# Curvature properties of a special type of pure radiation metrics

Absos Ali Shaikh∗1, Haradhan Kundu2, Musavvir Ali3 and Zafar Ahsan4 1,3
Department of Mathematics,
Aligarh Muslim University,
Aligarh-202002,
Uttar Pradesh, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>2 Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS>4 Faculty of Science and
Technology,
University of Islamic Sciences,
Nilai,Malaysia<EMAIL_ADDRESS>
###### Abstract.

A spacetime denotes a pure radiation field if its energy momentum tensor
represents a situation in which all the energy is transported in one direction
with the speed of light. In 1989, Wils and later in 1997 Ludwig and Edgar
studied the physical properties of pure radiation metrics, which are
conformally related to a vacuum spacetime. In the present paper we investigate
the curvature properties of special type of pure radiation metrics presented
by Ludwig and Edgar. It is shown that such a pure radiation spacetime is
semisymmetric, Ricci simple, $R$-space by Venzi and its Ricci tensor is
Riemann compatible. It is also proved that its conformal curvature 2-forms and
Ricci 1-forms are recurrent. We also present a pure radiation type metric and
evaluate its curvature properties along with the form of its energy momentum
tensor. It is interesting to note that such pure radiation type metric is
$Ein(3)$ and 3-quasi-Einstein. We also find out the sufficient conditions for
which this metric represents a generalized pp-wave, pure radiation and perfect
fluid. Finally we made a comparison between the curvature properties of Ludwig
and Edgar’s pure radiation metric and pp-wave metrics.

###### Key words and phrases:

pure radiation metric, pp-wave metric, generalized pp-wave metric, Einstein
field equation, Weyl conformal curvature tensor, semisymmetric type curvature
conditions, pseudosymmetric type curvature conditions, quasi-Einstein manifold

###### 2010 Mathematics Subject Classification:

53B20, 53B25, 53B30, 53B50, 53C15, 53C25, 53C35, 83C15

##  1\. Introduction

We consider a smooth connected semi-Riemannian manifold $(M^{n},g)$ with
$n\geq 3$ (this condition is assumed throughout the paper) equipped with the
semi-Riemannian metric $g$ with signature $(p,n-p)$ and the Levi-Civita
connection $\nabla$. We note that $M$ is Riemannian if $p=0$ or $n$ and $M$ is
Lorentzian if $p=1$ or $n-1$. Let us consider $R$, $\mathcal{R}$, $S$,
$\mathcal{S}$ and $\kappa$ respectively be the Riemann-Christoffel curvature
tensor of type $(0,4)$, the Riemann-Christoffel curvature tensor of type
$(1,3)$, the Ricci tensor of type $(0,2)$, the Ricci tensor of type $(1,1)$
and the scalar curvature of $M$.
A spacetime is a connected 4-dimensional Lorentzian manifold and it presents a
pure radiation if its energy momentum tensor $T$ is of the form

$T=\rho\eta\otimes\eta,$ (1.1)

where $\eta$ is a null vector and $\rho$ is the radiation density. Such a
spacetime describes some kinds of field that propagates at the speed of light
and it could represent a null electromagnetic field. It could also represent
an incoherent beam of photons or some kinds of massless neutrino fields. On
the other hand a spacetime presents a perfect fluid spacetime if its energy-
momentum tensor is of the form

$T=(\rho+p)\eta\otimes\eta+pg,$ (1.2)

where $\rho$ is the energy density, $p$ is the isotropic pressure and $\eta$
is the four-velocity of the fluid. Thus a pure radiation source can be
considered as a limiting case of a pressureless perfect fluid with null four-
velocity. For this reason a pure radiation source is sometimes referred as
“null dust”.
In [59] Wils investigated homogeneous and conformally Ricci flat solutions of
Einstein’s field equations for pure radiation case. Later in 1997, Ludwig and
Edgar [20] obtained exhausted class of conformally Ricci flat pure radiation
solutions of Einstein’s field equations. The line element of conformally Ricci
flat pure radiation spacetime in
$(x^{1};x^{2};x^{3};x^{4})=(u;r;x;y)$-coordinates with $x>0$ is given by [20]

$\displaystyle
ds^{2}=\left(-2xW^{\circ}-8\tilde{P}^{2}\frac{r^{2}}{x^{2}}\right)du^{2}+2dudr-\frac{4r}{x}dudx-\frac{1}{8\tilde{P}^{2}}(dx^{2}+dy^{2}),$

where $\tilde{P}$ is an arbitrary non-zero constant and $W^{\circ}$ is an
arbitrary function of the three non-radial coordinates $u,x$ and $y$. For
simplicity of symbols we write the metric as

$ds^{2}=\left(xw-p^{2}\frac{r^{2}}{x^{2}}\right)du^{2}+2dudr-\frac{4r}{x}dudx-\frac{1}{p^{2}}(dx^{2}+dy^{2}),$
(1.3)

where $w=w(u,x,y)=-2W^{\circ}$ and $p=2\sqrt{2}\tilde{P}=const.$
Now in $(u;r;x;y)$-coordinates with $x>0$, we consider the following metric

$ds^{2}=\left(xw+a\frac{r^{2}}{x^{2}}\right)du^{2}+2dudr+\frac{2br}{x}dudx+f(dx^{2}+dy^{2}),$
(1.4)

where $a_{1}$, $a_{2}$ are arbitrary non-zero constants and $f$ is a nowhere
vanishing function of $x$ and $y$. We note that if $a=-p^{2}$, $b=-2$ and
$f\equiv-\frac{1}{p^{2}}$, then the metric (1.4) reduces to pure radiation
metric (1.3). Hence we call the metric (1.4) as pure radiation type metric.
Again if $w(u,x,y)=\frac{-2h(u,x,y)}{x}$, $a=b=0$ and
$f(x,y)=-\frac{1}{2}F(x,y)$, then the metric (1.4) reduces to generalized pp-
wave metric ([29], [35], [50]), given by,

$ds^{2}=-2h(u,x,y)du^{2}+2dudr-\frac{1}{2}F(dx^{2}+dy^{2}).$ (1.5)

On the other hand if $w(u,x,y)=\frac{h(u,x,y)}{x}$, $a=b=0$ and $f\equiv 1$,
then the metric (1.4) reduces to pp-wave metric ([3], [50]), given by,

$ds^{2}=h(u,x,y)du^{2}+2dudr+(dx^{2}+dy^{2}).$ (1.6)

