# Conformal symmetries of the energy-momentum tensor of spherically symmetric
static spacetimes

Ugur Camci  Department of Chemistry and Physics, Roger Williams University,
One Old Ferry Road, Bristol, Rhode Island 02809, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>Khalid Saifullah  Department of Mathematics, Quaid-i-Azam
University, Islamabad, Pakistan<EMAIL_ADDRESS>
###### Abstract

Conformal matter collineations of the energy-momentum tensor for a general
spherically symmetric static spacetime are studied. The general form of these
collineations is found when the energy-momentum tensor is non-degenerate, and
the maximum number of independent conformal matter collineations is _fifteen_.
In the degenerate case of the energy-momentum tensor it is found that these
collineations have infinite degrees of freedom. In some subcases of degenerate
energy-momentum, the Ricci tensor is non-degenerate, that is, there exist non-
degenerate Ricci inheritance collineations.

###### pacs:

04.20-q, 04.20.Jb, 04.70.-s, 11.30.-j

##  I Introduction

Recent observations indicate that the universe contains black holes whose
horizons are rotating at a speed close to that of light. General relativity
(GR) suggests that the dynamics near the horizon of such black holes is
governed by a strong infinite-dimensional conformal symmetry - similar to the
one seen near the critical points of different condensed matter systems.
Researchers have explored possible observational consequences of such a
symmetry (AS, ).

Symmetries and conformal symmetries play a very important role in mathematical
physics. One of the fundamental symmetries on a Riemannian manifold is that of
the metric tensor $\mathbf{g}$ written mathematically as katzin

$\pounds_{\xi}{\bf g}=2\sigma{\bf g}.$ (1)

Here $\sigma$ is the conformal factor and $\pounds_{\xi}$ represents the Lie
derivative operator relative to the vector field ${\xi}$, which gives
isometries or Killing vectors (KVs) if the $\sigma$ is zero, homothetic
motions (HMs) if it is a constant, and conformal Killing vectors (CKVs) if it
is a function of the coordinates $x^{a}$. In component form we can write the
above equation as

$g_{ab,c}\xi^{c}+g_{ac}\xi_{,b}^{c}+g_{cb}\xi^{c}_{,a}=2\sigma g_{ab}.$ (2)

Apart from the metric tensor the other quantities fundamental to the Einstein
field equations (EFEs)

$R_{ab}-\frac{1}{2}Rg_{ab}=\kappa T_{ab},$ (3)

are the stress-energy tensor, ${\bf T}$, which describes the matter field in
the manifold, the Ricci tensor, $\bf R$, which is a contraction of the
curvature tensor, and the Ricci scalar $R$. In Eq. (3) $\kappa$ is the
coupling constant defined by $\kappa=8\pi G/c^{4}$, $G$ and $c$ are Newton’s
gravitational constant and the speed of light. Thus, the symmetries of both
the stress-energy tensor and the Ricci tensor, play a significant role. These
symmetries known as matter collineations (MCs) and Ricci collineations (RCs)
hall-book , respectively, satisfy the equations

$\pounds_{\xi}{\bf T}=0,\,\,\,\ \text{and}\,\,\,\ \pounds_{\xi}{\bf R}=0.$ (4)

Similarly, one can define collineations for the curvature and Weyl tensors
hall-book .

Solutions of EFEs can be classified by requiring these symmetries and thus a
complete list of metrics having certain symmetry can be obtained ESEFEs .
Spacetimes have been classified on the basis, for example, of KVs bq1987 ;
qz1988 ; qz1995 , HMs dz1997 , CKVs maartens1995 ; keane2004 , RCs tm1990 ;
bk1993 ; fqz1995 ; hall1996 ; qz1998 ; ziad2003 ; saif2003 ; ugur1 ; cb2002 ;
ugur2 ; ugur3 ; tsamparlis1  and MCs ugur4 ; sharif2003 ; sharif2004 ; ugur5 ;
ugur6 . This also provides a way to find new solutions of EFEs which are
otherwise very difficult to solve. These collineations have been generalized
to define what are called _conformal collineations_ (or _inheritance
collineations_ duggal1 ; duggal11 ; duggal2 ). Thus we obtain conformal matter
collineations (CMCs)

$\pounds_{\bf X}{\bf T}=2\psi(x^{a}){\bf T},$ (5)

or conformal Ricci collineations (CRCs) defined by

$\pounds_{\bf Y}{\bf R}=2\phi(x^{a}){\bf R}.$ (6)

Conformal symmetry is physically significant as CKVs, for example, generate
constants of motion along the null geodesics for massless particles which are
conserved quantities. On the other hand, it is of mathematical interest to
obtain classification by conformal collineations and to investigate their
relation with collineations. Though there has been a good amount of literature
on the study of CKVs, the interest in conformal collineations is relatively
recent. The complete classification of spherically symmetric static space-
times by their CRCs when the conformal factor $\phi$ is a constant has been
carried out in Ref. bkk2003 . The CRCs with a non-constant $\phi$ have been
studied for the Friedmann-Robertson-Walker spacetimes cb2002 , the general
static spherically symmetric spacetimes crcsph , the non-static spherically
symmetric spacetimes bokhari2020  and Kantowski-Sachs spacetimes hkb2018 . For
pp-waves relationship between CRCs and CKVs has been studied in Ref. keane2004
. Recently, Akhtar et al. bokhari2018  have classified static plane symmetric
space-times according to CRCs. Further, the CRCs for the Einstein-Maxwell
field equations in the case of non-null electromagnetic fields have been
investigated as well faridi .

In this paper we classify spherically symmetric static spacetimes by their
CMCs. The equation (5) for CMCs in component form can be written as

$T_{ab,c}X^{c}+T_{ac}X^{c}_{,b}+T_{cb}X^{c}_{,a}=2\psi T_{ab},$ (7)

where $\psi$ is the conformal factor which is a function of all the spacetime
coordinates $x^{a}=(x^{0},x^{1},x^{2},x^{3})$. In this paper we use the usual
component notation in local charts and a partial derivative will be denoted by
a comma. Note that the above equation gives MCs if $\psi=0$, thus the
classification of spherically symmetric static spacetimes by MCs sharif2003
becomes a special case of the classification obtained in this paper. We call a
CMC as _proper_ if it is neither a KV nor an MC. The set of all CMCs on the
manifold is a vector space, but it may be infinite dimensional. If $T_{ab}$ is
non-degenerate, i.e. $det(T_{ab})\neq 0$, then the Lie algebra of CMCs is
finite dimensional but if $T_{ab}$ is degenerate, it may be infinite. Thus, in
the case of a non-degenerate energy-momentum tensor, i.e. $det(T_{ab})\neq 0$,
we use the standard results on conformal symmetries to obtain the maximal
dimensions of the algebra of CMCs as 15. Since $T_{ab}$ describes the
distribution and motion of matter contents of a manifold, and mathematically
it is very similar to the the Ricci tensor, the study of MCs and CMCs has a
natural geometrical as well as physical significance.

In the next section, we setup the CMC equations for static spherically
symmetric spacetimes. In Section III these equations are solved when the
energy-momentum tensor is degenerate, while in Section IV we obtain results
when the tensor is non-degenerate. We find that the degenerate case always
gives infinite dimensional Lie algebras of CMCs. We conclude with a brief
summary and discussion in Section V. Throughout the paper, we will consider
four-dimensional space-times, and space-time indices will be denoted by small
Latin letters (e.g., $a$, $b$, $c$,…) and the metric has signature
$(+,-,-,-)$.

##  II Equations for Conformal Matter Collineations

We consider a general spherically symmetric static spacetime in the usual
spherical coordinates

$ds^{2}=e^{\nu(r)}dt^{2}-e^{\lambda(r)}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (8)

The non-vanishing components of the Ricci tensor $R_{ab}=R^{c}_{\,acb}$ for
this metric are given by

$\displaystyle R_{00}\equiv
R_{0}(r)=\frac{1}{4}e^{\nu-\lambda}\left(2\nu^{\prime\prime}+{\nu^{\prime}}^{2}-\nu^{\prime}\lambda^{\prime}+\frac{4}{r}\nu^{\prime}\right),$
(9) $\displaystyle R_{11}\equiv
R_{1}(r)=-\frac{1}{4}\left(2\nu^{\prime\prime}+{\nu^{\prime}}^{2}-\nu^{\prime}\lambda^{\prime}-\frac{4}{r}\lambda^{\prime}\right),$
(10) $\displaystyle R_{22}\equiv
R_{2}(r)=\frac{1}{2}e^{-\lambda}\left[r(\lambda^{\prime}-\nu^{\prime})-2\right]+1,$
(11) $\displaystyle R_{33}=\sin^{2}\theta R_{2},$ (12)

and the Ricci scalar $R=R^{a}_{\,\,a}$ is

$R=\frac{e^{-\lambda}}{2}\left[2\nu^{\prime\prime}+\nu^{\prime
2}-\nu^{\prime}\lambda^{\prime}+\frac{4}{r}(\nu^{\prime}-\lambda^{\prime})+\frac{4}{r^{2}}(1-e^{\lambda})\right],$
(13)

where the prime represents derivative with respect to the radial coordinate
$r$. Thus we can write the _Ricci tensor form_ as

$ds^{2}_{Ric}\equiv
R_{ab}dx^{a}dx^{b}=R_{0}(r)dt^{2}+R_{1}(r)dr^{2}+R_{2}(r)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (14)

The metric (8) has time-independent coefficients, and using the field
equations (3) with $c=1$ and $G=\frac{1}{8\pi}$, i.e., $\kappa=1$, the
components of the energy-momentum tensor $T_{ab}$ become

$\displaystyle T_{00}\equiv
T_{0}(r)=\frac{1}{r^{2}}e^{\nu-\lambda}\left(r\lambda^{\prime}+e^{\lambda}-1\right),$
(15) $\displaystyle T_{11}\equiv
T_{1}(r)=\frac{1}{r^{2}}\left(r\nu^{\prime}-e^{\lambda}+1\right),$ (16)
$\displaystyle T_{22}\equiv
T_{2}(r)=\frac{r^{2}}{4}e^{-\lambda}\left[2\nu^{\prime\prime}+{\nu^{\prime}}^{2}-\nu^{\prime}\lambda^{\prime}+\frac{2}{r}(\nu^{\prime}-\lambda^{\prime})\right],$
(17) $\displaystyle T_{33}\equiv T_{3}(r)=\sin^{2}\theta T_{2}.$ (18)

Similarly, the matter tensor form can be written as

$ds^{2}_{Matter}\equiv
T_{ab}dx^{a}dx^{b}=T_{0}(r)dt^{2}+T_{1}(r)dr^{2}+T_{2}(r)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (19)

Using the above energy-momentum tensor components, the Ricci curvature scalar
given in (13) can be cast into the form

$R=-e^{-\nu}T_{0}+e^{-\lambda}T_{1}+\frac{2}{r^{2}}T_{2}\,.$ (20)

Further, when we state the energy-momentum tensor components given above in
terms of the Ricci tensor components (9)-(11) we find that

$\displaystyle
T_{0}=\frac{1}{2}\left(R_{0}+e^{\nu-\lambda}\,R_{1}\right)+\frac{e^{\nu}}{r^{2}}R_{2},$
(21) $\displaystyle
T_{1}=\frac{1}{2}\left(e^{\lambda-\nu}R_{0}+R_{1}\right)-\frac{e^{\lambda}}{r^{2}}R_{2},$
(22) $\displaystyle
T_{2}=\frac{r^{2}}{2}\left(e^{-\nu}R_{0}-e^{-\lambda}R_{1}\right).$ (23)

In GR, physical fields are described by the symmetric tensor $T_{ab}$ which is
the energy-momentum tensor of the field. We have the covariant
decomposition/identity for $T_{ab}$ as follows tp2019 :

$T_{ab}=\rho u_{a}u_{b}-ph_{ab}+2q_{(a}u_{b)}+\pi_{ab}\,,$ (24)

where $h_{ab}=g_{ab}-u_{a}u_{b}$ is the projection tensor, and the quantities
$\rho,p,q_{a}$ and $\pi_{ab}$ are the physical variables representing the mass
density, the isotropic pressure, the heat flux and the traceless stress
tensor, respectively, as measured by the observers $u^{a}$. In the above
decomposition, $T_{ab}$ is described by two scalar fields ($\rho,p$), one
spacelike vector ($q^{a},\,q_{a}u^{a}=0$), and a traceless symmetric 2-tensor
($\pi_{ab},\,g_{ab}\pi^{ab}=0$). The irreducible parts of $T_{ab}$ are defined
as

$\displaystyle\rho=u^{a}u^{b}T_{ab}\,,$ (25) $\displaystyle
p=-\frac{1}{3}h^{ab}T_{ab}\,,$ (26) $\displaystyle q^{a}=h^{ab}T_{bc}u^{c}\,,$
(27)
$\displaystyle\pi_{ab}=\left(h_{a}^{\,c}h_{b}^{\,d}-\frac{1}{3}h_{ab}h^{cd}\right)T_{cd}\,,$
(28)

where $u^{a}$ is a timelike unit four vector field normalized by
$u^{a}u_{a}=1$. The energy-momentum tensor $T_{ab}$ given in (24) represents
the general anisotropic fluid, and reduces to an anisotropic fluid without
heat flux if $q^{a}=0$, an isotropic non-perfect fluid if $\pi^{ab}=0$, a
perfect fluid if $q^{a}=0$ and $\pi^{ab}=0$, and a dust if $p=0,q^{a}=0$ and
$\pi^{ab}=0$. GR is a classical theory of relativity, however, in the field
equations (3), the classical spacetime geometry is also related to the stress-
energy tensor of quantum matter. To overcome this inconsistency, we need to
embed GR (or its generalizations) within some quantum mechanical framework,
i.e., quantum gravity. For the metric ansatz (8), it is customary to choose
the fluid to be at rest because the spacetime is static, i.e.,
$u^{a}=u^{0}\delta^{a}_{0}$. Then, using the normalization condition of the
four-velocity, that is, $u^{a}u_{a}=1$, one can find $u^{0}=e^{-\nu/2}$. Thus,
we find from (27) that for this choice of observers the heat flux vanishes
($q^{a}=0$), which is expected from the symmetries of the metric. Also, under
the latter choice of observers, the remaining physical variables $\rho,p$ and
$\pi_{ab}$ that follow from (25), (26) and (28) are

