)

where $\begin{Bmatrix}c\\\ ab\end{Bmatrix}$ is the Christoffel symbol of
second kind. Hence the necessary and sufficient condition for an AC [from
(2.2)] is

${\Huge\eta}_{ab;c}=0,$ (2.3)

It may be noted, from equations (2.2) and (2.3), that every M is AC.

######  Definition 2.3.

Conformal Motion (Conf M) A spacetime is said to admit a conformal motion if
there exist a vector field $\xi^{a}$ such that

$\pounds_{\xi}\mathfrak{K}_{ab}=0,$ (2.4)

where $\mathfrak{K}$ ([34]) is defined by

$\mathfrak{K}=g^{-\frac{1}{4}}g_{ab}.$ (2.5)

Accordingly we have

$\eta_{ab}=2\phi g_{ab},$ (2.6)

where $\phi$ is scalar and we may express in the following form

$\phi=\dfrac{1}{4}\xi^{d}_{;d}.$ (2.7)

######  Definition 2.4.

Projective Collineation (PC) A spacetime is said to admit projective
collineation if there exist a vector $\xi^{a}$ such that

$\pounds_{\xi}\prod_{bc}^{a}=0,$ (2.8)

where the projective connection is defined as for $n=4$

$\prod_{bc}^{a}=\begin{Bmatrix}a\\\
bc\end{Bmatrix}-\dfrac{1}{5}\Bigg{(}\delta^{a}_{b}\begin{Bmatrix}h\\\
hc\end{Bmatrix}+\delta^{a}_{c}\begin{Bmatrix}h\\\ hb\end{Bmatrix}\Bigg{)}.$
(2.9)

From equations (2.8) and (2.9), we get

$\pounds_{\xi}\begin{Bmatrix}a\\\
bc\end{Bmatrix}=\delta^{a}_{b}\sigma_{;c}+\delta^{a}_{c}\sigma_{;b},$ (2.10)

where

$\sigma_{;c}=\frac{1}{5}\xi^{m}_{;mc}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}\sigma_{;b}=\frac{1}{5}\xi^{m}_{;mb}.$
(2.11)

Further, for every projective collineation, we have ([24])

$\pounds_{\xi}W^{h}_{bcd}=0,$ (2.12)

where the Weyl projective curvature tensor for $n=4$ is given by

$W^{h}_{bcd}=R^{h}_{bcd}-\frac{1}{3}(\delta^{h}_{d}R_{bc}-\delta^{h}_{c}R_{bd}).$
(2.13)

From equations (2.10), (2.11) and (2.12) it follows that every AC is PC.

######  Definition 2.5.

Conformal Collineation (Conf C) A spacetime is said to admit a conformal
collineation if there exist a vector $\xi^{a}$ such that

$\pounds_{\xi}\begin{Bmatrix}a\\\
bc\end{Bmatrix}=\delta^{a}_{b}\phi_{;c}+\delta^{a}_{c}\phi_{;b}-g_{bc}g^{am}\phi_{;m},$
(2.14)

where $\phi=\frac{1}{4}\xi^{d}_{;d}$.

Equations (2.6) and (2.14) may be expressed as ([24])

$\eta_{ab;c}=2\phi_{;c}g_{ab},$ (2.15)

and that every Conf C must satisfy (for explanation c.f., [36])

$\pounds_{\xi}C^{h}_{bcd}=0,$ (2.16)

where $C^{h}_{bcd}$ is conformal curvature tensor, which from equation (1.3)
for $n=4$, is given by

$C^{h}_{bcd}=R^{h}_{bcd}+\frac{1}{2}(\delta^{h}_{c}R_{bd}-\delta^{h}_{d}R_{bc}+g_{bd}R^{h}_{c}-g_{bc}R^{h}_{d})+\frac{R}{6}(\delta^{h}_{d}g_{bc}-\delta^{h}_{c}g_{bd}).$
(2.17)

######  Definition 2.6.

Curvature Collineation (CC) A spacetime is said to admit curvature
collineation if there is a vector field $\xi^{a}$ such that

$\pounds_{\xi}R^{h}_{bcd}=0,$ (2.18)

where Riemann curvature tensor is defined as ([20])

$R^{h}_{bcd}=\begin{Bmatrix}h\\\ bd\end{Bmatrix}_{,c}-\begin{Bmatrix}h\\\
bc\end{Bmatrix}_{,d}+{\begin{Bmatrix}m\\\ bd\end{Bmatrix}}{\begin{Bmatrix}h\\\
mc\end{Bmatrix}}-{\begin{Bmatrix}m\\\ bc\end{Bmatrix}}{\begin{Bmatrix}h\\\
md\end{Bmatrix}}.$ (2.19)

######  Definition 2.7.

Ricci Collineation (RC) A spacetime is said to admit Ricci collineation if
there is a vector field $\xi^{a}$ such that

$\pounds_{\xi}R_{ab}=0,$ (2.20)

where $R_{ab}$ is the Ricci tensor.