The physical properties of the pure radiation metric (1.3) are well known and
we refer the reader to see [20] and [59]. In the literature of differential
geometry there are many curvature restricted geometric structures on a semi-
Riemannian manifold, such as locally symmetric manifold [4], semisymmetric
manifold ([4], [51], [52], [53]), recurrent manifold ([30], [31], [32], [58]),
pseudosymmetric manifold ([1], [8] and also references therein) etc. The main
object of the present paper is to investigate such kinds of geometric
structures admitted by the pure radiation metric (1.3). It is noteworthy to
mention that the metric (1.3) is neither locally symmetric nor conformally
symmetric but semisymmetric and hence Ricci semisymmetric, conformally
semisymmetric and projective semisymmetric. It is also shown that the pure
radiation metric (1.3) is Ricci simple, weakly Ricci symmetric, weakly cyclic
Ricci symmetric, $R$-space by Venzi and its curvature 2-forms, Ricci 1-forms
and conformal curvature 2-forms are recurrent. Again the spacetime satisfies
the semisymmetric type conditions $C\cdot R=0$, $C\cdot C=0$, $C\cdot S=0$,
$Q(S,R)=0$, $Q(S,C)=0$, $P\cdot R=P\cdot C=0$ and also satisfies the
pseudosymmetric type conditions $P\cdot P=-\frac{1}{3}Q(S,P)$. It is shown
that its energy momentum tensor $T$ is semisymmetric and it is Codazzi type
(resp., cyclic parallel or covariantly constant) if $w_{33}+w_{44}$ is
independent of $x$ and $y$ (resp., constant or zero), where $w_{ij}$ denotes
the covariant derivative with respect to $x^{i}$ and $x^{j}$.
The paper is organized as follows. Section 2 deals with the preliminaries. In
section 3 we compute the components of various tensors of the metric (1.3) and
we state the main results on the geometric structures admitted by pure
radiation metric (1.3). Section 4 deals with the curvature properties of pure
radiation type metric (1.4). It is shown that such metric is $Ein(3)$ and
3-quasi-Einstein. We also obtain the conditions for which the metric is
2-quasi-Einstein, Ricci generalized pseudosymmetric and manifold of vanishing
scalar curvature. Finally, we made a comparison (similarities and
dissimilarities) between pure radiation metric and pp-wave metric. It is
interesting to mention that both are semisymmetric and weakly Ricci symmetric,
but generalized pp-wave metric is Ricci recurrent whereas pure radiation
metric is not so.

##  2\. Curvature Restricted Geometric Structures

It is wellknown that a curvature restricted geometric structure is a geometric
structure on a semi-Riemannian manifold $M$ obtained by imposing a restriction
on its curvature tensors by means of covariant derivatives of first order or
higher orders. We will now explain some useful notations and definitions of
various curvature restricted geometric structures.
For two symmetric $(0,2)$-tensors $A$ and $E$, their Kulkarni-Nomizu product
$A\wedge E$ is defined as (see e.g. [12], [17]):

$\displaystyle(A\wedge E)(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$
$\displaystyle A(X_{1},X_{4})E(X_{2},X_{3})+A(X_{2},X_{3})E(X_{1},X_{4})$
$\displaystyle-$ $\displaystyle
A(X_{1},X_{3})E(X_{2},X_{4})-A(X_{2},X_{4})E(X_{1},X_{3}),$

where $X_{1},X_{2},X_{3},X_{4}\in\chi(M)$, the Lie algebra of all smooth
vector fields on $M$. Throughout the paper we will consider
$X,Y,X_{1},X_{2},\cdots\in\chi(M)$.
Again for a symmetric $(0,2)$-tensor $A$, we get an endomorphism $\mathcal{A}$
defined by $g(\mathcal{A}X,Y)=A(X,Y)$. Then its $k$-th level tensor $A^{k}$ of
type $(0,2)$ is given by

$A^{k}(X,Y)=A(\mathcal{A}^{k-1}X,Y),$

where $\mathcal{A}^{k-1}$ is the endomorphism corresponding to $A^{k-1}$.
In terms of Kulkarni-Nomizu product the conformal curvature tensor $C$,
concircular curvature tensor $W$, conharmonic curvature tensor $K$ ([18],
[60]) and the Gaussian curvature tensor $\mathfrak{G}$ can respectively be
expressed as

$\displaystyle C$ $\displaystyle=$ $\displaystyle R-\frac{1}{n-2}(g\wedge
S)+\frac{r}{2(n-2)(n-1)}(g\wedge g),$ $\displaystyle W$ $\displaystyle=$
$\displaystyle R-\frac{r}{2n(n-1)}(g\wedge g),$ $\displaystyle K$
$\displaystyle=$ $\displaystyle R-\frac{1}{n-2}(g\wedge S),$
$\displaystyle\mathfrak{G}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(g\wedge
g).$

Again the projective curvature tensor $P$ is given by

$P(X_{1},X_{2},X_{3},X_{4})=R(X_{1},X_{2},X_{3},X_{4})-\frac{1}{n-1}[g(X_{1},X_{4})S(X_{2},X_{3})-g(X_{2},X_{4})S(X_{1},X_{3})].$

For a symmetric $(0,2)$-tensor $A$, $(0,4)$-tensor $D$ and a $(0,k)$-tensor
$H$, $k\geq 1$, one can define two $(0,k+2)$-tensors $D\cdot H$ and $Q(A,H)$
respectively as follows (see [9], [10], [14], [36], [38] and also references
therein):

$D\cdot
H(X_{1},X_{2},\cdots,X_{k},X,Y)=-H(\mathcal{D}(X,Y)X_{1},X_{2},\cdots,X_{k})-\cdots-H(X_{1},X_{2},\cdots,\mathcal{D}(X,Y)X_{k})$

and

$\displaystyle Q(A,H)(X_{1},X_{2},\ldots,X_{k},X,Y)$ $\displaystyle=$
$\displaystyle
A(X,X_{1})H(Y,X_{2},\cdots,X_{k})+\cdots+A(X,X_{k})H(X_{1},X_{2},\cdots,Y)$
$\displaystyle-$ $\displaystyle
A(Y,X_{1})H(X,X_{2},\cdots,X_{k})-\cdots-A(Y,X_{k})H(X_{1},X_{2},\cdots,X),$

where $\mathcal{D}$ is the corresponding $(1,3)$-tensor of $D$, given by
$D(X_{1},X_{2},X_{3},X_{4})=g(\mathcal{D}(X_{1},X_{2})X_{3},X_{4}).$

######  Definition 2.1.

A semi-Riemannian manifold $M$ is said to be $H$-symmetric ([4], [5]) if
$\nabla H=0$. In particular if $H=R$ (resp., $S$ and $C$), then the manifold
is called locally symmetric (resp., Ricci symmetric and conformally
symmetric).

######  Definition 2.2.

A symmetric $(0,2)$-tensor $E$ on $M$ is said to be cyclic parallel (resp,
Codazzi type) (see, [15], [16] and references therein) if

$(\nabla_{X_{1}}E)(X_{2},X_{3})=(\nabla_{X_{2}}E)(X_{1},X_{3})$
$\big{(}\mbox{resp.,}\
(\nabla_{X_{1}}E)(X_{2},X_{3})+(\nabla_{X_{2}}E)(X_{3},X_{1})+(\nabla_{X_{3}}E)(X_{1},X_{2})=0\big{)}.$

######  Definition 2.3.

$($[1], [5], [8], [38], [41], [42], [51]$)$ A semi-Riemannian manifold $M$ is
said to be $H$-semisymmetric type if $D\cdot H=0$ and it is said to be
$H$-pseudosymmetric type if $\left(\sum\limits_{i=1}^{k}c_{i}D_{i}\right)\cdot
H=0$ for some scalars $c_{i}$’s, where $D$ and each $D_{i}$, $i=1,\ldots,k$,
$(k\geq 2)$, are (0,4) curvature tensors.

######  Definition 2.4.

A semi-Riemannian manifold $M$ is said to be Einstein if its Ricci tensor is a
scalar multiple of the metric tensor $g$. Again $M$ is called quasi-Einstein
(resp., 2-quasi-Einstein and 3-quasi-Einstein) if at each point of $M$,
rank$(S-\alpha g)\leq 1$ (resp., $\leq 2$ and $\leq 3$) for a scalar $\alpha$.
In particular, if $\alpha=0$, then a quasi-Einstein manifold is called Ricci
simple.