$\displaystyle\rho=e^{-\nu}T_{0}\,,\qquad
p=\frac{1}{3}\left(e^{-\lambda}T_{1}+\frac{2}{r^{2}}T_{2}\right)\,,$ (29)
$\displaystyle\pi_{00}=0,\quad\pi_{11}=\frac{2}{3}\left(T_{1}-\frac{e^{\lambda}}{r^{2}}T_{2}\right)\,,\quad\pi_{22}=-\frac{r^{2}}{2}e^{-\lambda}\pi_{11}\,,\quad\pi_{33}=\sin^{2}\theta\,\pi_{22}\,.$
(30)

If the choice of observers as a timelike four-vector field
$u^{a}=u^{0}\delta^{a}_{0}$ is not appropriate for any reason, then we must
apply the normalization condition of the four-velocity with $u^{a}u_{a}=-1$ by
choosing the four-velocity as $u^{a}=u^{1}\delta^{a}_{1}$ which is a spacelike
four-vector, since one can always normalize the four-velocity to $\pm 1$. Then
we find for the metric (8) that $u^{1}=e^{-\lambda/2}$. For the spacelike
four-vector, the projection tensor has the form $h_{ab}=g_{ab}+u_{a}u_{b}$.
Then this choice effects the mass density, the isotropic pressure $p$ and the
traceless stress tensor $\pi_{ab}$ due to Eqs. (25), (26) and (28) such that

$\displaystyle\rho=e^{-\lambda}T_{1}\,,\qquad
p=-\frac{1}{3}\left(e^{-\nu}T_{0}-\frac{2}{r^{2}}T_{2}\right)\,,$ (31)
$\displaystyle\pi_{11}=0,\quad\pi_{00}=\frac{2}{3}\left(T_{0}+\frac{e^{\nu}}{r^{2}}T_{2}\right)\,,\quad\pi_{22}=\frac{r^{2}}{2}e^{-\nu}\pi_{00}\,,\quad\pi_{33}=\sin^{2}\theta\,\pi_{22}\,.$
(32)

In GR, it is conventional to restrict the possible energy-momentum tensors by
imposing energy conditions. The energy conditions for the energy-momentum
tensor $T_{ab}$ to represent some known matter fields are the conditions that
are coordinate-independent restrictions on $T_{ab}$. In literature, there are
five categories of the energy conditions. These are the trace energy
conditions (TEC), null energy conditions (NEC), weak energy conditions (WEC),
strong energy conditions (SEC) and dominant energy conditions (DEC). The TEC
means that the trace of the energy-momentum tensor $T=g^{ab}T_{ab}$ should
always be positive (or negative depending on the signature of metric). The NEC
mathematically states that $T_{ab}k^{a}k^{b}\geq 0$ for any null vector
$k^{a}$, i.e., $k^{a}k_{a}=$. On the other hand, the WEC requires that
$T_{ab}t^{a}t^{b}\geq 0$ for all timelike vectors $t^{a}$. The SEC states that
$T_{ab}t^{a}t^{b}\geq\frac{1}{2}Tt^{c}t_{c}$ for all timelike vectors $t^{a}$.
The DEC includes the WEC, as well as the additional requirement that
$T^{ab}t_{b}$ is a non-spacelike vector, i.e.,
$T_{ab}T^{b}_{\,\,c}t^{a}t^{c}\leq 0$ carrol ; capo2015 . For a perfect fluid,
the energy conditions are described as

$\displaystyle{\rm TEC:}\quad\rho-3p\geq 0\,;\qquad{\rm NEC:}\quad\rho+p\geq
0\,;\qquad{\rm WEC:}\quad\rho\geq 0\,,\,\,\rho+p\geq 0\,;$ (33)
$\displaystyle{\rm SEC:}\quad\rho+3p\geq 0\,,\,\,\rho+p\geq 0;\quad{\rm
DEC:}\quad\rho\geq 0\,,\,\,\mid p\mid\leq\rho\,.$ (34)

Thus it is seen that the energy conditions are simple constraints on various
linear combinations of the energy density $\rho$ and the pressure $p$. The
matter including both positive energy density and positive pressure, which is
called ”normal” matter, satisfies all the standard energy conditions. On the
contrary, ”exotic” matter violates any one of the energy conditions. For
example, the SEC is satisfied by ”all known forms of energy”, but not by the
dark energy where $p=-\rho$. We note that for the static and spherically
symmetric metric (8) one can find the null vector as
$k^{a}=e^{-\nu/2}\delta^{a}_{0}+e^{-\lambda/2}\delta^{a}_{1}$ by using the
condition $k^{a}k_{a}=0$. Then the NEC for the general anisotropic fluid (24)
becomes

$\rho+3p-\frac{2}{r^{2}}T_{2}\geq 0\,,$ (35)

which yields the perfect fluid energy condition taking $T_{2}=pr^{2}$.

For the perfect fluid, it is easily seen that $T_{0}=\rho
e^{\nu},\,\,T_{1}=pe^{\lambda},\,\,T_{2}=pr^{2}$, i.e.,
$T_{2}=r^{2}e^{-\lambda}T_{1}$, and $T_{3}=sin^{2}\theta T_{2}$, which yields
$\pi_{ab}=0$ as it should be, and these give rise to the following relations

$R_{0}=\frac{e^{\nu}}{2}(\rho+3p),\quad
R_{1}=\frac{e^{\lambda}}{2}(\rho-p),\quad R_{2}=\frac{r^{2}}{2}(\rho-p),$ (36)

where $\rho$ and $p$ are, respectively, density and pressure of the fluid,
which are

$\rho=\frac{e^{-\lambda}}{r^{2}}\left(r\lambda^{\prime}+e^{\lambda}-1\right),\qquad
p=\frac{e^{-\lambda}}{r^{2}}\left(r\nu^{\prime}-e^{\lambda}+1\right).$ (37)

Further, the condition of isotropy of the pressure for the perfect fluid
matter yields that

$2\nu^{\prime\prime}+\nu^{\prime
2}-\nu^{\prime}\lambda^{\prime}-\frac{2}{r}\left(\nu^{\prime}+\lambda^{\prime}\right)+\frac{4}{r^{2}}\left(e^{\lambda}-1\right)=0,$
(38)

which is equivalent with

$r^{2}R_{1}-e^{\lambda}R_{2}=0\,.$ (39)

In this case the energy conditions for a barotropic equation of state
$p=p(\rho)$ are given by

$\rho>0,\qquad 0\leq p\leq\rho,\qquad 0\leq\frac{dp}{d\rho}\leq 1\,.$ (40)

The linear form of a barotropic equation of state is given by $p=w\rho$ where
$w$ is the equation of state parameter. For $w=0,1/3$ and $1$, we get _dust,
incoherent radiation_ and _stiff matter_ , respectively.

For the spherically symmetric static spacetimes (8), Eq. (7) takes the form:

$\displaystyle T^{\prime}_{i}X^{1}+2T_{i}X^{i}_{,i}=2T_{i}\psi,\,\,\ i=0,1,2$
(41) $\displaystyle T_{2}\left(X^{2}_{,2}-\cot\theta
X^{2}-X^{3}_{,3}\right)=0,$ (42) $\displaystyle
T_{0}X^{0}_{,j}+T_{j}X^{j}_{,0}=0,\,\,\,\ j=1,2,3$ (43) $\displaystyle
T_{j}X^{j}_{,k}+T_{k}X^{k}_{,j}=0,\,\,\,j,k=1,2,3\,\,\,(j\neq k)$ (44)

In the above equations the summation convention is not assumed. For the non-
degenerate energy-momentum tensor $T_{ab}$, after some tedious calculations
similar to those done in Ref. crcsph  we see that the general solution of
equations (41)-(44) can be written as

$\displaystyle X^{0}$ $\displaystyle=$
$\displaystyle\frac{T_{2}}{T_{0}}\left[\sin\theta\left(A^{\prime}_{1}\sin\phi-A^{\prime}_{2}\cos\phi\right)+A^{\prime}_{3}\cos\theta\right]+A_{4}(t,r),$
(45) $\displaystyle X^{1}$ $\displaystyle=$
$\displaystyle\frac{T_{2}}{T_{1}}\left[\sin\theta\left(A^{\prime}_{1}\sin\phi-A^{\prime}_{2}\cos\phi\right)+A^{\prime}_{3}\cos\theta\right]+A_{5}(t,r),$
(46) $\displaystyle X^{2}$ $\displaystyle=$
$\displaystyle-\cos\theta\left[A_{1}\sin\phi-
A_{2}\cos\phi\right]+A_{3}\sin\theta+a_{1}\sin\phi-a_{2}\cos\phi,$ (47)
$\displaystyle X^{3}$ $\displaystyle=$ $\displaystyle-
csc\theta\left[A_{1}\cos\phi+A_{2}\sin\phi\right]+\cot\theta(a_{1}\cos\phi+a_{2}\sin\phi)+a_{3},$
(48)

with the conformal function given by

$\displaystyle\psi=\left(\frac{T^{\prime}_{2}}{2T_{2}}A^{\prime}_{1}+A_{1}\right)\sin\theta\sin\phi-\left(\frac{T^{\prime}_{2}}{2T_{2}}A^{\prime}_{2}+A_{2}\right)\sin\theta\cos\phi$
$\displaystyle\qquad+\left(\frac{T^{\prime}_{2}}{2T_{2}}A^{\prime}_{3}+A_{3}\right)\cos\theta+\frac{T^{\prime}_{2}}{2T_{2}}A_{5}(t,r),$
(49)

where $A_{\ell}(t,r),\,(\ell=1,2,3,4,5)$, are integration functions and
$a_{j}\,(j=1,2,3)$ are constant parameters, which give the _three_ KVs of
spherically symmetric spacetimes

$\displaystyle\mathbf{X_{1}}=\sin\phi\,\partial_{\theta}+\cos\phi\cot\theta\,\partial_{\phi},\quad\mathbf{X_{2}}=\cos\phi\,\partial_{\theta}-\sin\phi\cot\theta\,\partial_{\phi},\quad\mathbf{X_{3}}=\partial_{\phi}.$
(50)

Further, the functions $A_{\ell}(t,r)$ in the above equations (45)-(49) are
subject to the following constraint equations

$\displaystyle\left(\sqrt{\frac{T_{1}}{T_{2}}}A_{5}\right)^{\prime}=0,$ (51)
$\displaystyle T_{0}A_{4}^{\prime}+T_{1}\dot{A_{5}}=0,$ (52) $\displaystyle
2\dot{A_{4}}+\left(\frac{T^{\prime}_{0}}{T_{0}}-\frac{T^{\prime}_{2}}{T_{2}}\right)A_{5}=0,$
(53)
$\displaystyle\left(\sqrt{\frac{T_{2}}{T_{0}}}\dot{A}_{j}\right)^{\prime}=0,$
(54)
$\displaystyle\ddot{A}_{j}+\frac{T_{0}}{2T_{1}}\left(\frac{T^{\prime}_{0}}{T_{0}}-\frac{T^{\prime}_{2}}{T_{2}}\right)A^{\prime}_{j}-\frac{T_{0}}{T_{2}}A_{j}=0,$
(55) $\displaystyle
A^{\prime\prime}_{j}+\frac{1}{2}\left(\frac{T^{\prime}_{2}}{T_{2}}-\frac{T^{\prime}_{1}}{T_{1}}\right)A^{\prime}_{j}-\frac{T_{1}}{T_{2}}A_{j}=0,$
(56)

where the dot represents the derivative with respect to time $t$, and
$j=1,2,3$. When we solve the above constraint equations for possible cases of
non-degenerate energy-momentum tensor $T_{ab}$, we obtain the corresponding
CMCs for the spherically symmetric static spacetimes (see Section IV). In the
following section we find CMCs for the degenerate energy-momentum tensor of
the spherically symmetric static spacetimes.

##  III Conformal Matter Collineations for the Degenerate Matter Tensor

If the energy-momentum tensor is degenerate, that is, $det(T_{ab})=0$, then we
have the following _four_ possibilities: (D-A1) all of the
$T_{a}\,(a=0,1,2,3)$ are zero; (D-A2) one of the $T_{a}$ is nonzero; (D-A3)
two of the $T_{a}$ are nonzero; (D-A4) three of the $T_{a}$ are nonzero.

Case (D-A1). This case corresponds to the vacuum (such as the Schwarzschild)
spacetime in which every vector is a CMC.

Case (D-A2). In this case, we have the subcases such that: (D-A2-i) $T_{0}\neq
0,\,\,T_{j}=0,\,\,(j=1,2,3)$; (D-A2-ii) $T_{1}\neq
0,\,\,T_{k}=0,\,\,(k=0,2,3)$.

Subcase (D-A2-i). In this subcase we find

$\displaystyle X^{0}=X^{0}(t),\quad
X^{1}=(\psi-\dot{X}^{0})\frac{2T_{0}}{T^{\prime}_{0}}\,,\quad
X^{\alpha}=X^{\alpha}(x^{a})\,,$ (57)

where $T^{\prime}_{0}\neq 0$ and $\alpha=2,3$. If $T^{\prime}_{0}=0$, i.e.,
$T_{0}=c$ (a constant), then the CMCs takes the following form

$\displaystyle X^{0}=\int{\psi dt}+a,\quad X^{j}=X^{j}(x^{a}),$ (58)

where $a$ is an integration constant. The corresponding Lie algebra of the
vector fields in this subcase is infinite dimensional because the vector
fields given in (57) and (58) have arbitrary components. For this subcase,
using (21)-(23) we have

$T_{0}=-2R_{0},\quad R_{1}=e^{\lambda-\nu}R_{0},\quad
R_{2}=r^{2}e^{-\nu}R_{0},$ (59)

which shows that all Ricci tensor components are nonzero, e.g. the Ricci
tensor in this subcase is non-degenerate. Furthermore, in this subcase we have
the equation of state $p=0$ (dust) for the perfect fluid.