######  Definition 2.8.

Maxwell collineation (MC) The electromagnetic field inherits the symmetry
property of spacetime such that

$\pounds_{\xi}F_{ab}=F_{ab;c}\xi^{c}+F_{ac}\xi_{;b}^{c}+F_{bc}\xi_{;a}^{c}=0,$
(2.21)

where $F_{ab}$ is the electromagnetic field tensor. A point transformation
that leave $F_{ab}$ invariant, i.e., equation (2.21) is satisfied, is called a
Maxwell collineation ([18]).

##  3 Semiconformal Symmetry

For $n=4$, the semiconformal curvature tensor, from equation (1.2), is given
by

$P^{h}_{bcd}=-2BC^{h}_{bcd}+[A+2B]L^{h}_{bcd},$ (3.1)

where $C^{h}_{bcd}$ and $L^{h}_{bcd}$ are the conformal and conharmonic
curvature tensor respectively.

We now define a new symmetry for the spacetime manifold of general relativity
as

######  Definition 3.1.

Semiconformal Curvature Collineation (Semiconf CC) A spacetime $V_{4}$ is said
to admit a semiconformal curvature collineation if there exists a vector field
$\xi^{a}$ such that

$\pounds_{\xi}P^{h}_{bcd}=0,$ (3.2)

where $P^{h}_{bcd}$ is semiconformal curvature tensor is defined in equation
(1.2).

######  Definition 3.2.

Special Semiconformal Curvature Collineation (S Semiconf CC) A semiconformal
curvature collineation with the following condition

$\sigma_{;bc}=0,$ (3.3)

is called a special semiconformal curvature collineation.

where $\sigma_{;bc}=\dfrac{1}{4}\xi^{d}_{;dbc}$ and $\xi^{a}$ is Killing
vector field.

We also define

######  Definition 3.3.

Conharmonic Curvature Collineation (Conh CC) A spacetime $V_{4}$ is said to
admit a conharmonic curvature collineation if there exists a vector $\xi^{a}$
such that

$\pounds_{\xi}L^{h}_{bcd}=0,$ (3.4)

where conharmonic curvature tensor $L^{h}_{bcd}$ is defined by ([2], [31] )

$L^{h}_{bcd}=R^{h}_{bcd}+\frac{1}{2}(\delta^{h}_{c}R_{bd}-\delta^{h}_{d}R_{bc}+g_{bd}R^{h}_{c}-g_{bc}R^{h}_{d}).$
(3.5)

######  Definition 3.4.

Concircular Curvature Collineation (Conc CC) A spacetime $V_{4}$ is said to
admit a concircular curvature collineation if there exists a vector $\xi^{a}$
such that

$\pounds_{\xi}M^{h}_{bcd}=0,$ (3.6)

where concircular curvature tensor $M^{h}_{bcd}$ is defined by ([2], [14])

$M^{h}_{bcd}=R^{h}_{bcd}-\dfrac{R}{12}(\delta^{h}_{d}g_{bc}-\delta^{h}_{c}g_{bd}).$
(3.7)

Main results of the section 3.

######  Theorem 3.1.

The necessary and sufficient condition for a semiconformal curvature
collineation (Semiconf CC) to be a curvature collineation (CC) is that

$\phi_{;bc}=0,$

where $\phi=\dfrac{1}{4}\xi^{d}_{;d}$ is a scalar function.

######  Theorem 3.2.

The necessary and sufficient condition for a semiconformal curvature
collineation to be a Weyl projective curvature collineation is that

$\sigma_{;bc}=0.$

where $\sigma=\dfrac{1}{5}\xi^{m}_{;m}.$

######  Theorem 3.3.

A spacetime $V_{4}$ admits semiconformal curvature collineation along a vector
field ${\xi}^{a}$ provided that ${\xi}^{a}$ is Killing.

##  4 Semiconformal symmetry in empty spacetime

The Einstein field equations are given by

$R_{bc}-\dfrac{1}{2}g_{bc}R=-kT_{bc}$ (4.1)

where $R_{bc}$ is the Ricci tensor, $g_{bc}$ is the metric tensor, $T_{bc}$ is
the energy momentum tensor, $R$ is the scalar curvature tensor and $k$ is the
constant.
Multiplying by$g^{bc}$ and using $g^{bc}g_{bc}=4$, equation (4.1) takes the
form

$R=kT.$ (4.2)

from the equations (4.1) and (4.2), we get

$R_{bc}=k(T_{bc}-\dfrac{1}{2}g_{bc}T)$ (4.3)

If $T_{bc}=0$, then $T=g^{bc}T_{bc}=0$, equation (4.3) yields

$R_{bc}=0,$ (4.4)

these equations are the field equations for empty spacetime.