We note that Som-Raychaudhuri metric [40] and Robinson-Trautman metric [34]
are 2-quasi-Einstein whereas Gödel metric [15] is Ricci simple.

######  Definition 2.5.

([2], [39]) A semi-Riemannian manifold $M$ is said to be $Ein(2)$, $Ein(3)$
and $Ein(4)$ respectively if

$S^{2}+\lambda_{1}S+\lambda_{2}g=0,$
$S^{3}+\lambda_{3}S^{2}+\lambda_{4}S+\lambda_{5}g=0\ \mbox{and}$
$S^{4}+\lambda_{6}S^{3}+\lambda_{7}S^{2}+\lambda_{8}S+\lambda_{9}g=0$

holds for some scalars $\lambda_{i},\,1\leq i\leq 9$.

######  Definition 2.6.

Let $D$ be a $(0,4)$-tensor and $E$ be a symmetric $(0,2)$-tensor on $M$. Then
$E$ is said to be $D$-compatible ([11], [21], [22]) if

$D(\mathcal{E}X_{1},X,X_{2},X_{3})+D(\mathcal{E}X_{2},X,X_{3},X_{1})+D(\mathcal{E}X_{3},X,X_{1},X_{2})=0$

holds, where $\mathcal{E}$ is the endomorphism corresponding to $E$ defined as
$g(\mathcal{E}X_{1},X_{2})=E(X_{1},X_{2})$. Again an 1-form $\Pi$ is said to
be $D$-compatible if $\Pi\otimes\Pi$ is $D$-compatible.

Generalizing the concept of recurrent manifold ([30], [31], [32], [58]),
recently Shaikh et al. [49] introduced the notion of super generalized
recurrent manifold along with its characterization and existence by proper
example.

######  Definition 2.7.

A semi-Riemannian manifold $M$ is said to be super generalized recurrent
manifold ([38], [44], [49]) if

$\nabla R=\Pi\otimes R+\Omega\otimes(S\wedge S)+\Theta\otimes(g\wedge
S)+\omega\otimes(g\wedge g)$

holds on $\\{x\in M:R\neq 0\mbox{ and any one of }S\wedge S,g\wedge S\mbox{ is
non-zero at $x$}\\}$ for some 1-forms $\Pi$, $\Omega$, $\Theta$ and $\omega$,
called the associated 1-forms. Especially, if $\Omega=\Theta=\omega=0$ (resp.,
$\Theta=\omega=0$ and $\Omega=\omega=0$), then the manifold is called
recurrent ([30], [31], [32], [58]) (resp., weakly generalized recurrent ([33],
[47]) and hyper generalized recurrent ([46], [48])) manifold.

Again as a generalization of locally symmetric manifold and recurrent
manifold, Tam$\acute{\mbox{a}}$ssy and Binh [55] introduced the notion of
weakly symmetric manifolds.

######  Definition 2.8.

Let $D$ be a (0, 4)-tensor on a semi-Riemannian manifold $M$. Then $M$ is said
to be weakly $D$-symmetric manifold ([55], [37]) if

$\displaystyle\nabla_{X}D(X_{1},X_{2},X_{3},X_{4})=\Pi(X)D(X_{1},X_{2},X_{3},X_{4})+\Omega(X_{1})D(X_{1},X_{2},X_{3},X_{4})$
$\displaystyle\hskip
56.9055pt+\overline{\Omega}(X_{2})D(X_{1},X,X_{3},X_{4})+\Theta(X_{3})D(X_{1},X_{2},X,X_{4})+\overline{\Theta}(X_{4})D(X_{1},X_{2},X_{3},X)$

holds $\forall~{}X,X_{i}\in\chi(M)$ $(i=1,2,3,4)$ and some 1-forms
$\Pi,\Omega,\overline{\Omega},\Theta$ and $\overline{\Theta}$ on $\\{x\in
M:R_{x}\neq 0\\}$. Such a manifold is called as weakly $D$-symmetric manifold
with solution $(\Pi,\Omega,\overline{\Omega},\Theta,\overline{\Theta})$. In
particular, if the solution is of the form $(2\Pi,\Pi,\Pi,\Pi,\Pi)$, then the
manifold is called Chaki $D$-pseudosymmetric manifold [6]. Again if the
solution is of the form $(\Pi,0,0,0,0)$ then the manifold is called
$D$-recurrent manifold ([30], [31], [32], [58]).

######  Definition 2.9.

Let $Z$ be a (0, 2)-tensor on a semi-Riemannian manifold $M$. Then $M$ is said
to be weakly $Z$-symmetric ([56], [37]) if

$\displaystyle(\nabla_{X}Z)(X_{1},X_{2})=\Pi(X)\,Z(X_{1},X_{2})+\Omega(X_{1})\,Z(X,X_{2})+\Theta(X_{2})\,Z(X_{1},X)$

holds $\forall~{}X,X_{1},X_{2}\in\chi(M)$ and some 1-forms $\Pi,\Omega$ and
$\Theta$ on $U_{Z}=\\{x\in M:Z\neq 0\ \mbox{ at }x\\}$. Such a manifold is
called as weakly $Z$-symmetric manifold with solution $(\Pi,\Omega,\Theta)$.
Especially, if the solution is of the form $(2\Pi,\Pi,\Pi)$ then the manifold
is called Chaki pseudo $Z$-symmetric manifold [7]. Again if the solution is of
the form $(\Pi$, $0$, $0)$ then the manifold is called $Z$-recurrent [27].

For details about the defining condition of weak symmetry and the
interrelation between weak symmetry and Deszcz psudosymmetry, we refer the
reader to see [36] and also references therein.

######  Definition 2.10.

A Riemannian manifold $M$ is said to be weakly cyclic Ricci symmetric [45] if
its Ricci tensor satisfies the condition

$\displaystyle(\nabla_{X}S)(X_{1},X_{2})+(\nabla_{X_{1}}S)(X,X_{2})+(\nabla_{X_{2}}S)(X_{1},X)$
$\displaystyle=\Pi(X)\,S(X_{1},X_{2})+\Omega(X_{1})\,S(X,X_{2})+\Theta(X_{2})\,S(X_{1},X),$

for three 1-forms $\Pi$, $\Omega$ and $\Theta$ on $M$. Such a manifold is
called weakly cyclic Ricci symmetric manifold with solution
$(\Pi,\Omega,\Theta)$.

It is noteworthy to mention that the solution of weakly cyclic Ricci symmetric
structure is not always unique.

######  Definition 2.11.

Let $D$ be a $(0,4)$ tensor and $Z$ be a $(0,2)$-tensor on $M$. Then the
corresponding curvature 2-forms $\Omega_{(D)l}^{m}$ ([2], [19]) are said to be
recurrent if and only if ([23], [24], [25])

$\displaystyle(\nabla_{X_{1}}D)(X_{2},X_{3},X,Y)+(\nabla_{X_{2}}D)(X_{3},X_{1},X,Y)+(\nabla_{X_{3}}D)(X_{1},X_{2},X,Y)=$
$\displaystyle\hskip
72.26999pt\Pi(X_{1})D(X_{2},X_{3},X,Y)+\Pi(X_{2})D(X_{3},X_{1},X,Y)+\Pi(X_{3})D(X_{1},X_{2},X,Y),$

and the 1-forms $\Lambda_{(Z)l}$ [54] are said to be recurrent if and only if

$(\nabla_{X_{1}}Z)(X_{2},X)-(\nabla_{X_{2}}Z)(X_{1},X)=\Pi(X_{1})Z(X_{2},X)-\Pi(X_{2})Z(X_{1},X)$

for an 1-form $\Pi$.