Subcase (D-A2-ii). Considering the constraints of this subcase, we obtain that

$\displaystyle X^{1}=\frac{1}{\sqrt{\mid
T_{1}\mid}}\left[\int{\psi\,\sqrt{\mid T_{1}\mid}dr}+a_{1}\right],\qquad
X^{k}=X^{k}(x^{a}),$ (60)

where $a_{1}$ is a constant of integration. Here, we again have infinite
dimensional Lie algebra of vector fields. Using (21)-(23), we find

$R_{0}=e^{\nu-\lambda}R_{1}\,,\qquad R_{2}=-r^{2}e^{-\lambda}R_{1}\,,\qquad
T_{1}=2R_{1}\,,$ (61)

which are independent relations for any form of the energy-momentum tensor. It
is obvious that for this case the choice of observers as a timelike four-
vector field $u^{a}=e^{-\nu/2}\delta^{a}_{0}$ is not appropriate, since it
gives $\rho=0$ due to the condition $T_{0}=0$. Thus we use the spacelike four-
velocity $u^{a}=e^{-\lambda/2}\delta^{a}_{1}$ of the observers for the metric
(8), which implies that Eqs. (31) and (32) give

$\rho=e^{-\lambda}T_{1}\,,\quad p=0\,,\quad\pi_{ab}=0\,.$ (62)

Thus, the dust fluid is also allowed in this subcase. Furthermore, all Ricci
tensor components for this subcase are non-zero. This means that the Ricci
tensor is non-degenerate even though the matter tensor is degenerate.

Case (D-A3). For this case, the possible subcases are given by (D-A3-i)
$T_{p}\neq 0,\,\,T_{q}=0,\,\,(p=0,1\,\,{\rm and}\,\,q=2,3)$ and (D-A3-ii)
$T_{p}=0,\,\,T_{q}\neq 0$.

Subcase (D-A3-i). Here, by choosing a timelike four-velocity, the conditions
$T_{0}\neq 0\neq T_{1}$ and $T_{2}=0=T_{3}$ mean that the fluid represents an
anisotropic fluid without heat flux, and then the physical quantities become

$\displaystyle\rho=e^{-\nu}T_{0}\,,\qquad p=\frac{1}{3}e^{-\lambda}T_{1}\,,$
(63)
$\displaystyle\pi_{11}=2e^{\lambda}p\,,\quad\pi_{22}=-pr^{2}\,,\quad\pi_{33}=-pr^{2}\sin^{2}\theta\,.$
(64)

Using the transformations $d\bar{r}=\psi\sqrt{\mid T_{1}\mid}dr$, where
$\bar{r}=\bar{r}(t,r)$, one finds

$X^{0}=-\int{\frac{(\dot{\bar{r}}+\dot{f})}{\psi T_{0}}g(t)d\bar{r}},\qquad
X^{1}=\frac{\bar{r}+f(t)}{\sqrt{\mid T_{1}\mid}},\qquad X^{q}=X^{q}(x^{a}),$
(65)

where $f(t)$ and $g(t)$ are functions of integration, and the conformal factor
$\psi$ has the form

$\psi=\frac{2T_{0}}{(\bar{r}+f-2T_{0})}\left[\int{\left(\frac{\dot{\bar{r}}+\dot{f}}{\psi}\right)^{.}\frac{d\bar{r}}{T_{0}}}-\dot{g}(t)\right].$
(66)

For this case, it follows from the condition $T_{q}=0$ that

$\displaystyle R_{0}=\frac{e^{\nu-\lambda}}{2r}(\nu+\lambda)^{\prime}\,,\qquad
R_{1}=\frac{1}{2r}(\nu+\lambda)^{\prime}\,,$ (67) $\displaystyle
T_{0}+e^{\nu-\lambda}T_{1}=2R_{0}\,,\qquad
e^{-\nu}T_{0}-e^{-\lambda}T_{1}=\frac{2}{r^{2}}R_{2}\,.$ (68)

We find from Eqs. (63) and (68) that

$\rho+3p=2e^{-\nu}R_{0}\,,\qquad\rho-3p=\frac{2}{r^{2}}R_{2}\,.$ (69)

These equations show that it should be satisfied the conditions $R_{0}\geq
e^{\nu}/2$ and $R_{2}\geq 0$ in order to be valid the SEC and TEC,
respectively. There is only one constraint equation
$2\nu^{\prime\prime}+{\nu^{\prime}}^{2}-\nu^{\prime}\lambda^{\prime}+\frac{2}{r}(\nu^{\prime}-\lambda^{\prime})=0$
following from the condition $T_{2}=0$ for this subcase. Also, all the Ricci
tensor components are again non-zero, i.e. $\det(R_{ab})\neq 0$, even if the
matter tensor is degenerate.

Subcase (D-A3-ii). For this subcase, considering the constraint
$T_{0}=0=T_{1}$, we find

$\lambda^{\prime}=\frac{1}{r}\left(1-e^{\lambda}\right)=-\nu^{\prime}\,,$ (70)

which yields the following solution

$\nu(r)=\ln\left(1-\frac{\lambda_{0}}{r}\right)\,,\qquad\lambda(r)=\ln\left(\frac{1}{1-\frac{\lambda_{0}}{r}}\right)\,,$
(71)

where $\lambda_{0}$ is an integration constant. The above solution is just the
Schwarzschild metric which gives $R_{ab}=0$, i.e., all $R_{i}$’s and $T_{i}$’s
vanish identically. Therefore there is a contradiction with the condition
$T_{2}\neq 0\neq T_{3}$ as an assumption that is not possible in this subcase.

Case (D-A4). In this case, the possible subcases are (D-A4-i)
$T_{0}=0,\,\,T_{j}\neq 0,\,\,(j=1,2,3)$ and (D-A4-ii) $T_{1}=0,\,\,T_{k}\neq
0,\,\,(k=0,2,3)$.

Subcase (D-A4-i). In this subcase, the constraint $T_{0}=0$ gives

$\lambda^{\prime}=\frac{1}{r}\left(1-e^{\lambda}\right)\,,$ (72)

and

$\displaystyle
R_{0}=-2e^{\nu}\left(e^{-\lambda}R_{1}+\frac{1}{r^{2}}R_{2}\right),\quad
T_{1}=-\frac{1}{2}\left(R_{1}+\frac{4}{r^{2}}e^{\lambda}R_{2}\right),\quad
T_{2}=-\frac{3}{2}r^{2}e^{-\lambda}R_{1}-R_{2},$ (73)

where $T_{1}=\frac{1}{r}\left(\nu+\lambda\right)^{\prime}$. For this case the
choice of a timelike four velocity of the observers is not allowed since
$T_{0}=0$ gives $\rho=0$. So we need to choose a spacelike four-velocity such
as $u^{a}=e^{-\lambda/2}\delta^{a}_{r}$. Using Eqs. (31) and (32), this choice
gives rise to an anisotropic fluid without heat flux as follows:

$\displaystyle\rho=e^{-\lambda}T_{1}\,,\qquad
p=\frac{2}{3r^{2}}T_{2}\,,\quad\pi_{00}=pe^{\lambda}\,,\quad\pi_{11}=0\,,\quad\pi_{22}=\frac{1}{2}pr^{2}\,,\quad\pi_{33}=\sin^{2}\theta\,\pi_{22}\,.$
(74)

Here Eq. (72) has the following solution

$\lambda(r)=\ln\left(\frac{1}{1-\frac{\lambda_{1}}{r}}\right)\,,$ (75)

where $\lambda_{1}$ is a constant of integration. The equations given in (73)
are second order ordinary differential equations in terms of $\nu$. Then,
using the $\lambda(r)$ given by (75) in any of the three equations of (73),
one can solve the obtained second order differential equation to find $\nu(r)$
as

$\nu(r)=2\ln\left\\{-\frac{\nu_{0}}{4}\sqrt{1-\frac{\lambda_{1}}{r}}+\frac{\nu_{1}}{2}\left[r-3\lambda_{1}+\frac{3}{2}\lambda_{1}\sqrt{1-\frac{\lambda_{1}}{r}}\ln\left(r+\sqrt{r(r-\lambda_{1})}-\frac{\lambda_{1}}{2}\right)\right]\right\\}\,,$
(76)

where $\nu_{0}$ and $\nu_{1}$ are constants of integration. The Ricci scalar
of the obtained metric given by (75) and (76) is

$R=\frac{8\nu_{1}}{-\nu_{0}\sqrt{r(r-\lambda_{1})}+\nu_{1}\left[2r(r-3\lambda_{1})+3\lambda_{1}\sqrt{r(r-\lambda_{1})}\ln\left(r+\sqrt{r(r-\lambda_{1})}-\frac{\lambda_{1}}{2}\right)\right]}\,,$
(77)

and the $R_{i}$’s and $T_{i}$’s for this solution are

$R_{0}=\frac{1}{4r^{2}}\left\\{-\nu_{0}\sqrt{r(r-\lambda_{1})}+\nu_{1}\left[2r(r-3\lambda_{1})+3\lambda_{1}\sqrt{r(r-\lambda_{1})}\ln\left(r+\sqrt{r(r-\lambda_{1})}-\frac{\lambda_{1}}{2}\right)\right]\right\\}\,,$
(78)
$R_{1}=0\,,\,R_{2}=\frac{2\nu_{1}r^{2}}{\nu_{0}\sqrt{r(r-\lambda_{1})}-\nu_{1}\left[2r(r-3\lambda_{1})+3\lambda_{1}\sqrt{r(r-\lambda_{1})}\ln\left(r+\sqrt{r(r-\lambda_{1})}-\frac{\lambda_{1}}{2}\right)\right]}\,,$
(79) $T_{1}=\frac{2}{r(\lambda_{1}-r)}R_{2}\,,\qquad T_{2}=-R_{2}\,.$ (80)

Further, the matter density and pressure that come from Eqs. (74), (75) and
(80) are

$\rho=\frac{2}{r^{2}}T_{2}\,,\qquad p=\frac{1}{3}\rho\,,$ (81)

which implies that the equation of state parameter $w$ yields an incohorent
radiation, i.e., $w=1/3$. Finally, we conclude that the Ricci tensor is also
degenerate in this case.

For this subcase, we obtain that $X^{0}=X^{0}(x^{a})$ and
$X^{j}=X^{j}(r,\theta,\phi)$ where the form of $X^{j}$ is the same as in
equations (46) - (48), and the constraint equations (51) - (56) yield the
following solution

$A_{j}=b_{j}\cosh\bar{r}+d_{j}\sinh\bar{r},\qquad
A_{5}=\ell\sqrt{\frac{T_{2}}{T_{1}}},$ (82)

where $b_{j},d_{j}$ and $\ell$ are constants. In this case, the conformal
factor is given by

$\displaystyle\psi=\left(\frac{T_{{2},\bar{r}}}{2T_{2}}A_{1,\bar{r}}+A_{1}\right)\sin\theta\sin\phi-\left(\frac{T_{{2},\bar{r}}}{2T_{2}}A_{2,\bar{r}}+A_{2}\right)\sin\theta\cos\phi$
$\displaystyle\qquad+\left(\frac{T_{2,\bar{r}}}{2T_{2}}A_{3,\bar{r}}+A_{3}\right)\cos\theta+\ell\frac{T_{2,\bar{r}}}{2T_{2}},$
(83)

where we have used the transformation $dr=(T_{2}/T_{1})^{1/2}d\bar{r}$. By
considering (79) and (80), the latter transformation yields
$\bar{r}=-\sqrt{2}\ln\left(r+\sqrt{r(r-\lambda_{1})}-\frac{\lambda_{1}}{2}\right)$.
Thus, in addition to the three KVs given in (50), it follows that the
remaining CMCs and corresponding conformal factors are

$\displaystyle\mathbf{X}_{4}=\sinh\bar{r}\cos\,\theta\partial_{\bar{r}}+\cosh\bar{r}\sin\theta\,\partial_{\theta}\,,\quad\psi_{4}=\cos\theta\left(\frac{T_{{2},\bar{r}}}{2T_{2}}\sinh\bar{r}+\cosh\bar{r}\right)\,,$
$\displaystyle\mathbf{X}_{5}=\cosh\bar{r}\cos\theta\,\partial_{\bar{r}}+\sinh\bar{r}\sin\theta\,\partial_{\theta}\,,\quad\psi_{5}=\cos\theta\left(\frac{T_{{2},\bar{r}}}{2T_{2}}\cosh\bar{r}+\sinh\bar{r}\right)\,,$
$\displaystyle\mathbf{X}_{6}=\sinh\bar{r}\sin\theta\sin\phi\,\partial_{\bar{r}}-\cosh\bar{r}\,{\xi_{1}}\,,\quad\psi_{6}=\sin\theta\sin\phi\left(\frac{T_{{2},\bar{r}}}{2T_{2}}\sinh\bar{r}+\cosh\bar{r}\right)\,,$
$\displaystyle\mathbf{X}_{7}=\sinh\bar{r}\sin\theta\cos\phi\,\partial_{\bar{r}}-\cosh\bar{r}\,{\xi_{2}}\,,\quad\psi_{7}=\sin\theta\cos\phi\left(\frac{T_{{2},\bar{r}}}{2T_{2}}\sinh\bar{r}+\cosh\bar{r}\right)\,,$
(84)
$\displaystyle\mathbf{X}_{8}=\cosh\bar{r}\sin\theta\sin\phi\,\partial_{\bar{r}}-\sinh\bar{r}\,{\xi_{1}}\,,\quad\psi_{8}=\sin\theta\sin\phi\left(\frac{T_{{2},\bar{r}}}{2T_{2}}\cosh\bar{r}+\sinh\bar{r}\right)\,,$
$\displaystyle\mathbf{X}_{9}=\cosh\bar{r}\sin\theta\cos\phi\,\partial_{\bar{r}}-\sinh\bar{r}\,{\xi_{2}}\,,\quad\psi_{9}=\sin\theta\cos\phi\left(\frac{T_{{2},\bar{r}}}{2T_{2}}\cosh\bar{r}+\sinh\bar{r}\right)\,,$
$\displaystyle\mathbf{X}_{10}=\partial_{\bar{r}}\,,\quad\qquad\qquad\qquad\qquad\qquad\quad\,\,\,\psi_{10}=\frac{T_{{2},\bar{r}}}{2T_{2}}\,,$
$\displaystyle\mathbf{X}_{11}=F(t,r,\theta,\phi)\partial_{t}\,,\quad\qquad\qquad\qquad\quad\psi_{11}=0\,,$

where $F(t,r,\theta,\phi)$ is an arbitrary function, and we have defined
${\xi}_{1}$ and ${\xi_{2}}$ as follows