Main results of the section 4.

######  Theorem 4.1.

In an Empty spacetime $V^{0}_{4}$ Lie derivative of semiconformal curvature
tensor is proportional to Lie derivative of Riemann curvature tensor.

######  Theorem 4.2.

In empty spacetime $V^{0}_{4}$ the Lie derivatives of semiconformal curvature
and Weyl projective curvature tensors are proportional.

##  5 Semiconformal collineation and electromagnetic fields

It is known that in general relativity, the electromagnetism can be described
through Maxwell’s equation

$F_{[ab;c]}=0,~{}~{}~{}~{}F^{ab}_{;b}=J^{a},$ (5.1)

where the skew-symmetric tensor $F_{ab}$ represents the electromagnetic field
tensor and $J^{a}$ the current density. Moreover, we have defined the Einstein
field equations in equation (4.1) and in presence of matter in equation (4.3)
The energy - momentum tensor for an electromagnetic field is given by

$T_{ab}=-F_{ac}F^{c}_{b}+\dfrac{1}{4}g_{ab}F_{pq}F^{pq},$ (5.2)

which is symmetric tensor. Equation (5.2) leads to $T^{a}_{a}=T=0$ and thus
the Einstein equation for a purely electromagnetic distribution is given by

$R_{ab}=kT_{ab},$ (5.3)

The geometrical symmetry defined by equation (2.21) along with the symmetry
given by equation (2.1) has been the subject of interest for quite some time.
Thus for example, for non-null electromagnetic fields Woolley ([35]) has shown
that if equation (2.1) holds then $F_{ab}$ satisfies
$\pounds_{\xi}F_{ab}=k(\alpha)F_{ab}$ form some constant $k(\alpha),$
$\alpha=1,2,..........r;$ while Michalski and Wainwright ([29]) have shown
that $\pounds_{\xi}g_{ab}=0\implies\pounds_{\xi}F_{ab}=0$ for non-null fields.
On the other hand, for non-null fields, Duggal ([19]) has proved the converse
part under certain conditions. Maxwell collineations have also been studied by
Ahsan and Ahsan and Husain ([13], [3], [5]). It is seen that for null
electromagnetic fields neither MC is a consequence of Motion nor Motion is a
consequence of Maxwell collineation. Moreover, using Newman-Penrose formalism,
Ahsan ([5]) has obtained the conditions under which a null electromagnetic
field may admit Maxwell collineation and Motion. The concept of Maxwell
collineation was further extended as Maxwell Inheritance (MI) by Ahsan and
Ahsan ([8]), who applied this concept to (i) the spacetime solution
corresponding to strong gravitational waves propagating in generalised
electromagnetic universes and (ii) the algebraically general twist-free
solution of Einstein-Maxwell equation for non-radiative electromagnetic
fields.

In 1986 Khlebnikov ([26]) has obtained the solutions for Einstein-Maxwell
equations corresponding to the strong gravitational waves in the generalized
electromagnetic universe. He used the technique of N.P. formalism ([30]) to
obtain the solution in non-radiative electromagnetic fields. In his solution
he took the first real null tetrad vector $l^{a}$ as geodetic and shear-free
and the tetrad as parallelly propagated along $l^{a}$ also proved that the
solution does not admit the Maxwell collineation. As another example we can
refer the twist-free algebraically general solution given by Tariq and Tupper
([32]). For this solution of Einstein-Maxwell equations in non-null
electromagnetic fields together with the condition of coupling theorem Tariq
and Tupper proved that their solutions also do not admit the Maxwell
collineation.

Motivated by above discussions, in this section, we shall investigate the role
of semiconformal curvature collineation to the non-null and null
electromagnetic fields.

######  Lemma 5.1.

([29]) In a non-null electromagnetic field, the Lie derivative of
electromagnetic field tensor $F_{ab}$ with respect to a vector field $\xi$
vanishes, if $\xi$ is Killing vector.

Main results of section 5.

######  Theorem 5.1.

A non-null electromagnetic field admits semiconformal curvature collineation
along a Killing vector field.

######  Theorem 5.2.

A non-null electromagnetic field admits semiconformal curvature collineation
if it admits Maxwell collineation.

######  Theorem 5.3.

A null electromagnetic field admits semiconformal curvature collineation along
a propagation (polarization) vector if propagation (polarization) vector is
killing and expansion-free.

Note: Proof of the main results will be uploaded after publication of the
article.

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DEPARTMENT OF MATHEMATICS
ALIGARH MUSLIM UNIVERSITY
ALIGARH-202002, INDIA
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