######  Definition 2.12.

$($[28], [41], [57]$)$ Let $\mathcal{L}(M)$ be the vector space formed by all
1-forms $\Theta$ on $M$ satisfying

$\Theta(X_{1})D(X_{2},X_{3},X_{4},X_{5})+\Theta(X_{2})D(X_{3},X_{1},X_{4},X_{5})+\Theta(X_{3})D(X_{1},X_{2},X_{4},X_{5})=0,$

where $D$ is a $(0,4)$-tensor. Then $M$ is said to be a $D$-space by Venzi if
$dim[\mathcal{L}(M)]\geq 1$.

From definition of recurrency of curvature 2-forms $\Omega_{(R)l}^{m}$ and
second Bianchi identity it is clear that on a semi-Riemannian manifold
$\Omega_{(R)l}^{m}$ are recurrent if and only if it is a $R$ space by Venzi.

##  3\. Curvature properties of pure radiation metric

The metric tensor of pure radiation metric (1.3) is given by

$\displaystyle
g=\left(\begin{array}[]{cccc}\left(xw-p^{2}\frac{r^{2}}{x^{2}}\right)&1&\frac{-2r}{x}&0\\\
1&0&0&0\\\ \frac{-2r}{x}&0&-\frac{1}{p^{2}}&0\\\ 0&0&0&-\frac{1}{p^{2}}\\\
\end{array}\right).$

Then the non-zero components (upto symmetry) of its Riemann-Christoffel
curvature tensor $R$, Ricci tensor $S$, scalar curvature $\kappa$, conformal
curvature tensor $C$ and projective curvature tensor $P$ are given by

$R_{1313}=-\frac{w_{33}x}{2},\ \ R_{1314}=-\frac{w_{34}x}{2},\ \
R_{1414}=-\frac{w_{44}x}{2},$
$S_{11}=-\frac{1}{2}p^{2}\left(w_{33}+w_{44}\right)x,\ \ \ \kappa=0,$
$-C_{1313}=C_{1414}=\frac{1}{4}\left(w_{33}-w_{44}\right)x,\ \
C_{1314}=-\frac{w_{34}x}{2},$
$P_{1211}=-\frac{1}{6}p^{2}\left(w_{33}+w_{44}\right)x,\ \
P_{1311}=\frac{1}{3}p^{2}r\left(w_{33}+w_{44}\right),$
$P_{1313}=-\frac{1}{6}\left(2w_{33}-w_{44}\right)x,\ \
-P_{1314}=P_{1341}=-P_{1413}=P_{1431}=\frac{w_{34}x}{2},$
$P_{1331}=\frac{w_{33}x}{2},\ \
P_{1414}=\frac{1}{6}\left(w_{33}-2w_{44}\right)x,\ \
P_{1441}=\frac{w_{44}x}{2}.$

Then from above it is easy to check that $R\cdot R=Q(S,R)=R\cdot C=Q(S,C)=0$.
Now the non-zero components (upto symmetry) of $\nabla R$, $\nabla S$, $\nabla
C$ are given by

$R_{1213,1}=\frac{p^{2}w_{33}}{2},\ \ R_{1214,1}=\frac{p^{2}w_{34}}{2},\ \
R_{1313,1}=-\frac{x^{2}w_{133}+2p^{2}rw_{33}}{2x},\ \
R_{1414,1}=\frac{2p^{2}rw_{44}-x^{2}w_{144}}{2x},$
$R_{1313,3}=\frac{1}{2}\left(w_{33}-w_{333}x\right),\ \
R_{1313,4}=-\frac{w_{334}x}{2},\ \ R_{1314,1}=-\frac{w_{134}x}{2},\ \
R_{1414,3}=\frac{1}{2}\left(w_{44}-w_{344}x\right),$
$R_{1314,3}=\frac{1}{2}\left(w_{34}-w_{334}x\right),\ \
R_{1314,4}=-\frac{w_{344}x}{2},\ \ R_{1334,1}=\frac{w_{34}}{2},\ \
R_{1414,4}=-\frac{w_{444}x}{2},\ \ R_{1434,1}=\frac{w_{44}}{2};$
$S_{11,1}=\frac{p^{2}\left(-x^{2}w_{144}-x^{2}w_{133}+2p^{2}rw_{33}+2p^{2}rw_{44}\right)}{2x},$
$S_{11,3}=\frac{1}{2}p^{2}\left(-w_{333}x-w_{344}x+w_{33}+w_{44}\right),$
$S_{11,4}=-\frac{1}{2}p^{2}\left(w_{334}+w_{444}\right)x,\ \
S_{13,1}=\frac{1}{2}p^{2}\left(w_{33}+w_{44}\right);$
$C_{1213,1}=\frac{1}{4}p^{2}\left(w_{33}-w_{44}\right),\ \
C_{1214,1}=\frac{p^{2}w_{34}}{2},$
$C_{1313,1}=-\frac{-x^{2}w_{144}+x^{2}w_{133}+2p^{2}rw_{33}-2p^{2}rw_{44}}{4x},$
$C_{1313,3}=-C_{1414,3}=\frac{1}{4}\left(-w_{333}x+w_{344}x+w_{33}-w_{44}\right),$
$-C_{1313,4}=C_{1414,4}=\frac{1}{4}\left(w_{334}-w_{444}\right)x,\ \
C_{1314,1}=-\frac{w_{134}x}{2},$
$C_{1314,3}=\frac{1}{2}\left(w_{34}-w_{334}x\right),\ \
C_{1314,4}=-\frac{w_{344}x}{2},\ \ C_{1334,1}=\frac{w_{34}}{2},$
$C_{1414,1}=-\frac{x^{2}w_{144}-x^{2}w_{133}+2p^{2}rw_{33}-2p^{2}rw_{44}}{4x},\
\ C_{1434,1}=-\frac{1}{4}\left(w_{33}-w_{44}\right).$

According to Einstein’s field equations, the energy momentum tensor $T$ for
zero cosmological constant is related to the Ricci tensor and the metric
tensor as

$T=\frac{c^{4}}{8\pi G}\left[S-\left(\frac{\kappa}{2}\right)g\right],$

where $c=$ speed of light in vacuum and $G=$ gravitational constant. Thus the
non-zero components of the energy momentum tensor of the pure radiation metric
(1.3) is given by:

$T_{11}=-\frac{c^{4}\left(p^{2}w_{33}x+p^{2}w_{44}x\right)}{16\pi Gx^{2}}.$

Obviously, $T=\rho(\eta\otimes\eta)$, where $\eta=\\{1,0,0,0\\}$ and the
radiation density $\rho=T_{11}$. It is easy to check that $||\eta||=0$.
Now the non-zero components of covariant derivative of $T$ are given by

$T_{11,1}=\frac{c^{4}p^{2}\left(-x^{2}w_{144}-x^{2}w_{133}+2p^{2}rw_{33}+2p^{2}rw_{44}\right)}{16\pi
Gx},\ \ T_{13,1}=\frac{c^{4}p^{2}\left(w_{33}+w_{44}\right)}{16\pi G}$
$T_{11,3}=\frac{c^{4}p^{2}\left(-w_{333}x-w_{344}x+w_{33}+w_{44}\right)}{16\pi
G},\ \ T_{11,4}=-\frac{c^{4}p^{2}\left(w_{334}+w_{444}\right)x}{16\pi G}.$

From the value of the local components (presented in Section 3) of various
tensors of the pure radiation metric (1.3), we can conclude that the pure
radiation metric (1.3) fulfills the following curvature restricted geometric
structures.