$\displaystyle{\xi_{1}}=\cos\theta\sin\phi\,\partial_{\theta}+\csc\theta\cos\phi\,\partial_{\phi}\,,\qquad{\xi_{2}}=\cos\theta\cos\phi\,\partial_{\theta}-\csc\theta\sin\phi\,\partial_{\phi}\,.$
(85)

In order to construct a closed algebra for vector fields (84), we find that
$F=F(t)$. Hence we have finite dimensional Lie algebra of CMCs which has the
following non-vanishing commutators

$\displaystyle\left[\bf{X}_{1},\bf{X}_{2}\right]=\bf{X}_{3},\qquad\left[\bf{X}_{1},\bf{X}_{3}\right]=-\bf{X}_{2},\quad\left[\bf{X}_{1},\bf{X}_{4}\right]=-\bf{X}_{6},\quad\left[\bf{X}_{1},\bf{X}_{5}\right]=\bf{X}_{8},\qquad\left[\bf{X}_{1},\bf{X}_{6}\right]=\bf{X}_{4},$
$\displaystyle\left[\bf{X}_{1},\bf{X}_{8}\right]=\bf{X}_{5},\qquad\left[\bf{X}_{2},\bf{X}_{3}\right]=\bf{X}_{1},\qquad\left[\bf{X}_{2},\bf{X}_{4}\right]=-\bf{X}_{7},\quad\left[\bf{X}_{2},\bf{X}_{5}\right]=-\bf{X}_{9},\quad\left[\bf{X}_{2},\bf{X}_{7}\right]=\bf{X}_{4},$
$\displaystyle\left[\bf{X}_{2},\bf{X}_{9}\right]=\bf{X}_{5},\qquad\left[\bf{X}_{3},\bf{X}_{6}\right]=\bf{X}_{7},\qquad\left[\bf{X}_{3},\bf{X}_{7}\right]=-\bf{X}_{6},\quad\left[\bf{X}_{3},\bf{X}_{8}\right]=\bf{X}_{9},\quad\,\,\,\,\left[\bf{X}_{3},\bf{X}_{9}\right]=-\bf{X}_{8},$
$\displaystyle\left[\bf{X}_{4},\bf{X}_{5}\right]=-\bf{X}_{10},\quad\left[\bf{X}_{4},\bf{X}_{6}\right]=\bf{X}_{1},\qquad\left[\bf{X}_{4},\bf{X}_{7}\right]=\bf{X}_{2},\quad\,\,\,\left[\bf{X}_{4},\bf{X}_{10}\right]=-\bf{X}_{5},\,\,\,\,\left[\bf{X}_{5},\bf{X}_{8}\right]=-\bf{X}_{1},$
(86)
$\displaystyle\left[\bf{X}_{5},\bf{X}_{9}\right]=-\bf{X}_{2},\,\,\quad\left[\bf{X}_{5},\bf{X}_{10}\right]=-\bf{X}_{4},\quad\left[\bf{X}_{6},\bf{X}_{7}\right]=-\bf{X}_{3},\,\,\,\left[\bf{X}_{6},\bf{X}_{8}\right]=-\bf{X}_{10},\,\,\,\,\left[\bf{X}_{6},\bf{X}_{10}\right]=-\bf{X}_{8},$
$\displaystyle\left[\bf{X}_{7},\bf{X}_{9}\right]=-\bf{X}_{10},\quad\left[\bf{X}_{7},\bf{X}_{10}\right]=-\bf{X}_{9},\quad\left[\bf{X}_{8},\bf{X}_{9}\right]=\bf{X}_{3},\quad\,\,\left[\bf{X}_{8},\bf{X}_{10}\right]=-\bf{X}_{6},\,\,\,\,\left[\bf{X}_{9},\bf{X}_{10}\right]=-\bf{X}_{7}.$

Subcase (D-A4-ii). For this subcase, where $T_{1}=0$, we have
$\nu^{\prime}=\frac{1}{r}\left(e^{\lambda}-1\right)$ and thus
$T_{0}=\frac{1}{r}e^{\nu-\lambda}\left(\lambda+\nu\right)^{\prime}$. Also,
$T_{0}$ and $T_{2}$ in terms of $R_{i}$’s ($i=0,1,2$) become

$T_{0}=R_{0}+e^{\nu-\lambda}R_{1}\,,\qquad
T_{2}=\frac{r^{2}}{2}\left(e^{-\nu}R_{0}-e^{-\lambda}R_{1}\right)\,,$ (87)

and

$R_{2}=\frac{r^{2}}{2}\left(e^{-\nu}R_{0}+e^{-\lambda}R_{1}\right)\,.$ (88)

In this subcase one can choose a timelike four-velocity of the observers such
that $u^{a}=e^{-\nu}\delta^{a}_{t}$, which yields an anisotropic fluid without
heat flux as

$\rho=e^{-\nu}T_{0}\,,\quad
p=\frac{2}{3r^{2}}T_{2}\,,\quad\pi_{11}=-e^{\lambda}p\,,\quad\pi_{22}\frac{r^{2}}{2}p\,,\quad\pi_{33}=\sin^{2}\theta\,\pi_{22}\,.$
(89)

It is interesting to point out that we have a variable equation of state
parameter $w=\frac{2}{3\,r^{2}}e^{\nu}$, i.e.,
$p=\frac{2}{3\,r^{2}}e^{\nu}\rho$ when $T_{0}=T_{2}$.

If $T_{0}=T_{2}$, it follows from the constraint equations (51)-(56) that
$A_{j}$ and $A_{4}$ have the following solutions

$A_{j}=b_{j}\cosh\bar{r}+d_{j}\sinh\bar{r}\,,\qquad A_{4}=\ell\,,$ (90)

where $b_{j},d_{j}$ and $\ell$ are integration constants. Then, the components
of the CMC vector field are obtained as

$\displaystyle
X^{0}=\sin\theta\left[A^{\prime}_{1}\sin\phi-A^{\prime}_{2}\cos\phi\right]+A^{\prime}_{3}\cos\theta+\ell\,,$
$\displaystyle
X^{1}=\frac{2T_{0}}{T^{\prime}_{0}}\left[\psi-\sin\theta(A_{1}\sin\phi-
A_{2}\cos\phi)-A_{3}\cos\theta\right]\,,$ (91) $\displaystyle
X^{2}=-\cos\theta(A_{1}\sin\phi-A_{2}\cos\phi)+A_{3}\sin\theta+a_{1}\sin\phi-
a_{2}\cos\phi\,,$ $\displaystyle
X^{3}=-\csc\theta(A_{1}\cos\phi+A_{2}\sin\phi)+\cot\theta(a_{1}\cos\phi+a_{2}\sin\phi)+a_{3}\,,$

where $T^{\prime}_{0}\neq 0$, and $\psi$ is an arbitrary conformal factor,
that is, the component $X^{1}$ is an arbitrary function of the coordinates and
so we have _infinite_ dimensional algebra of CMCs. If $T_{0}\neq T_{2}$, then,
in addition to the three KVs given in (50), we have the following CMCs

$\displaystyle\mathbf{X_{4}}=\sin\theta\sin\phi\,G(r)\,\partial_{r}-{\xi}_{1}\,,\quad\,\,\,\psi_{4}=\sin\theta\sin\phi\,G(r)\,,$
$\displaystyle\mathbf{X_{5}}=\sin\theta\cos\phi\,G(r)\,\partial_{\bar{r}}+{\xi}_{2}\,,\quad\,\,\,\psi_{5}=\sin\theta\cos\phi\,G(r)\,,$
(92)
$\displaystyle\mathbf{X_{6}}=\cos\theta\,G(r)\,\partial_{\bar{r}}+\sin\theta\,\partial_{\theta}\,,\quad\,\,\,\,\psi_{6}=\cos\theta\,G(r)\,,$
$\displaystyle\mathbf{X_{7}}=\dot{f}(t)\,G(r)\,\partial_{\bar{r}}+f(t)\,\partial_{t}\,,\qquad\,\,\psi_{7}=\dot{f}(t)\left[1-G(r)\right]\,,$

where $G(r)\equiv\frac{2T_{0}/T_{2}}{(T_{0}/T_{2})^{\prime}}$,
$(T_{0}/T_{2})^{\prime}\neq 0$, and $f(t)$ is an integration function. Thus,
we again have an _finite_ dimensional Lie algebra of CMCs, and non-zero
commutators of the Lie algebra have the following form:

$\displaystyle\left[\bf{X}_{1},\bf{X}_{2}\right]=\bf{X}_{3}\,,\qquad\left[\bf{X}_{1},\bf{X}_{3}\right]=-\bf{X}_{2}\,,\,\,\quad\left[\bf{X}_{1},\bf{X}_{4}\right]=\bf{X}_{6},\quad\,\,\,\,\,\left[\bf{X}_{1},\bf{X}_{6}\right]=-\bf{X}_{4}\,,$
$\displaystyle\left[\bf{X}_{2},\bf{X}_{3}\right]=\bf{X}_{1}\,,\qquad\left[\bf{X}_{2},\bf{X}_{5}\right]=\bf{X}_{6}\,,\qquad\left[\bf{X}_{2},\bf{X}_{6}\right]=-\bf{X}_{5}\,,\quad\left[\bf{X}_{3},\bf{X}_{4}\right]=\bf{X}_{5}\,,$
(93)
$\displaystyle\left[\bf{X}_{3},\bf{X}_{5}\right]=-\bf{X}_{4}\,,\quad\left[\bf{X}_{4},\bf{X}_{5}\right]=-\bf{X}_{3}\,,\quad\,\,\left[\bf{X}_{4},\bf{X}_{6}\right]=-\bf{X}_{1}\,,\quad\left[\bf{X}_{5},\bf{X}_{6}\right]=-\bf{X}_{2}\,.$

##  IV Conformal Matter Collineations for the Non-degenerate Matter Tensor

In this section, we consider the CMCs in non-degenerate case, i.e.
$\det(T_{ab})\neq 0$, admitted by the static spherically symmetric spacetimes.
Here we consider the following _five_ possibilities of the non-degenerate
matter tensor.

Case (ND-A). For this case, where none of the $T^{\prime}_{a}\,(a=0,1,2,3)$ is
zero, applying the transformation $dr=\sqrt{T_{2}/T_{1}}d\bar{r}$, we find
that the the number of CMCs is _fifteen_ such that there are three minimal KVs
given in (50), and the remaining ones are

$\displaystyle{\bf X}_{4}=f_{1}(t)\left[\sin\theta\sin\phi\,h_{1}(\bar{r}){\bf
Y}-h_{2}(\bar{r})\xi_{1}\right]\,,\qquad\quad\,\,\,\psi_{4}=f_{1}(t)H_{1}(\bar{r})\sin\theta\sin\phi\,,$
$\displaystyle{\bf X}_{5}=f_{2}(t)\left[\sin\theta\sin\phi\,h_{1}(\bar{r}){\bf
Y}-h_{2}(\bar{r})\xi_{1}\right]\,,\qquad\quad\,\,\,\psi_{5}=f_{2}(t)H_{1}(\bar{r})\sin\theta\sin\phi\,,$
$\displaystyle{\bf
X}_{6}=\frac{1}{b}\left[\sin\theta\sin\phi\,h_{2}(\bar{r}){\bf
Y}-h_{1}(\bar{r})\xi_{1}\right]\,,\qquad\qquad\quad\psi_{6}=\frac{H_{2}(\bar{r})}{b}\sin\theta\sin\phi\,,$
$\displaystyle{\bf
X}_{7}=f_{1}(t)\left[-\sin\theta\cos\phi\,h_{1}(\bar{r}){\bf
Y}+h_{2}(\bar{r})\xi_{2}\right]\,,\qquad\,\,\,\psi_{7}=-f_{1}(t)H_{1}(\bar{r})\sin\theta\cos\phi\,,$
$\displaystyle{\bf
X}_{8}=f_{2}(t)\left[-\sin\theta\cos\phi\,h_{1}(\bar{r}){\bf
Y}+h_{2}(\bar{r})\xi_{2}\right]\,,\qquad\,\,\,\psi_{8}=-f_{2}(t)H_{1}(\bar{r})\sin\theta\cos\phi\,,$
$\displaystyle{\bf
X}_{9}=\frac{1}{b}\left[-\sin\theta\cos\phi\,h_{2}(\bar{r}){\bf
Y}+h_{1}(\bar{r})\xi_{2}\right]\,,\qquad\qquad\psi_{9}=-\frac{H_{2}(\bar{r})}{b}\sin\theta\cos\phi\,,$
(94) $\displaystyle{\bf X}_{10}=f_{1}(t)\left[\cos\theta\,h_{1}(\bar{r}){\bf
Y}+h_{2}(\bar{r})\sin\theta\,\partial_{\theta}\right]\,,\qquad\quad\psi_{10}=f_{1}(t)H_{1}(\bar{r})\cos\theta\,,$
$\displaystyle{\bf X}_{11}=f_{2}(t)\left[\cos\theta\,h_{1}(\bar{r}){\bf
Y}+h_{2}(\bar{r})\sin\theta\,\partial_{\theta}\right]\,,\qquad\quad\psi_{11}=f_{2}(t)H_{1}(\bar{r})\cos\theta\,,$
$\displaystyle{\bf X}_{12}=\frac{1}{b}\left[\cos\theta\,h_{2}(\bar{r}){\bf
Y}+h_{1}(\bar{r})\sin\theta\,\partial_{\theta}\right]\,,\qquad\qquad\,\,\,\psi_{12}=\frac{H_{2}(\bar{r})}{b}\cos\theta\,,$
$\displaystyle{\bf
X}_{13}=-\ddot{f}_{1}\frac{\sinh\bar{r}}{a\,h_{2}(\bar{r})}\partial_{t}+\dot{f}_{1}\,\partial_{\bar{r}}\,,\qquad\qquad\qquad\qquad\quad\,\psi_{13}=\dot{f}_{1}\frac{T_{2,\bar{r}}}{2\sqrt{T_{1}T_{2}}}\,,$
$\displaystyle{\bf
X}_{14}=-\ddot{f}_{2}\frac{\sinh\bar{r}}{a\,h_{2}(\bar{r})}\partial_{t}+\dot{f}_{2}\,\partial_{\bar{r}}\,,\qquad\qquad\qquad\qquad\quad\,\psi_{14}=\dot{f}_{2}\frac{T_{2,\bar{r}}}{2\sqrt{T_{1}T_{2}}}\,,$
$\displaystyle{\bf
X}_{15}=\partial_{t}\,,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\,\psi_{15}=0\,,$

where
$H_{1}(\bar{r}),H_{2}(\bar{r}),h_{1}(\bar{r}),h_{2}(\bar{r}),f_{1}(t),f_{2}(t)$
and ${\bf Y}$ are defined as follows:

$\displaystyle
H_{1}(\bar{r})=h_{1}(\bar{r})\frac{T_{1}T_{2,\bar{r}}}{2T_{2}^{2}}+h_{2}(\bar{r})\,,\quad
H_{2}(\bar{r})=h_{2}(\bar{r})\frac{T_{1}T_{2,\bar{r}}}{2T_{2}^{2}}+h_{1}(\bar{r})\,,$
(95) $\displaystyle h_{1}(\bar{r})=a\sinh\bar{r}+b\cosh\bar{r}\,,\qquad
h_{2}(\bar{r})=a\cosh\bar{r}+b\sinh\bar{r}\,,$ (96) $\displaystyle
f_{1}(t)=\left\\{\begin{array}[]{l}\frac{1}{\alpha}\sinh(\alpha
t)\,,\quad\,\,\,{\rm for}\,\,\alpha^{2}>0,\\\
\frac{1}{|\alpha|}\sin(|\alpha|t)\,,\quad{\rm
for}\,\,\alpha^{2}<0,\end{array}\right.$ (99) $\displaystyle
f_{2}(t)=\left\\{\begin{array}[]{l}\frac{1}{\alpha}\cosh(\alpha
t)\,,\qquad\,\,{\rm for}\,\,\alpha^{2}>0,\\\
-\frac{1}{|\alpha|}\cos(|\alpha|t)\,,\quad{\rm
for}\,\,\alpha^{2}<0,\end{array}\right.$ (102) $\displaystyle{\bf
Y}=\frac{\sqrt{T_{1}T_{2}}}{T_{0}}\partial_{t}+\sqrt{\frac{T_{2}}{T_{1}}}\partial_{r}=\frac{\sqrt{T_{1}T_{2}}}{T_{0}}\partial_{t}+\partial_{\bar{r}}\,,$
(103)

and $T_{0}=h_{2}(\bar{r})^{2}T_{2}$ , $a$ and $b$ are integration constants,
and $\alpha$ is a constant of separation such that

$\alpha^{2}=\left\\{\begin{array}[]{l}a^{2}-b^{2}\,,\quad{\rm
for}\,\,\alpha^{2}>0\,,\\\ b^{2}-a^{2}\,,\quad{\rm
for}\,\,\alpha^{2}<0\,.\end{array}\right.$ (104)

For $\alpha^{2}=0$, after solving the constraint equations (51)-(56), we find
_twelve_ CMCs as follows: the KVs ${\bf X}_{1},{\bf X}_{2},{\bf X}_{3}$ given
in (50), and

$\displaystyle{\bf
X}_{4}=e^{\beta\bar{r}}\left[\beta\,\sin\theta\sin\phi\,{\bf
Y}-\xi_{1}\right]\,,\qquad\psi_{4}=H(\bar{r})e^{\beta\bar{r}}\sin\theta\sin\phi\,;\qquad{\bf
X}_{5}=c_{0}\,t\,{\bf X}_{4}\,,\quad\psi_{5}=c_{0}\,t\,\psi_{4}\,;$
$\displaystyle{\bf
X}_{6}=e^{\beta\bar{r}}\left[-\beta\,\sin\theta\cos\phi\,{\bf
Y}+\xi_{2}\right]\,,\quad\psi_{6}=-e^{\beta\bar{r}}H(\bar{r})\sin\theta\cos\phi\,;\quad{\bf
X}_{7}=c_{0}\,t\,{\bf X}_{6}\,,\quad\psi_{7}=c_{0}\,t\,\psi_{6}\,;$
$\displaystyle{\bf X}_{8}=e^{\beta\bar{r}}\left[\beta\,\cos\theta\,{\bf
Y}+\sin\theta\,\partial_{\theta}\right]\,,\qquad\psi_{8}=e^{\beta\bar{r}}H(\bar{r})\cos\theta\,;\qquad{\bf
X}_{9}=c_{0}\,t\,{\bf X}_{8}\,,\quad\psi_{9}=c_{0}\,t\,\psi_{8}\,;$ (105)
$\displaystyle{\bf
X}_{10}=\frac{1}{2}\left(\frac{e^{-2\beta\bar{r}}}{\beta\,c_{0}^{2}}-\beta\,t^{2}\right)\partial_{t}+t\,\partial_{\bar{r}}\,,\qquad\psi_{10}=\frac{t\,T_{2,\bar{r}}}{2\sqrt{T_{1}T_{2}}}\,,$
$\displaystyle{\bf X}_{11}=-\beta
t\,\partial_{t}+\partial_{\bar{r}}\,,\quad\psi_{11}=\frac{T_{2,\bar{r}}}{2\sqrt{T_{1}T_{2}}}\,;\quad{\bf
X}_{12}=\partial_{t}\,,\quad\ \psi_{12}=0\,,$

where $H(\bar{r})$ is defined as

$H(\bar{r})=\beta\frac{T_{1}T_{2,\bar{r}}}{2T_{2}^{2}}+1\,,$ (106)

and $T_{0}=c^{2}_{0}e^{2\beta\bar{r}}T_{2}$,
$T_{1}=(c_{1}e^{-2\beta\bar{r}}+\beta)T_{2}$, $\beta$ is a separation constant
such that $\beta=\pm 1$, $c_{0}$ and $c_{1}$ are integration constants.

Case (ND-B). Three of the $T^{\prime}_{a}$ are zero. In this case we have the
possibilities: (ND-B-i) $T^{\prime}_{0}\neq
0,\,\,T^{\prime}_{j}=0,\,\,(j=1,2,3)$ and (ND-B-ii) $T^{\prime}_{1}\neq
0,\,\,T^{\prime}_{k}=0,\,\,(k=0,2,3)$.

Subcase (ND-B-i). For this subcase we have 15 CMCs which are the same form as
(94) together with the KVs ${\bf X}_{1},{\bf X}_{2},{\bf X}_{3}$ given in
(50), under the transformations $\bar{r}\rightarrow k\,r$,
$h_{1}(\bar{r})\rightarrow h_{1}(r)=a\sinh(kr)+b\cosh(kr)$,
$h_{2}(\bar{r})\rightarrow h_{2}(r)=a\cosh(kr)+b\sinh(kr)$,
$H_{1}(\bar{r})\rightarrow h_{2}(r)$ and $H_{2}(\bar{r})\rightarrow h_{1}(r)$,
where $k=c_{1}/c_{2}$, and $a,b,c_{1},c_{2}$ are non-zero constants, and

$\displaystyle T_{1}=\pm c_{1}^{2},\quad\,\,T_{2}=\pm
c_{2}^{2},\quad\,\,\alpha^{2}=\pm(a^{2}-b^{2})/c_{2}^{2}\,,\qquad
T_{0}=\left[a\cosh(k\,r)+b\sinh(k\,r)\right]^{2}\,.$ (107)

Here we note that the vector fields ${\bf X}_{13},{\bf X}_{14}$ and ${\bf
X}_{15}$ are MCs since the scale factors for those are zero, i.e.,
$\psi_{13}=0,\psi_{14}=0$ and $\psi_{15}=0$.

When $\alpha^{2}=0$, one finds that there are 15 CMCs which are the KVs ${\bf
X}_{1},{\bf X}_{2},{\bf X}_{3}$ given in (50) and

$\displaystyle{\bf X}_{4}=\beta\,K_{{}_{-}}(t,r)\,\sin\theta\sin\phi\,{\bf
Y}-K_{+}(t,r)\,\xi_{1}\,,\qquad\quad\,\,\,\psi_{4}=K_{+}(t,r)\sin\theta\sin\phi\,,$
$\displaystyle{\bf
X}_{5}=\frac{c_{0}\,\beta}{c_{1}}e^{\beta\,r}t\left[\sin\theta\sin\phi\,{\bf
Y}-\xi_{1}\right]\,,\qquad\qquad\qquad\quad\psi_{5}=\frac{c_{0}\,\beta}{c_{1}}e^{\beta\,r}t\,\sin\theta\sin\phi\,,$
$\displaystyle{\bf X}_{6}=e^{\beta\,r}\left[\sin\theta\sin\phi\,{\bf
Y}-\xi_{1}\right]\,,\qquad\qquad\qquad\qquad\quad\,\psi_{6}=e^{\beta\,r}\sin\theta\sin\phi\,,$
$\displaystyle{\bf X}_{7}=-\beta\,K_{{}_{-}}(t,r)\sin\theta\cos\phi\,{\bf
Y}+K_{+}(t,r)\xi_{2}\,,\qquad\,\,\,\,\,\psi_{7}=-K_{+}(t,r)\sin\theta\cos\phi\,,$
$\displaystyle{\bf
X}_{8}=\frac{c_{0}\,\beta}{c_{1}}e^{\beta\,r}t\left[-\sin\theta\cos\phi\,{\bf
Y}+\xi_{2}\right]\,,\qquad\qquad\quad\,\,\,\,\,\psi_{8}=-\frac{c_{0}\,\beta}{c_{1}}e^{\beta\,r}t\sin\theta\cos\phi\,,$
$\displaystyle{\bf X}_{9}=e^{\beta\,r}\left[-\sin\theta\cos\phi\,{\bf
Y}+\xi_{2}\right]\,,\qquad\qquad\qquad\quad\,\,\,\,\,\,\psi_{9}=-e^{\beta\,r}\sin\theta\cos\phi\,,$
(108) $\displaystyle{\bf X}_{10}=\beta\,K_{{}_{-}}(t,r)\cos\theta\,{\bf
Y}+K_{+}(t,r)\sin\theta\,\partial_{\theta}\,,\qquad\quad\psi_{10}=K_{+}(t,r)\cos\theta\,,$
$\displaystyle{\bf
X}_{11}=\frac{c_{0}\,\beta}{c_{1}}e^{\beta\,r}t\left[\cos\theta\,{\bf
Y}+\sin\theta\,\partial_{\theta}\right]\,,\qquad\qquad\qquad\,\,\psi_{11}=\frac{c_{0}\,\beta}{c_{1}}e^{\beta\,r}t\,\cos\theta\,,$
$\displaystyle{\bf X}_{12}=e^{\beta\,r}\left[\cos\theta\,{\bf
Y}+\sin\theta\,\partial_{\theta}\right]\,,\qquad\qquad\qquad\qquad\,\,\,\psi_{12}=e^{\beta\,r}\cos\theta\,,$
$\displaystyle{\bf
X}_{13}=\frac{1}{2}\left(\frac{T_{1}\,e^{-2\beta\,r}}{\beta\,c_{0}^{2}}-\beta\,t^{2}\right)\partial_{t}+t\,\partial_{r}\,,\qquad\qquad\quad\,\,\psi_{13}=0\,,$
$\displaystyle{\bf
X}_{14}=-\beta\,t\,\partial_{t}+\partial_{r}\,,\qquad\qquad\qquad\qquad\qquad\qquad\,\,\,\,\,\psi_{14}=0\,,$
$\displaystyle{\bf
X}_{15}=\partial_{t}\,,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\,\psi_{15}=0\,,$

where $\beta$ is an integration constant, $T_{0}=c_{0}^{2}e^{2\beta
r},T_{1}=\pm c_{1}^{2},\beta^{2}=T_{1}/T_{2}$, ${\bf Y}$ is given in (103),
and $K_{\pm}(t,r)$ is defined as

$K_{\pm}(t,r)=\frac{1}{2}\left(\frac{c_{0}\,\beta}{c_{1}}e^{\beta\,r}t^{2}\pm\frac{c_{1}}{c_{0}\,\beta}e^{-\beta\,r}\right)\,.$
(109)

It is explicitly seen from (108) that in addition to the KVs ${\bf X}_{1},{\bf
X}_{2},{\bf X}_{3}$ given in (50), the vector fields ${\bf X}_{13},{\bf
X}_{14}$ and ${\bf X}_{15}$ are MCs.

Subcase (ND-B-ii). In this case, where $T_{0},T_{2}$ and $T_{3}$ are
constants, one can easily find that the number of CMCs is _six_ , and these
reduce to the MCs which are given by three KVs ${\bf X}_{1},{\bf X}_{2},{\bf
X}_{3}$ given in (50), and the remaining ones

$\displaystyle{\bf X}_{4}=\partial_{t}\,,\qquad{\bf
X}_{5}=\partial_{\bar{r}}\,,\qquad{\bf
X}_{6}=\bar{r}\partial_{t}-t\partial_{\bar{r}}\,,$ (110)

where we have used the rescaling $d\bar{r}=\sqrt{T_{1}}dr$.

Case (ND-C). Two of the $T^{\prime}_{a}$ are zero. In this case, the possible
subcases are (ND-C-i) $\ T^{\prime}_{p}\neq
0,\,\,T^{\prime}_{q}=0,\,\,(p=0,1\,\,{\rm and}\,\,q=2,3)$ and (ND-C-ii)
$T^{\prime}_{p}=0,\,\,T^{\prime}_{q}\neq 0$.