######  Theorem 3.1.

The pure radiation metric (1.3) possesses the following curvature properties:

  1. (i)

Its Ricci tensor is neither Codazzi type nor cyclic parallel but the scalar
curvature is zero and hence $R=W$ and $C=K$.

  2. (ii)

It is a $R$-space (also $C$-space, $P$-space) by Venzi for the associated
1-form $\Pi=\\{c,0,0,0\\}$, $c$ being arbitrary scalar. Hence the curvature
2-forms $\Omega_{(R)l}^{m}$ are recurrent for $\Pi$ as the 1-forms of
recurrency.

  3. (iii)

It is neither locally symmetric nor conformally symmetric but semisymmetric.
Hence it satisfies $R\cdot S=0$, $R\cdot C=0$ and $R\cdot P=0$.

  4. (iv)

It satisfies the semisymmetric type condition $C\cdot R=0$ and hence $C\cdot
S=0$, $C\cdot C=0$, $C\cdot P=0$ and $P\cdot S=0$.

  5. (v)

$R$ or $C$ of the space is not a scalar multiple of $S\wedge S$, but
$Q(S,R)=0$, $Q(S,C)=0$. Hence $P\cdot R=0$ and $P\cdot C=0$, although
$P\cdot\mathcal{R}\neq 0$ but $P\cdot\mathcal{S}=0$.

  6. (vi)

It is not Einstein but Ricci simple, since $S=\beta(\eta\otimes\eta)$, where
$\beta=-\frac{p^{2}~{}x~{}(w_{33}+w_{44})}{2}$ and $\eta=\\{1,0,0,0\\}$
(moreover $||\eta||=0$ and $\nabla\eta\neq 0$). Hence $S\wedge S=0$ and
$S^{2}=0$.

  7. (vii)

Here $P\cdot P\neq 0$ but $P\cdot P=-\frac{1}{3}Q(S,P)$.

  8. (viii)

If $w_{33}+w_{44}$ is nowhere zero, then the metric is neither recurrent nor
Ricci recurrent but Ricci 1-forms are recurrent with associated 1-form

$\left\\{1,0,\frac{w_{333}+w_{344}}{w_{33}+w_{44}},\frac{w_{334}+w_{444}}{w_{33}+w_{44}}\right\\}.$

  9. (ix)

If $4w_{34}^{2}+\left(w_{44}-w_{33}\right)^{2}$ is nowhere zero, then the
metric is not conformally recurrent but conformal 2-forms are recurrent with
associated 1-form $\Pi$, given by

$\Pi_{1}=1,\ \ \ \Pi_{2}=0,$
$\Pi_{3}=\frac{2w_{34}\left(w_{334}+w_{444}\right)-\left(w_{44}-w_{33}\right)\left(w_{333}+w_{344}\right)}{4w_{34}^{2}+\left(w_{44}-w_{33}\right)^{2}},$
$\Pi_{4}=\frac{2w_{34}\left(w_{333}+w_{344}\right)+\left(w_{44}-w_{33}\right)\left(w_{334}+w_{444}\right)}{4w_{34}^{2}+\left(w_{44}-w_{33}\right)^{2}}.$

  10. (x)

The general form of the compatible tensor for $R$, $C$ and $P$ are
respectively given by

$\left(\begin{array}[]{cccc}a_{(1,1)}&a_{(1,2)}&a_{(1,3)}&a_{(1,4)}\\\
a_{(2,1)}&0&0&0\\\ a_{(3,1)}&0&a_{(3,3)}&a_{(3,4)}\\\
a_{(4,1)}&0&a_{(4,3)}&a_{(3,3)}+\frac{w_{44}a_{(3,4)}}{w_{34}}-\frac{w_{33}a_{(4,3)}}{w_{34}}\end{array}\right),$
$\left(\begin{array}[]{cccc}a_{(1,1)}&a_{(1,2)}&a_{(1,3)}&a_{(1,4)}\\\
a_{(2,1)}&0&0&0\\\ a_{(3,1)}&0&a_{(3,3)}&a_{(3,4)}\\\
a_{(4,1)}&0&-\frac{2w_{34}a_{(3,3)}}{w_{44}-w_{33}}-a_{(3,4)}+\frac{2w_{34}a_{(4,4)}}{w_{44}-w_{33}}&a_{(4,4)}\end{array}\right)\
\mbox{and}$
$\left(\begin{array}[]{cccc}a_{(1,1)}&a_{(1,2)}&a_{(1,3)}&a_{(1,4)}\\\
a_{(2,1)}&0&0&0\\\
a_{(3,1)}&0&-\frac{\left(2w_{44}-w_{33}\right)a_{(3,4)}}{3w_{34}}-\frac{\left(w_{44}-2w_{33}\right)a_{(4,3)}}{3w_{34}}+a_{(4,4)}&a_{(3,4)}\\\
a_{(4,1)}&0&a_{(4,3)}&a_{(4,4)}\end{array}\right),$

where $a_{(i,j)}$ are arbitrary scalars.

  11. (xi)

Ricci tensor of this spacetime is not Codazzi type but is compatible for $R$,
$C$ and $P$.

  12. (xii)

If $w_{33}+w_{44}$ is nowhere vanishing, then the metric is weakly Ricci
symmetric with infinitely many solutions $(\Pi,\Omega,\Theta)$, given by

$\Pi=\left\\{\Pi_{1},0,\frac{w_{333}x+w_{344}x-w_{33}-w_{44}}{\left(w_{33}+w_{44}\right)x},\frac{w_{334}+w_{444}}{w_{33}+w_{44}}\right\\},$
$\Omega=\left\\{\Omega_{1},0,-\frac{1}{x},0\right\\}\ \ \mbox{and}$
$\Theta=\left\\{\frac{x^{2}w_{144}x^{2}w_{133}-2p^{2}rw_{33}-2p^{2}rw_{44}}{\left(w_{33}+w_{44}\right)x^{2}}-\Pi_{1}-\Omega_{1},0,-\frac{1}{x},0\right\\},$

where $\Pi_{1}$ and $\Omega_{1}$ are arbitrary scalars.

  13. (xiii)

If $w_{33}+w_{44}$ is nowhere vanishing, then the metric is weakly cyclic
Ricci symmetric with infinitely many solutions $(\Pi,\Omega,\Theta)$, given by

$\Pi=\left\\{\Pi_{1},0,\frac{\left(w_{333}+w_{344}\right)x-3w_{33}-3w_{44}}{\left(w_{33}+w_{44}\right)x},\frac{w_{334}+w_{444}}{w_{33}+w_{44}}\right\\},$
$\Omega=\left\\{\Omega_{1},0,\frac{\left(w_{333}+w_{344}\right)x-3w_{33}-3w_{44}}{\left(w_{33}+w_{44}\right)x},\frac{w_{334}+w_{444}}{w_{33}+w_{44}}\right\\}\
\ \mbox{and}$
$\displaystyle\Theta=\left\\{\frac{3\left(x^{2}\left(w_{144}+w_{133}\right)-2p^{2}r\left(w_{33}+w_{44}\right)\right)}{\left(w_{33}+w_{44}\right)x^{2}}-\Pi_{1}-\Omega_{1},0,\right.$
$\displaystyle\hskip
173.44756pt\left.\frac{\left(w_{333}+w_{344}\right)x-3w_{33}-3w_{44}}{\left(w_{33}+w_{44}\right)x},\frac{w_{334}+w_{444}}{w_{33}+w_{44}}\right\\},$

where $\Pi_{1}$ and $\Omega_{1}$ are arbitrary scalars.

  14. (xiv)

It is not weakly symmetric for $R$, $C$, $P$, $W$ and $K$ and hence not Chaki
pseudosymmetric for $R$, $C$ or $P$.

  15. (xv)

$divR\neq 0$, $divC\neq 0$, $divP\neq 0$.

Now from the values of the non-zero components of $\nabla T$, we get

$\left.\begin{array}[]{l}T_{11,1}+T_{11,1}+T_{11,1}=\frac{3c^{4}p^{2}\left(-x^{2}w_{144}-x^{2}w_{133}+2p^{2}rw_{33}+2p^{2}rw_{44}\right)}{16\pi
Gx},\\\
T_{11,3}+T_{13,1}+T_{31,1}=\frac{c^{4}p^{2}\left(-w_{333}x-w_{344}x+3w_{33}+3w_{44}\right)}{16\pi
G},\\\
T_{11,4}+T_{14,1}+T_{41,1}=-\frac{c^{4}p^{2}\left(w_{334}+w_{444}\right)x}{16\pi
G}\end{array}\right\\}$ (3.2)

and

$\left.\begin{array}[]{l}T_{13,1}-T_{11,3}=\frac{c^{4}p^{2}\left(w_{333}+w_{344}\right)x}{16\pi
G},\\\ T_{14,1}-T_{11,4}=\frac{c^{4}p^{2}\left(w_{334}+w_{444}\right)x}{16\pi
G}.\end{array}\right\\}$ (3.3)

Again since $R\cdot S=0$ and $T$ is a linear combination of $S$ and $g$, so
$R\cdot T=0$. Hence we can state the following:

######  Theorem 3.2.

The energy-momentum tensor $T$ of the pure radiation metric (1.3) is
(i) semisymmertric i.e., $R\cdot T=0$,
(ii) Codazzi type if $w_{33}+w_{44}$ is independent of $x$ and $y$,
(iii) cyclic parallel if $w_{33}+w_{44}$ is independent of $u$, $x$ and $y$,
(iv) covariantly constant if $w_{33}+w_{44}=0$.

##  4\. Curvature properties of pure radiation type metric

We now consider the pure radiation type metric (1.4). Its metric tensor is
given by

$\displaystyle
g=\left(\begin{array}[]{cccc}\left(xw+a\frac{r^{2}}{x^{2}}\right)&1&\frac{br}{x}&0\\\
1&0&0&0\\\ \frac{br}{x}&0&f&0\\\ 0&0&0&f\\\ \end{array}\right),$

where $a,b$ are arbitrary non-zero constants and $w=w(u,x,y)$, $f=f(x,y)$ are
nowhere vanishing functions. Then the non-zero components (upto symmetry) of
$R$, $S$ and $\kappa$ are given by

$R_{1212}=-\frac{4af-b^{2}}{4fx^{2}},\ \
R_{1213}=\frac{r\left(8af+b^{3}\right)}{4fx^{3}},\ \
R_{1334}=-\frac{b^{2}f_{4}r}{4fx^{2}},$ $\displaystyle
R_{1313}=-\frac{1}{4fx^{4}}$
$\displaystyle\left[ab^{2}fr^{2}+2abf_{3}r^{2}x+2af_{3}r^{2}x+12afr^{2}-b^{4}r^{2}+b^{2}fwx^{3}\right.$
$\displaystyle\left.+2bfw_{3}x^{4}+2bfwx^{3}-f_{3}w_{3}x^{5}+f_{4}w_{4}x^{5}+2fw_{33}x^{5}-f_{3}wx^{4}+4fw_{3}x^{4}\right],$
$R_{1314}=-\frac{2abf_{4}r^{2}+2af_{4}r^{2}+bfw_{4}x^{3}-f_{4}w_{3}x^{4}-f_{3}w_{4}x^{4}+2fw_{34}x^{4}-f_{4}wx^{3}+2fw_{4}x^{3}}{4fx^{3}},$
$R_{1323}=-\frac{b\left(bf+f_{3}x+2f\right)}{4fx^{2}},\ \
R_{1324}=R_{1423}=-\frac{bf_{4}}{4fx},$
$R_{1414}=-\frac{-2abf_{3}r^{2}-2af_{3}r^{2}+f_{3}w_{3}x^{4}-f_{4}w_{4}x^{4}+2fw_{44}x^{4}+f_{3}wx^{3}}{4fx^{3}},$
$R_{1424}=\frac{bf_{3}}{4fx},\ \ R_{1434}=\frac{b^{2}f_{3}r}{4fx^{2}},\ \
R_{3434}=-\frac{-f_{3}^{2}-f_{4}^{2}+ff_{33}+ff_{44}}{2f},$
$S_{11}=\frac{2a^{2}fr^{2}-3ab^{2}r^{2}-8abr^{2}+2afwx^{3}-6ar^{2}-b(bw+w_{3}x+w)x^{3}-(w_{33}+w_{44})x^{5}-2w_{3}x^{4}}{-2fx^{4}},$
$S_{12}=-\frac{2af-b^{2}-b}{2fx^{2}},\ \
S_{44}=-\frac{bff_{3}+f_{3}^{2}x+f_{4}^{2}x-ff_{33}x-ff_{44}x}{2f^{2}x},\ \
S_{34}=\frac{bf_{4}}{2fx},$
$S_{13}=\frac{r\left(4af+b^{3}+b^{2}\right)}{2fx^{3}},\ \
S_{33}=\frac{b^{2}f^{2}+2bf^{2}+bf_{3}fx+f_{33}fx^{2}+f_{44}fx^{2}-f_{3}^{2}x^{2}-f_{4}^{2}x^{2}}{2f^{2}x^{2}},$
$\kappa=\frac{-4af^{3}+b(3b+4)f^{2}+2\left(f_{33}+f_{44}\right)fx^{2}-2\left(f_{3}^{2}+f_{4}^{2}\right)x^{2}}{2f^{3}x^{2}}.$

Again for zero cosmological constant, the non-zero components (upto symmetry)
of the energy momentum tensor are given by

$\displaystyle T_{11}=\frac{c^{4}}{32\pi f^{3}Gx^{4}}$
$\displaystyle\left[3ab^{2}f^{2}r^{2}+12abf^{2}r^{2}+12af^{2}r^{2}+2af_{3}^{2}r^{2}x^{2}+2af_{4}^{2}r^{2}x^{2}-2aff_{33}r^{2}x^{2}\right.$
$\displaystyle-2aff_{44}r^{2}x^{2}-b^{2}f^{2}wx^{3}+2bf^{2}w_{3}x^{4}-2bf^{2}wx^{3}+2f^{2}w_{33}x^{5}+2f^{2}w_{44}x^{5}$
$\displaystyle\left.+4f^{2}w_{3}x^{4}+2f_{3}^{2}wx^{5}+2f_{4}^{2}wx^{5}-2ff_{33}wx^{5}-2ff_{44}wx^{5}\right],$
$T_{12}=-\frac{c^{4}\left(b^{2}f^{2}+2bf^{2}+2f_{33}fx^{2}+2f_{44}fx^{2}-2f_{3}^{2}x^{2}-2f_{4}^{2}x^{2}\right)}{32\pi
f^{3}Gx^{2}},$
$T_{13}=\frac{c^{4}r\left(4abf^{3}+8af^{3}+b^{3}\left(-f^{2}\right)-2b^{2}f^{2}+2bf_{3}^{2}x^{2}+2bf_{4}^{2}x^{2}-2bff_{33}x^{2}-2bff_{44}x^{2}\right)}{32\pi
f^{3}Gx^{3}},$
$T_{33}=\frac{c^{4}\left(4af^{2}+b^{2}(-f)+2bf_{3}x\right)}{32\pi fGx^{2}},\ \
T_{34}=\frac{bc^{4}f_{4}}{16\pi fGx},\ \
T_{44}=\frac{c^{4}\left(4af^{2}-3b^{2}f-2bf_{3}x-4bf\right)}{32\pi fGx^{2}}.$