Subcase (ND-C-i). For this subcase, where $T_{2},T_{3}$ are constants, we find
15 CMCs that are similar to the ones given in (94). Here, the functions
$f_{1}(t)$ and $f_{2}(t)$ are respectively the same form given in Eqs. (99)
and (102), and the vector field ${\bf Y}$ has the form ${\bf
Y}=\frac{\sqrt{T_{1}}}{T_{0}}\partial_{t}+\partial_{\bar{r}}$. Also, the
functions $h_{1}(\bar{r}),h_{2}(\bar{r}),H_{1}(\bar{r})$ and $H_{2}(\bar{r})$
in this subcase have the following form

$\displaystyle
h_{1}(\bar{r})=a\sinh\left(\frac{\bar{r}}{c_{2}}\right)+b\cosh\left(\frac{\bar{r}}{c_{2}}\right)\,,\quad
h_{2}(\bar{r})=\frac{1}{c_{2}}\left[a\cosh\left(\frac{\bar{r}}{c_{2}}\right)+b\sinh\left(\frac{\bar{r}}{c_{2}}\right)\right]\,,$
(111) $\displaystyle H_{1}(\bar{r})=h_{2}(\bar{r})\,,\quad
H_{2}(\bar{r})=h_{1}(\bar{r})\,,$ (112)

with
$dr=d\bar{r}/\sqrt{{|T_{1}|}},T_{0}=T_{2}\,h_{2}(\bar{r})^{2}\,,T_{2}=c_{2}^{2}$
and $\alpha^{2}=\pm(a^{2}-b^{2})/T_{2}$. The CMCs ${\bf X}_{13},{\bf X}_{14}$
and ${\bf X}_{15}$ of this subcase reduce to MCs since the scale factors of
these vector fields are zero.

When $\alpha^{2}=0$, there are twelve CMCs which are in the form (105), by
replacing $\beta{\bf Y}\rightarrow{\bf
Y}=\frac{\sqrt{T_{1}}}{T_{0}}\partial_{t}+\partial_{\bar{r}}$ and
$H(\bar{r})\rightarrow 1$. For this subcase, the CMCs ${\bf X}_{10},{\bf
X}_{11}$ and ${\bf X}_{12}$ become MCs due to $\psi_{10}=0=\psi_{11}$ and
$\psi_{12}=0$.

Subcase (ND-C-ii). In this subcase, where $T_{0},T_{1}$ are constants, there
appears a constant of separation $\alpha$ that is given by the following
constraint equation

$\frac{T_{0}}{2T_{1}\sqrt{T_{2}}}\left(\frac{T_{2}^{\prime}}{\sqrt{T_{2}}}\right)^{\prime}=\alpha^{2}\,.$
(113)

Then, it follows for $\alpha^{2}\neq 0$ that there are 15 CMCs such that the
KVs ${\bf X}_{1},{\bf X}_{2},{\bf X}_{3}$ given in (50) and

$\displaystyle{\bf
X}_{4}=-\sqrt{T_{0}}\,f_{3}(t)\left(\frac{T_{2,\bar{r}}}{2}\sin\theta\sin\phi\,{\bf
Z}+\frac{1}{\sqrt{T_{2}}}\xi_{1}\right)\,,\quad\,\,\,\psi_{4}=f_{3}(t)H_{3}(\bar{r})\sin\theta\sin\phi\,,$
$\displaystyle{\bf
X}_{5}=-\sqrt{T_{0}}\,f_{4}(t)\left(\frac{T_{2,\bar{r}}}{2}\sin\theta\sin\phi\,{\bf
Z}+\frac{1}{\sqrt{T_{2}}}\xi_{1}\right)\,,\quad\,\,\,\psi_{5}=f_{4}(t)H_{3}(\bar{r})\sin\theta\sin\phi\,,$
$\displaystyle{\bf X}_{6}=-T_{1}\sqrt{b}\sin\theta\sin\phi\,{\bf
Z}-\frac{bT_{2,\bar{r}}}{\sqrt{T_{2}}}\xi_{1}\,,\qquad\qquad\qquad\quad\psi_{6}=H_{4}(\bar{r})\sin\theta\sin\phi\,,$
$\displaystyle{\bf
X}_{7}=\sqrt{T_{0}}\,f_{3}(t)\left(\frac{T_{2,\bar{r}}}{2}\sin\theta\cos\phi\,{\bf
Z}+\frac{1}{\sqrt{T_{2}}}\xi_{2}\right)\,,\qquad\,\,\,\psi_{7}=-f_{3}(t)H_{3}(\bar{r})\sin\theta\cos\phi\,,$
$\displaystyle{\bf
X}_{8}=\sqrt{T_{0}}\,f_{4}(t)\left(\frac{T_{2,\bar{r}}}{2}\sin\theta\cos\phi\,{\bf
Z}+\frac{1}{\sqrt{T_{2}}}\xi_{2}\right)\,,\qquad\,\,\psi_{8}=-f_{4}(t)H_{3}(\bar{r})\sin\theta\cos\phi\,,$
$\displaystyle{\bf X}_{9}=T_{1}\sqrt{b}\sin\theta\cos\phi\,{\bf
Z}+\frac{bT_{2,\bar{r}}}{\sqrt{T_{2}}}\xi_{2}\,,\qquad\qquad\qquad\quad\,\,\,\,\psi_{9}=-H_{4}(\bar{r})\sin\theta\cos\phi\,,$
(114) $\displaystyle{\bf
X}_{10}=\sqrt{T_{0}}\,f_{3}(t)\left(-\frac{T_{2,\bar{r}}}{2}\cos\theta\,{\bf
Z}+\frac{\sin\theta}{\sqrt{T_{2}}}\,\partial_{\theta}\right)\,,\qquad\,\,\,\,\psi_{10}=f_{3}(t)H_{3}(\bar{r})\cos\theta\,,$
$\displaystyle{\bf
X}_{11}=\sqrt{T_{0}}\,f_{4}(t)\left(-\frac{T_{2,\bar{r}}}{2}\cos\theta\,{\bf
Z}+\frac{\sin\theta}{\sqrt{T_{2}}}\,\partial_{\theta}\right)\,,\qquad\,\,\,\,\psi_{11}=f_{4}(t)H_{3}(\bar{r})\cos\theta\,,$
$\displaystyle{\bf X}_{12}=-T_{1}\sqrt{b}\cos\theta\,{\bf
Z}+\frac{bT_{2,\bar{r}}}{\sqrt{T_{2}}}\sin\theta\,\partial_{\theta}\,,\qquad\qquad\qquad\psi_{12}=H_{4}(\bar{r})\cos\theta\,,$
$\displaystyle{\bf
X}_{13}=\left(\frac{a}{2}f_{1}-\frac{T_{1}}{T_{0}}\ddot{f}_{1}\bar{r}\right)\partial_{t}+T_{2}\dot{f}_{1}\,\partial_{\bar{r}}\,,\qquad\qquad\qquad\,\,\,\,\,\psi_{13}=\dot{f}_{1}\frac{T_{2,\bar{r}}}{2}\,,$
$\displaystyle{\bf
X}_{14}=\left(\frac{a}{2}f_{2}-\frac{T_{1}}{T_{0}}\ddot{f}_{2}\bar{r}\right)\partial_{t}+T_{2}\dot{f}_{2}\,\partial_{\bar{r}}\,,\qquad\qquad\qquad\,\,\,\,\,\psi_{14}=\dot{f}_{2}\frac{T_{2,\bar{r}}}{2}\,,$
$\displaystyle{\bf
X}_{15}=\partial_{t}\,,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,\psi_{15}=0\,,$

where $H_{3}(\bar{r}),H_{4}(\bar{r}),f_{3}(t),f_{4}(t)$ and ${\bf Z}$ are
defined as follows:

$\displaystyle
H_{3}(\bar{r})=\sqrt{\frac{T_{0}}{T_{2}}}\left(1-\frac{T_{2,\bar{r}}^{2}}{4T_{2}}\right)\,,\quad
H_{4}(\bar{r})=\left(b-\frac{T_{1}\sqrt{b}}{T_{2}}\right)\frac{T_{2,\bar{r}}}{\sqrt{T_{2}}}\,,$
(115) $\displaystyle
f_{3}(t)=\left\\{\begin{array}[]{l}\frac{1}{\alpha}\sin(\alpha
t)\,,\qquad\,\,\,\,{\rm for}\,\,\alpha^{2}>0,\\\
\frac{1}{|\alpha|}\sinh(|\alpha|t)\,,\quad{\rm
for}\,\,\alpha^{2}<0,\end{array}\right.$ (118) $\displaystyle
f_{4}(t)=\left\\{\begin{array}[]{l}\frac{1}{\alpha}\cos(\alpha
t)\,,\qquad\quad\,\,\,{\rm for}\,\,\alpha^{2}>0,\\\
-\frac{1}{|\alpha|}\cosh(|\alpha|t)\,,\quad{\rm
for}\,\,\alpha^{2}<0,\end{array}\right.$ (121) $\displaystyle{\bf
Z}=\frac{1}{T_{0}}\partial_{t}+\frac{\sqrt{T_{2}}}{T_{1}}\partial_{\bar{r}}\,.$
(122)

Here, Eq. (113) becomes

$\frac{T_{0}T_{2,\bar{r}\bar{r}}}{2T_{1}}=\alpha^{2}\,,$ (123)

by using the transformation $dr=d\bar{r}/\sqrt{|T_{2}|}$, and it has a
solution $T_{2}=\frac{\alpha^{2}T_{1}}{T_{0}}\bar{r}^{2}+a\,\bar{r}+b$, where
$a$ and $b$ are integration constants. Further, we have a relation
$T_{0}=4b\,\alpha^{2}T_{1}/(a^{2}-4T_{1})$ from the constraint equation (56).

If we consider the possibility $\alpha^{2}=0$ in (123), it yields
$T_{2}=a\,\bar{r}+b$. Then one can find 15 CMCs which are three KVs ${\bf
X}_{1},{\bf X}_{2},{\bf X}_{3}$ given in (50), and the following proper CMCs

$\displaystyle{\bf
X}_{4}=-\sqrt{T_{0}}\,t\left(\frac{a}{2}\sin\theta\sin\phi\,{\bf
Z}+\frac{1}{\sqrt{T_{2}}}\xi_{1}\right)\,,\,\,\,\psi_{4}=tH_{3}(\bar{r})\sin\theta\sin\phi\,;\,\,{\bf
X}_{5}=\frac{t}{2}{\bf X}_{4}\,,\,\,\psi_{5}=\frac{t}{2}\psi_{4}\,$
$\displaystyle{\bf X}_{6}=-\frac{1}{2a}\sin\theta\sin\phi\,{\bf
Z}-\frac{1}{\sqrt{T_{2}}}\xi_{1}\,,\qquad\qquad\psi_{6}=H_{5}(\bar{r})\sin\theta\sin\phi\,,$
$\displaystyle{\bf
X}_{7}=\sqrt{T_{0}}\,t\left(\frac{a}{2}\sin\theta\cos\phi\,{\bf
Z}+\frac{1}{\sqrt{T_{2}}}\xi_{2}\right)\,,\,\,\,\psi_{7}=-tH_{3}(\bar{r})\sin\theta\cos\phi\,;\,\,{\bf
X}_{8}=\frac{t}{2}{\bf X}_{7}\,,\,\,\psi_{8}=\frac{t}{2}\psi_{7}\,,$
$\displaystyle{\bf X}_{9}=\frac{1}{2a}\sin\theta\cos\phi\,{\bf
Z}+\frac{1}{\sqrt{T_{2}}}\xi_{2}\,,\qquad\qquad\psi_{9}=-H_{5}(\bar{r})\sin\theta\cos\phi\,,$
(124) $\displaystyle{\bf
X}_{10}=\sqrt{T_{0}}\,t\left(-\frac{a}{2}\cos\theta\,{\bf
Z}+\frac{\sin\theta}{\sqrt{T_{2}}}\,\partial_{\theta}\right)\,,\,\,\,\psi_{10}=tH_{3}(\bar{r})\cos\theta\,;\,\,{\bf
X}_{11}=\frac{t}{2}{\bf X}_{10}\,,\,\,\psi_{11}=\frac{t}{2}\psi_{10}\,,$
$\displaystyle{\bf X}_{12}=-\frac{1}{2a}\cos\theta\,{\bf
Z}+\frac{\sin\theta}{\sqrt{T_{2}}}\,\partial_{\theta}\,,\qquad\qquad\psi_{12}=H_{5}(\bar{r})\cos\theta\,,$
$\displaystyle{\bf
X}_{13}=\frac{a}{2}\left(\frac{t^{2}}{2}-\frac{\bar{r}}{\sqrt{T_{0}}}\right)\partial_{t}+\frac{2T_{2}}{a}t\,\partial_{\bar{r}}\,,\quad\psi_{13}=t\,,$
$\displaystyle{\bf
X}_{14}=t\,\partial_{t}+\frac{2T_{2}}{a}\,\partial_{\bar{r}}\,,\quad\psi_{14}=1\,;\,\,{\bf
X}_{15}=\partial_{t}\,,\,\,\,\psi_{15}=0\,,$

where $r=\frac{2}{a}\sqrt{a\,\bar{r}+b},T_{1}=a^{2}/4$, and $H_{5}({\bar{r}})$
is defined by

$\displaystyle
H_{5}(\bar{r})=\frac{1}{\sqrt{T_{2}}}\left(1-\frac{a^{3}}{4T_{2}}\right)\ .$
(125)

Case (ND-D). One of the $T^{\prime}_{a}$ is zero. In this case, the
possibilities are (ND-D-i) $T^{\prime}_{0}=0,\,\,T^{\prime}_{j}\neq 0$,
$(j=1,2,3)$ and (ND-D-ii) $\ T^{\prime}_{1}=0,\,\,T^{\prime}_{k}\neq
0,\,\,(k=0,2,3)$.

In the subcase (ND-D-i), if $\alpha^{2}\neq 0$, then one obtains 15 CMCs as
given by (50) and (94) in the case (ND-A), in which the differences are the
conditions $T_{0}=c_{0}^{2}$, $T_{2}=[a\cosh\bar{r}+b\sinh\bar{r}]^{-2}$, and
$\alpha^{2}=\pm T_{0}(a^{2}-b^{2})$ which is a separation constant. Also, when
$\alpha^{2}=0$, we find 12 CMCs which are the same form as (105) in the case
(ND-A), where the differences are $T_{0}=c_{0}^{2}$ and
$T_{2}=c_{2}^{2}e^{-2\beta\bar{r}}$.

For the subcase (ND-D-ii), we again find 15 CMCs for $\alpha^{2}\neq 0$, which
are the same form as given in (50) and (94), and the differences come from the
constraints as
$T_{1}=c_{1}^{2},T_{0}=T_{2}\left[a\cosh\bar{r}+b\sinh\bar{r}\right]^{2}$, and
$\alpha^{2}=\pm(a^{2}-b^{2})/T_{1}$. Further, if $\alpha^{2}=0$, then we find
12 CMCs which are the same form as in (50) and (105) together with the
constraints $T_{1}=c_{1}^{2},T_{0}=c_{0}^{2}T_{2}e^{2\beta\bar{r}}$ and
$\beta^{2}=1$.

Case (ND-E). All $T^{\prime}_{a}$ are zero. In this case, the constraints are
$T_{0}=c_{0},\,T_{1}=c_{1},\,T_{2}=c_{2}$, where $c_{0},c_{1}$ and $c_{2}$ are
non-zero constants. Using these constraints, it follows that in addition to
the three KVs given in (50) there are three additional CMCs such as

${\bf X}_{4}=t\,\partial_{r}-\frac{c_{1}}{c_{0}}r\,\partial_{t}\,,\qquad{\bf
X}_{5}=\partial_{r}\,,\qquad{\bf X}_{6}=\partial_{t}\,,$ (126)

with $\psi_{4},\psi_{5},\psi_{6}=0$, which means that these CMCs reduce to
MCs.

Here, using the constraints $T_{0}=c_{0},\,T_{1}=c_{1},\,T_{2}=c_{2}$ in Eqs.
(15), (16) and (17), we have the following ordinary differential equations:

$\displaystyle\lambda^{\prime}=\frac{1}{r}\left(1-e^{\lambda}\right)+c_{0}\,r\,e^{\lambda-\nu}\,,$
(127)
$\displaystyle\nu^{\prime}=\frac{1}{r}\left(e^{\lambda}-1\right)+c_{1}\,r\,,$
(128) $\displaystyle 2\nu^{\prime\prime}+\nu^{\prime
2}-\nu^{\prime}\lambda^{\prime}+\frac{2}{r}(\nu^{\prime}-\lambda^{\prime})-\frac{4}{r^{2}}c_{2}e^{\lambda}=0\,.$
(129)

Then, putting $\lambda^{\prime}$ and $\nu^{\prime}$ given above into (129),
yields

$e^{2\lambda-\nu}-(c_{1}\,r^{2}+1)e^{\lambda-\nu}+(3c_{1}\,r^{2}-4c_{2})\frac{e^{\lambda}}{c_{0}\,r^{2}}+\frac{c_{1}}{c_{0}}(c_{1}r^{2}+1)=0\,,$
(130)

which gives

$e^{\nu}=\frac{c_{0}\,e^{\lambda}\left[e^{\lambda}-(c_{1}\,r^{2}+1)\right]}{\left(\frac{4c_{2}}{r^{2}}-3c_{1}\right)e^{\lambda}-c_{1}(c_{1}\,r^{2}+1)}\,.$
(131)

Thus, the physical variables $\rho,p$ and $\pi_{ab}$ for this case are

$\rho=\frac{\left(\frac{4c_{2}}{r^{2}}-3c_{1}\right)-c_{1}(c_{1}\,r^{2}+1)\,e^{-\lambda}}{e^{\lambda}-c_{1}\,r^{2}-1}\,,\qquad
p=\frac{1}{3}\left(c_{1}\,e^{-\lambda}+\frac{2c_{2}}{r^{2}}\right)\,,$ (132)
$\pi_{11}=\frac{2}{3}\left(c_{1}-\frac{c_{2}}{r^{2}}e^{\lambda}\right)\,,\qquad\pi_{22}=-\frac{r^{2}}{2}e^{-\lambda}\pi_{11}\,,\qquad\pi_{33}=\sin^{2}\theta\pi_{22}\,,$
(133)

with the choice of timelike observers. Note that the prefect fluid which
requires $\pi_{ab}=0$ is not allowed in this case since Eqs. (127) - (129) are
not equivalently satisfied for $e^{\lambda(r)}=c_{1}\,r^{2}/c_{2}$ and
$\nu(r)$ that comes from (131).

The vector fields for cases (ND-A)-(ND-D) are CMCs of the original metric (8)
which are _fifteen_ for $\alpha^{2}\neq 0$ and _twelve_ for $\alpha^{2}=0$ in
almost all cases. Also, we can employ an anisotropic fluid without heat flux
for all cases in this section.

##  V Conclusions

Symmetries of the metric tensor on a manifold, like KVs, HMs and CKVs have
finite dimensional Lie algebras as the metric tensor is always non-degenerate.
The maximum dimension for the Killing algebras (in four dimensional space) is
10, for HMs it is 11 and for CKVs 15. But there is no such guarantee for other
tensors which can be degenerate as well as non-degenerate. Thus the Lie
algebras for RCs, MCs etc. can be finite as well as infinite. When the tensor
is non-degenerate the algebra of RCs and MCs is finite and maximum dimension
is 10, but for the degenerate tensor it can be finite as well as infinite. For
conformal collineations the maximum dimension is 15 for the non-degenerate
tensor. For the degenerate case finite dimensionality is not guaranteed.

In this work, we have completely classified CMCs for static spherically
symmetric spacetimes which are not of Bertotti-Robinson type. We have seen in
Section III that if the energy-momentum tensor is degenerate, i.e.,
$\det(T_{ab})=0$, then the CMCs have infinite degrees of freedom. In Section
IV in which the non-degenerate case where $\det(T_{ab})\neq 0$ is considered,
there are _fifteen_ CMCs if the separation constant is not zero, or _twelve_
CMCs if the separation constant vanishes. In the cases (ND-B-ii) and (ND-E) of
_six_ dimensional Lie algebras the conformal factor comes out to be zero, and
thus, they are actually MCs and not CMCs.

We point out that in degenerate case (D-A4-i) we found an exact spherically
symmetric solution of the form

$\displaystyle
ds^{2}=\left[-\frac{\nu_{0}}{4}\sqrt{1-\frac{2M}{r}}+\frac{\nu_{1}}{2}\left\\{r-6M+3M\sqrt{1-\frac{2M}{r}}\ln\left(r+\sqrt{r(r-2M)}-M\right)\right\\}\right]^{2}dt^{2}$
$\displaystyle\qquad\qquad-\frac{dr^{2}}{1-\frac{2M}{r}}-r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right)\,,$
(134)

where we have taken $\lambda_{1}=2M$. This new metric is radiation-dominated
fluid solution for EFEs. It is interesting to note from Eq. (35) that the NEC
for the above solution reads $p\geq 0$. For this metric the Ricci scalar
becomes

$R=\frac{8\nu_{1}}{-\nu_{0}\sqrt{r(r-2M)}+2\nu_{1}\left[r(r-6M)+3M\sqrt{r(r-2M)}\ln\left(r+\sqrt{r(r-2M)}-M\right)\right]}\,.$
(135)

This metric reduces to the well-known form of Schwarzschild solution if
$\nu_{0}=-4$ and $\nu_{1}=0$. The above solution admits _eleven_ CMCs, which
are given in (50) and (84), and they have finite dimensional Lie algebra.

For the sake of completeness, we shortly touch on the possible extension of
similar studies in generic dimensions. It is well-known that the spherically
symmetric metrics continue to exist in space-times of dimension greater than
four. However, it will be much more complicated to solve the symmetry
equations even for static and spherically symmetric space-times in higher
dimensions.

## Acknowledgements

KS acknowledges a research grant from the Higher Education Commission of
Pakistan.

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# Distributed Twins in Edge Computing: Blockchain and IOTA

Anwar Sadad 1, Muazzam A. Khan 1, Baraq Ghaleb 2, Fadia Ali Khan 3, Maha
Driss4, Wadii Boulila5, Jawad Ahmad 2
1 Department of Computer Science, Quaid-i-Azam University, Islamabad,
Pakistan.  2 School of Computing, Engineering & The Built Environment,
Edinburgh Napier University, Edinburgh, UK.  3Department of Mechatronics
Engineering, Wah Engineering College, University of Wah, Wah, Pakistan
4Security Engineering Laboratory, CCIS, Prince Sultan University, Riyadh,
Saudi Arabia  5Robotics and Internet-of-Things Laboratory, Prince Sultan
University, Riyadh, Saudi Arabia

###### Abstract

Blockchain (BC) and Information for Operational and Tactical Analysis (IOTA)
are distributed ledgers that record a huge number of transactions in multiple
places at the same time using decentralized databases. Both BC and IOTA
facilitate Internet-of-Things (IoT) by overcoming the issues related to
traditional centralized systems, such as privacy, security, resources cost,
performance, and transparency. Still, IoT faces the potential challenges of
real-time processing, resource management, and storage services. Edge
computing (EC) has been introduced to tackle the underlying challenges of IoT
by providing real-time processing, resource management, and storage services
nearer to IoT devices on the network’s edge. To make EC more efficient and
effective, solutions using BC and IOTA have been devoted to this area.
However, BC and IOTA came with their pitfalls. This survey outlines the
pitfalls of BC and IOTA in EC and provides research directions to be
investigated further.

###### Index Terms:

Blockchain, IOTA, Distributed Ledger, Edge Computing, IoT, Bitcoin

##  I Introduction

††footnotetext: This work is supported by Prince Sultan University in Saudi
Arabia

IoTs is an emerging technology capable of connecting the real world with
devices that can generate or transmit data, communicate with one another, and
remotely control objects via the Internet in the absence of humans. With the
help of Wireless Sensors Networks and Radio Frequency Identification [1], IoT
is extensively evolved in healthcare, industries, education, smart homes,
smart cities, surveillance, and smart agriculture [2, 3, 4]. For data
processing of IoT, cloud computing (CC) serves as a backbone due to its
scalability and flexibility. However, there are significant issues associated
with CC, like real-time processing, resource allocation, and security of data
[5]. To provide real-time data processing, EC has been introduced, which
allows data processing on the edge of the network. However, issues associated
with its security still exist, such as leakage of sensitive information,
denial of service attacks, access control, and privacy of data.

Distributed immutable ledgers, such as BCs and IOTA, have recently shown
feasibility in overcoming the aforementioned security issues of IoT. BCs use a
peer-to-peer (P2P) network for communication which allows one to directly
initiate payment and send it to another party without the involvement of any
trusted third parties [6]. In other words, it is a distributed database system
linked in a P2P network through advanced cryptographic techniques that use a
distributed ledger to ensure the security of messages or transactions
exchanged over the network [7].

BC (e.g., bitcoin and ethereum) faces challenges when used in small day-to-day
transactions (e.g., micropayments). Therefore, the research community proposed
IOTA to tackle such problems [8]. IOTA differs significantly from bitcoin
because it is not developed based on BC technology. To facilitate low-cost
micropayments, the developers of IOTA have developed an entirely different
architecture using directed acyclic graph (DAG) known as tangling [9].

To overcome the underlying challenges of EC, including security, privacy,
latency, accessibility, data leakage, scalability, resource handling,
throughput, reliability, control management, and energy efficiency, many
contributions based on BC and IOTA have been introduced in the last few years.
To the best of our knowledge there is no survey or work exist in the
literature that highlights the contributions of BC and IOTA in EC. Therefore,
the purpose of this study is to present a survey that overviews recently
proposed methods based on BC and IOTA in EC and conclude which technology (BC
or IOTA) serves best based on chosen parameters for EC.

The main contributions of the proposed work are as follows.

  * •

In literature, there is no work that discusses the contributions of BC and
IOTA in EC.

  * •

We have discussed that how BC and IOTA cope with underlying challenges of EC.

  * •

We have comparatively analyze the role of BC and IOTA in EC.

  * •

Finally, we conclude with future direction in EC using BC and IOTA.

The rest of the paper is organized as follows. Section 2 overviews both BC and
IOTA and highlights contributions being made in EC using BC and IOTA. Section
3 comparatively analyzes proposed methods in terms of chosen parameters.
Finally, Section 4 concludes the study with some recommendations.

##  II Literature Survey

This section briefly discusses BC and IOTA, and explores the contributions
being made in EC using BC and IOTA.

###  II-A Blockchain and IOTA

This section introduces BC and IOTA along with their strength and weaknesses.

The BC technology, as roughly shown in Figure 1, is known for its security
which is protected based on three aspects such as, decentralization [10],
advanced cryptographic algorithms [11], and consensus protocols [12].

As depicted in Figure 1, each system in BC technology can be represented as
nodes. Whereas, each node hold a ledger that consists of information like
block number, header, timestamp, hash to previous block, own hash and the
record of transactions made in the chain.

Figure 1: Rough Presentation of BC Technology

BC uses a distributed ledger that facilitates P2P communication without a
trusted third party. It is evident that BC is a secure technology based on the
principles of hashing and digital signatures. However, due to the progress
made in quantum computing, BC security is at risk as it has been shown
recently that some of its underlying cryptographic algorithms can easily be
cracked by Grover’s and Shor’s [13] algorithms. In addition, there are various
shortcomings of using BC in IoT, including the unfair/high transaction fee and
the constrained nature of machine-to-machine communication. Therefore, new
distributed ledger has emerged called IOTA [14], which overcomes the issues of
BC concerning IoT and provide secure communication based on a concept called
DAG. The underlying features provided by IOTA are scalability, zero fee
transaction, quantum immune, security, and low resource requirements [15].