From above we have the following:

######  Theorem 4.1.

The pure radiation type metric (1.4) has the following curvature properties:
(i) It is a 3-quasi-Einstein manifold, since $S-\frac{b+b^{2}-2af}{2fx^{2}}g$
is of rank 3.
(ii) For zero cosmological constant, its energy momentum tensor $T$ is of the
form

$T=\alpha g+\beta_{{}_{1}}(e_{1}\otimes e_{1})+\beta_{{}_{3}}(e_{3}\otimes
e_{3})+\beta_{{}_{4}}(e_{4}\otimes e_{4})+\sigma_{{}_{1}}(e_{1}\otimes
e_{3}+e_{3}\otimes e_{1})+\sigma_{{}_{2}}(e_{3}\otimes e_{4}+e_{4}\otimes
e_{3}),$

where

$e_{1}=\\{1,0,0,0\\},\ \ e_{3}=\\{0,0,1,0\\},\ \ e_{4}=\\{0,0,0,1\\},$
$\alpha=-\frac{c^{4}\left[b(b+2)f^{2}+2\left(f_{33}+f_{44}\right)fx^{2}-2\left(f_{3}^{2}+f_{4}^{2}\right)x^{2}\right]}{32\pi
f^{3}Gx^{2}},$ $\beta_{{}_{1}}=T_{11}-\alpha g_{11},\ \
\beta_{{}_{3}}=T_{33}-\alpha g_{33},\ \ \beta_{{}_{4}}=T_{44}-\alpha g_{44},\
\ \sigma_{{}_{1}}=T_{13}-\alpha g_{13},\ \ \sigma_{{}_{2}}=T_{34}-\alpha
g_{34}.$

(iii) It is an $Ein(4)$ manifold, such that

$\displaystyle S^{4}$ $\displaystyle=$
$\displaystyle\frac{\left(-2af+b^{2}+b\right)^{2}}{16f^{8}x^{7}}\left[b(b+2)f^{3}\left(\left(f_{33}+f_{44}\right)x-bf_{3}\right)\right.$
$\displaystyle+f^{2}x\left(\left(f_{33}+f_{44}\right){}^{2}x^{2}-2b(b+1)\left(f_{3}^{2}+f_{4}^{2}\right)\right)$
$\displaystyle\left.-2\left(f_{3}^{2}+f_{4}^{2}\right)\left(f_{33}+f_{44}\right)fx^{3}+\left(f_{3}^{2}+f_{4}^{2}\right)^{2}x^{3}\right]g$
$\displaystyle-$
$\displaystyle\frac{\left(-2af+b^{2}+b\right)}{8f^{7}x^{6}}\left[f^{4}\left(b^{2}(b+1)(b+2)-4a\left(f_{33}+f_{44}\right)x^{2}\right)\right.$
$\displaystyle+2f^{3}x\left(x\left(2af_{4}^{2}+b(2b+3)\left(f_{33}+f_{44}\right)\right)+2af_{3}^{2}x-(b+2)b^{2}f_{3}\right)$
$\displaystyle-2ab(b+2)f^{5}+2f^{2}x^{2}\left(\left(f_{33}+f_{44}\right)^{2}x^{2}-3b(b+1)\left(f_{3}^{2}+f_{4}^{2}\right)\right)$
$\displaystyle\left.-4\left(f_{3}^{2}+f_{4}^{2}\right)\left(f_{33}+f_{44}\right)fx^{4}+2\left(f_{3}^{2}+f_{4}^{2}\right)^{2}x^{4}\right]S$
$\displaystyle+$
$\displaystyle\frac{1}{4f^{6}x^{4}}\left[4a^{2}f^{6}+f^{4}\left(b^{2}(b+1)(3b+5)-8a\left(f_{33}+f_{44}\right)x^{2}\right)\right.$
$\displaystyle+f^{3}x\left(x\left(8af_{4}^{2}+b(5b+6)\left(f_{33}+f_{44}\right)\right)+8af_{3}^{2}x-(b+2)b^{2}f_{3}\right)$
$\displaystyle-4ab(2b+3)f^{5}+f^{2}x^{2}\left(\left(f_{33}+f_{44}\right){}^{2}x^{2}-6b(b+1)\left(f_{3}^{2}+f_{4}^{2}\right)\right)$
$\displaystyle\left.-2\left(f_{3}^{2}+f_{4}^{2}\right)\left(f_{33}+f_{44}\right)fx^{4}+\left(f_{3}^{2}+f_{4}^{2}\right)^{2}x^{4}\right]S^{2}$
$\displaystyle+$
$\displaystyle\frac{1}{2f^{3}x^{2}}\left[f^{2}(4af-b(3b+4))+2\left(f_{3}^{2}+f_{4}^{2}-f\left(f_{33}+f_{44}\right)\right)x^{2}\right]S^{3},$

######  Remark 4.1.

Since for zero cosmological constant $S=\frac{\kappa}{2}g+\frac{8\pi
G}{c^{4}}T$, hence from above theorem we can state that the Ricci tensor of
(1.4) is of the form

$\displaystyle S$ $\displaystyle=$
$\displaystyle(S_{12})g+(S_{11}-S_{12}g_{11})(e_{1}\otimes
e_{1})+(S_{33}-S_{12}g_{33})(e_{3}\otimes
e_{3})+(S_{44}-S_{12}g_{44})(e_{4}\otimes e_{4})$
$\displaystyle+(S_{13}-S_{12}g_{13})(e_{1}\otimes e_{3}+e_{3}\otimes
e_{1})+(S_{34}-S_{12}g_{34})(e_{3}\otimes e_{4}+e_{4}\otimes e_{3}).$

We can easilly check that $||e_{1}||=0$ and the nonzero components of $\nabla
e_{1}$ are given by

$\nabla_{{}_{1}}e_{1}=\frac{ar}{x^{2}},\ \ \nabla_{{}_{3}}e_{1}=\frac{b}{2x}.$

We know that a spacetime is called generalized pp-wave ([29], [35]) if there
exists a covariantly constant null vector field. Hence we can state the
following:

######  Theorem 4.2.

The pure radiation type metric (1.4) represents generalized pp-wave if
$a=b=0$.

Now a spacetime is 2-quasi-Einstein if Rank$(S-\alpha g)=0$. Hence the metric
(1.4) becomes 2-quasi-Einstein if one of the following condition holds

$(i)\ (S_{34}-S_{12}g_{34})=(S_{44}-S_{12}g_{44})=0,$ $(ii)\
(S_{13}-S_{12}g_{13})=(S_{33}-S_{12}g_{33})=(S_{34}-S_{12}g_{34})=0,$ $(iii)\
(S_{11}-S_{12}g_{11})=(S_{13}-S_{12}g_{13})=0.$

Simplifying the above conditions we can state the following:

######  Theorem 4.3.

The pure radiation type metric (1.4) becomes 2-quasi-Einstein if any one the
following condition holds

$\begin{array}[]{l}(i)\
b=2af^{3}-\left(f_{3}^{2}+f_{4}^{2}-f\left(f_{33}+f_{44}\right)\right)x^{2}=0,\\\
(ii)\
f_{4}=2af^{3}+xf\left(xf_{33}-bf_{3}\right)-b(b+1)f^{2}-x^{2}f_{3}^{2}=0,\\\
(iii)\ a=b=-f_{3}^{2}+f_{4}^{2}+f\left(f_{33}+f_{44}\right)=0,\\\ (iv)\
f_{4}=a=bf_{3}+\frac{bf}{x}+xf_{33}-\frac{xf_{3}^{2}}{f}=0,\\\ (v)\
f_{4}=b-2=\frac{2af^{2}}{x}+xf_{33}-2f_{3}-\frac{xf_{3}^{2}}{f}-\frac{2f}{x}=0,\\\
(vi)\ a=(b+2)w_{3}+w_{33}x+w_{44}x=0,\\\ (vii)\
b-2=w_{33}+w_{44}=0.\end{array}$

######  Example 4.1.

If we consider the metric (1.4) with
$f(x,y)=\frac{e^{\frac{1}{3}x^{3}}}{x^{2/3}}$, $b=-2$ and $a=0$, then (1.4)
becomes a 2-quasi-Einstein manifold.

Now a spacetime is perfect fluid if it satisfies (1.2). Hence the metric (1.4)
represents perfect fluid if one of the following condition holds

$\mbox{(i) }\beta_{{}_{1}}=\beta_{{}_{3}}=\sigma_{{}_{1}}=\sigma_{{}_{2}}=0,$
$\mbox{(ii) }\beta_{{}_{1}}=\beta_{{}_{4}}=\sigma_{{}_{1}}=\sigma_{{}_{2}}=0,$
$\mbox{(iii)
}\beta_{{}_{3}}=\beta_{{}_{4}}=\sigma_{{}_{1}}=\sigma_{{}_{2}}=0.$

Simplifying the above conditions we can state the following:

######  Theorem 4.4.

The pure radiation type metric (1.4) represents perfect fluid if any one the
following condition holds

$\begin{array}[]{l}(i)\
a=b=2w_{3}+xw_{33}+xw_{44}=f_{3}^{2}+f_{4}^{2}-f\left(f_{33}+f_{44}\right)=0,\\\
(ii)\
b+2=f_{4}=w_{33}+w_{44}=2af^{3}-x^{2}f_{3}^{2}+xf\left(xf_{33}-2f_{3}\right)-2f^{2}=0,\\\
(iii)\
a=f_{4}=(b+2)w_{3}+w_{33}x+w_{44}x=xf\left(bf_{3}+xf_{33}\right)+bf^{2}-x^{2}f_{3}^{2}=0,\\\
(iv)\
a=f_{4}=xf\left(bf^{\prime}+xf^{\prime\prime}\right)+bf^{2}-x^{2}f^{\prime
2}=xf\left(bf^{\prime}-xf^{\prime\prime}\right)+b(b+1)f^{2}+x^{2}f^{\prime
2}=0,\\\ (v)\ b+2=f-\frac{1}{a}=0.\end{array}$

######  Example 4.2.

Consider the metric (1.4), where $w(u,x,y)=uxy$,
$f(x,y)=\frac{e^{\frac{1}{3}x^{3}}}{x^{2/3}}$, $b=-2$ and $a=0$, then (1.4)
represents a perfect fluid spacetime. The energy momentum tensor of this
metric can be expressed as

$T=-\frac{c^{4}e^{-\frac{x^{3}}{3}}\left(3x^{3}+1\right)}{24\pi
Gx^{4/3}}g+\frac{c^{4}\left(3x^{3}-2\right)}{12\pi Gx^{2}}(e_{4}\otimes
e_{4}).$

Moreover in this case the metric is quasi-Einstein and $Ein(2)$.

Again a spacetime is a pure radiation spacetime if it satisfies (1.1). Now
$||e_{1}||=0$ and hence the metric (1.4) represents perfect fluid if
$\alpha=\beta_{{}_{3}}=\beta_{{}_{4}}=\sigma_{{}_{1}}=\sigma_{{}_{2}}=0$.
Simplifying these conditions we can state the following:

######  Theorem 4.5.

The pure radiation type metric (1.4) represents pure radiation if
$f\equiv\frac{1}{a}$ and $b=-2$.

Again from above we can easily calculate (large but straightforward) the
components of $R\cdot R$, $Q(g,R)$ and $Q(S,R)$. Then we have the following:

######  Theorem 4.6.

The pure radiation type metric (1.4) is
(i) Ricci generalized pseudosymmetric ($R\cdot R=Q(S,R)$) if $b=-2$ and $f$ is
constant. And in this case the metric is $R$-space by Venzi for the associated
1-form $\\{1,0,0,0\\}$,
(ii) a manifold of vanishing scalar curvature if $f\equiv\frac{b(3b+4)}{4a}$,
(iii) semisymmetric if $b=-2$ and $f\equiv\frac{1}{a}$.

From (1.3), (1.4) and (1.5), we see that both the pure radiation metric and
the pp-wave metric are special cases of the metric (1.4). We refer the reader
to see [26], [43] and also references therein for recent works on pp-wave
metric. We now draw a comparison (similarities and dissimilarities) between
the curvature properties of pure radiation metric and pp-wave metric.
A. Similarities:

  1. (1)

Both the metrics are of vanishing scalar curvature,

  2. (2)

both are $R$-space by Venzi as well as $C$-space by Venzi,

  3. (3)

both are semisymmetric and semisymmetric due to conformal curvature tensor,

  4. (4)

for both the metrics $Q(S,R)=Q(S,C)=0$,

  5. (5)

both the metrics are Ricci simple,

  6. (6)

Ricci tensors of both metrics are Riemann compatible as well as conformal
compatible,

  7. (7)

Ricci 1-forms $\Lambda_{(Z)l}$ of both metrics are recurrent,

  8. (8)

conformal 2-forms $\Omega_{(C)l}^{m}$ of both metrics are recurrent,

  9. (9)

for both the metrics $P\cdot R=0$ but $P\cdot\mathcal{R}\neq 0$.

  10. (10)

both are weakly Ricci symmetric and hence weakly cyclic Ricci symmetric,

  11. (11)

both the metrics satisfy $P\cdot P=-\frac{1}{3}Q(S,P)$,

  12. (12)

the energy-momentum tensors of both the metrics are semisymmetric.

B. Dissimilarities:

  1. (1)

For zero cosmological constant, the energy-momentum tensors of both the
metrics are of rank one, but the associated 1-form for radiation metric is
null, and for pp-wave metric it is null as well as covariantly constant,

  2. (2)

pp-wave metric is Ricci recurrent but pure radiation metric is not so,

  3. (3)

for pp-wave metric energy-momentum tensor is cyclic parallel if and only if it
is parallel but this fact is not true for pure radiation metric.

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