###  II-B Blockchain in Edge Computing

This section presents the role and contributions of BC in EC, as depicted in
Table I. The widespread success of CC is evident; however, it is not an all-
in-one solution with the major issues including centralization of resources,
high latency and jitter in gaming, augmented reality, and e-health [16]. EC
was introduced to overcome the above mentioned issues by providing computing
power and storage near to edge network [5]. However, EC alone cannot solve the
issues regarding privacy and security of data in IoT. Hence, the focus shifted
to exploring how distributed ledger-based technologies, including BC and IOTA,
can be used to address such issues. Considering such importance, the authors
in [17] presented a survey on the integration of BC and EC showing how
problems of data privacy, integrity, leakage, and access control can be solved
with such integration. For secured access control, the study in [10] proposed
a distributed trusted authentication system using BC in EC. The study
guarantees a secured access control and achieve activity traceability of
terminals based on dynamic name resolution and elliptic curve cryptography.
However, it stores authentication and logs data using a practical Byzantine
fault tolerance consensus algorithm, compromise on authentication data may
provide unintended access to the network by exploiting desired authentication
or log data.

TABLE I: Proposed solutions based on BC to tackle underlying challenges of EC No. |  Authors | SEC | DP | LAT | ACC | DL | SCA | RH | THR | REL | CM | EE
---|---|---|---|---|---|---|---|---|---|---|---|---
1 |  Guo, S., et al [10] | ✓ | ✗ | ✓ | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓
2 |  Jayasinghe, U., et al [18] | ✓ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ | ✗
3 |  Zhang, L., et al [19] | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ | ✗
4 |  Yuan, L., et al [11] | ✓ | ✓ | ✓ | ✗ | ✓ | ✗ | ✗ | ✓ | ✗ | ✓ | ✗
5 |  Huang, Y., et al [20] | ✗ | ✗ | ✓ | ✗ | ✗ | ✓ | ✗ | ✓ | ✓ | ✓ | ✓
6 |  Miao, Q., et al [21] | ✗ | ✓ | ✓ | ✗ | ✓ | ✗ | ✗ | ✗ | ✓ | ✗ | ✗
7 |  Tzenetopoulos, A., et al [22] | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✓ | ✓ | ✓ | ✓
8 |  Ahmad, A., et al [12] | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✓ | ✓ | ✗ | ✓
9 |  Abdi, A.I., et al [5] | ✓ | ✓ | ✓ | ✓ | ✗ | ✓ | ✗ | ✗ | ✓ | ✓ | ✓
ABBREVATIONS
SEC: Security, DP: Data Privacy, LAT: Latency, ACC: Accessibility, DL: Data
Leakage, SCA: Scalability,
RH: Resource Handling, THR: Throughput, REL: Reliability, CM: Control
Management,
EE: Energy Efficiency

To provide efficient authentication in IOTs, the authors in [18] proposed a
method called SCAB-IoTA based on BC which employs a secure mechanism to make
clusters based on angular distance. To become a member of cluster, each IoT
device is required to authenticate itself if it is in the radius of the
desired cluster. However, encryption and decryption of SCAB-IoTA consume more
energy and the scalability has been set aside respectively. BC restricts
adversaries from alteration of data but may compromise stakeholders’ data. To
preserve the privacy of parties involved in chains, the study in [19]
presented a TrustChain solution by combining BC with trust concepts to exclude
problems associated with traditional BC architectures. They defined trust as a
qualitative or quantitative property of a trustee measured by a trustor for a
given task in a specific context and in a specific time period. They have used
a trust-based consensus management protocol to evaluate trust based on nodes’
knowledge, experience, and reputation. One major issue related to such a
technique is starvation because communication with the desired node depends on
trust. Nodes with high trust can frequently communicate, whereas nodes with
low trust will never be able to communicate with other nodes. To tackle
resource allocation problem in EC, the study in [11] proposed a three-tier
architecture based on BC technology. It consists of a group-agent strategy
with trust computing, a stacked task sorting and ranking mechanism, and a
secured and efficient content model. Fake edge devices can compromise a
group’s trustworthiness, and tasks with low ranks may face starvation.
Uploading cipher text to the cloud and indexing it using the BC may result in
overhead in the case of large-scale data. Multi-access EC [5] introduced an
extended form of CC that allows storage services at the network edge to
provide low-latency data retrieval. However, trust and incentive are two major
problems in collaborative edge storage. To overcome such issues, the authors
in [20] have proposed novel collaborative edge storage based on BC to address
incentive and trust evaluation using the historical performance of edge
servers. Edge server (data off-loader) can publish a task for which other edge
servers contend for the task. Reliable edge servers with a good reputation and
guaranteed response time can be selected. One major issue in such an approach
is when a new edge server wants to enter the ecosystem with no previous
performance record. BC can also help in the efficient utilization of resources
in edge environments. For the utilization problem, the authors in [21] have
proposed a system that can ensure fair and efficient utilization of resources
on edge devices rendering it more scalable. They have proposed a data
migration algorithm with consensus having low energy consumption in edge
devices along with a new proof of stake mechanism. Due to high mobility in the
edge environment, the nodes are moving in a small range, but the network
topology remains the same. To adapt to topological changes, a migration
algorithm has been designed to reallocate the data and block storage to
devices dynamically. They have also used a cache mechanism to provide recent
block allocation, which can reduce the overhead of missing blocks. One issue
in such approach is the overflow of cache memory and other is the node having
cache is down? Obviously, IoT devices are intended to share massive data with
each other to impose quality-of-service [23]. However, data sharing may result
in data leakage of providers’ data. Such concern is realized in [24], which
presented a data sharing model based on a secured data mechanism called
”BP2P-FL”. It is team-based data sharing with reward and punishment mechanisms
that are used to ensure data sharing with high quality and reliability. Team-
based data sharing provides a collaborative environment in which a team
sponsor initiates a task and assigns it to its members. Team-sponsor evaluates
the capability of members based on their contribution in task completion, and
members with high efforts get rewarded, whereas poor participation of members
is punished by excluding them from the team. The experimental results show
that the proposed method exhibited high accuracy and enhanced privacy in IoT.

###  II-C IOTA in Edge Computing

This section addresses the contributions of IOTA in EC as depicted in Table
II. As illustrated in Figure 2, each system in IOTA can be presented as node.
There are three phases namely, Tip, Unconfirmed, and Fully Confirmed through
which each node passes. Initially, when the nodes are added it passes through
Tip phase, then Confirmed and when all nodes confirm the authenticity of the
added node, it goes to Fully Confirmed phase and become part of the network.
There are various challenges in integrating of BC with EC, such as
scalability, latency, high energy consumption, fairness, and sensitivity to
quantum computing. Therefore, IOTA, which uses DAGs, comes in to tackle issues
introduced exist in BC. To address scalability in IoT, the authors in [14]
proposed a Scalable Distributed Intelligence Tangle-based approach to allow
the integration of IoT devices across various applications. They also have
presented a new proof-of-work (PoW) that enhanced energy efficiency. The
experimental results show the achievement of scalability along with
maintaining energy efficiency. For PoW, they have used an isolated server that
performs heavy tasks to minimize energy consumption. However, this study
suffers from the isolated server’s single-point-of-failure problem and the
data processing management.

Figure 2: Rough Presentation of IOTA Technology

For energy efficiency problem the authors in [25] have proposed a novel
Mobile-Agent Distributed Intelligence Tangle-Based approach to manage
resources and deploy IoT applications that are both scalable and energy
efficient. They have performed a PoW on IoT devices to reduce energy
consumption on resource-intensive devices. The proposed approach facilitates
local interaction, collection, aggregation of transactional data, and an
efficient route plan. One potential problem in such an approach is the
performance of PoW on dedicated servers, which may increase the cost of
maintenance.

TABLE II: Proposed solutions based on IOTA to tackle underlying problems No. |  Authors | SEC | DP | LAT | ACC | DL | SCA | RH | THR | REL | CM | EE
---|---|---|---|---|---|---|---|---|---|---|---|---
1 |  Alsboui, T., et al [14] | ✓ | ✓ | ✓ | ✗ | ✗ | ✓ | ✗ | ✗ | ✓ | ✗ | ✓
2 |  Alsboui, T., et al [25] | ✓ | ✗ | ✓ | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓
3 |  Hellani, H., et al [15] | ✓ | ✓ | ✗ | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗
4 |  Abdullah, S., et al [26] | ✓ | ✓ | ✗ | ✓ | ✗ | ✗ | ✗ | ✓ | ✓ | ✗ | ✗
5 |  Fang, Z., et al [27] | ✓ | ✓ | ✗ | ✗ | ✓ | ✗ | ✗ | ✓ | ✓ | ✗ | ✓
6 |  Chen, Y., et al [28] | ✓ | ✗ | ✗ | ✓ | ✓ | ✓ | ✗ | ✗ | ✓ | ✗ | ✗
7 |  Carelli, A., et al [29] | ✓ | ✓ | ✗ | ✗ | ✓ | ✓ | ✗ | ✗ | ✓ | ✗ | ✓

IOTA classifies nodes into full and light nodes and can manually connect light
nodes with full nodes using the IOTA client balancer. This overcharges full
nodes and degrades the performance of the IoT platform. To cope with such an
issue, the authors in [15] introduced a mechanism that fairly distributes the
task among all nodes. They have developed an enhanced resource allocation
algorithm called weight least connection which has improved the balancing of
data traffic among full nodes based on their weights and active connections.
For secure data sharing, a framework-based solution in [26] has presented
using masked authentication messaging (MAM) with a tangle. For the transfer of
data to distributed ledgers, MAM provides an additional layer of security with
cryptographic functionalities, which maintains the integrity, authenticity,
and confidentiality of data. Since IoT devices are restricted in terms of
memory and computing power, such a solution is quite expensive for IoT.
However, the focus of security analysis in IoT is limited to the node level,
while the interactive nature of the device has been ignored. To cover up such
a hole, the authors in [27] have proposed a framework that is used to monitor
and detect potential danger to IoT devices. IOTA has been used for generating
the attack graphs, probably pop-up resources, along with attack traces that
can be compromised. They have also identified the dependencies between various
devices by which adversaries can employ severe attacks. The proposed IOTA
based model has been tested against 37 syntactic smart home systems, showing
that it is highly effective and efficient. However, using such a model in edge
environments will result in high energy consumption, which needs to be
minimized for deployment in highly scalable networks. Furthermore, an attack
called Parasite Chain Attack (PCA) [30] in which the attacker invincibly
builds a sub-tangle that results in double-spending. The consequences of such
an attack may be catastrophic in terms of finance because it can affect the
entire network by double-spending if it is not adequately prevented.
Therefore, the study [28] presented a scheme to tackle the PCA attack. They
have proposed an algorithm for the prevention of PCA based on price splitting
to slow the formation of the parasite chain effectively. However, effective
detection and prevention mechanisms are required to operate in a scalable
environment consuming less energy. Sensors are primary elements in IoTs, for
sensors’ security and privacy, the study in [29] proposed L2sec, a
cryptographic protocol to secure source data exchange over the IOTA. Obtained
results have shown better performance in terms of effectiveness and
scalability.

##  III Comparative Analysis

This section comparatively analyzes both BC and IOTA in terms of security,
privacy, latency, accessibility, data leakage, scalability, resource handling,
throughput, reliability, control management, and energy efficiency.

The power and flexibility of IoT enable the creation of smart environments and
reduce human efforts across many areas; namely, smart health, smart homes,
smart cities, and smart vehicles [28]. To overcome the underlying challenges
of EC, the concept of distributed ledgers (BC and IOTA) has been deployed in
EC [12, 27]. It is obvious that the integration of BC and IOTA in EC is itself
a challenge that offers more issues related to scalability, energy efficiency,
and security. This survey is conducted with the purpose of studying the
integration effects of BC and IOTA with EC.

Solutions based on BC in EC are depicted in Figure 3, which shows the
distribution of defined parameters that BC tries to tackle. Challenges offered
by EC have been intelligently overcome using BC. However, BC itself has
various limitations such as scalability, latency, power consumption, fee
fairness, and privacy of stakeholders [27]. The latest proposed papers on BC
in edge networks from 2019 to 2020 have been scrutinized in which it is found
that energy efficiency, scalability, and resource handling are still set
aside, which are serious issues related to EC. Issues related to privacy, data
leakage, accessibility, and latency have been considered, and plenty of work
is devoted to scalability, energy efficiency, and scalability. BC is resource
intensive and designed for large transactions, whereas IoT consists of devices
with limited resources, which makes it difficult for IoT to incorporate BC
efficiently. Since IoT is a scalable network that connects devices ranging
from hundreds to thousands in number, the new node requires days of delay to
be part of the ledger. To be integrated with EC, BC requires more intelligent
and lightweight solutions that tackle scalability, energy efficiency, and
security preservation.

Figure 3: Distribution of proposed solutions using BC in EC

Challenges offered by BC can be resolved by using IOTA [26] which is based on
DAG, it is more secure and scalable than BC. It tackles scalability, fee
fairness, micro-payments, and energy efficiency problems effectively [22]. It
is more resistant to quantum attacks as compared to BC [7]. Fig 4 depicts the
contributions deployed in EC in recent years, from 2020 to 2022. The use of
IOTA in EC tackled scalability, accessibility, and energy efficiency problems
[12]. Plenty of recent works deal with privacy, data leakage, accessibility,
scalability, and energy efficiency, where resource handling and latency still
require a prominent solution. However, IOTA is also prone to attacks, namely
conflicting transactions, blowball, lazy tips, API, and social engineering
attacks (detailed study can be found in [31]). Considering such attacks, more
robust and resistive techniques are required to be introduced. Furthermore, to
facilitate EC, IOTA based solutions are required to efficiently use energy and
provide intelligent resource handling mechanisms for enhancing the utilization
of resources and improving the over-all performance of the IoT.

Figure 4: Distribution of proposed solutions using IOTA in EC

Integration of BC with EC requires prominent solutions to tackle resource
handling, scalability, and energy efficiency problems. For tackling such
issues, IOTA is a good option for deployment in EC. However, the latency
problems need to be considered and addressed to improve the system’s overall
performance. Data accumulation, backup, monitoring of devices, and control
management [22] are also significant issues in EC that need to be addressed
using BC and IOTA.

##  IV  Conclusion and Future Work

In this survey, we have studied recently proposed solutions deployed in EC
based on BC and IOTA. The aim was to inspect the solutions in terms of
security, privacy, latency, accessibility, data leakage, scalability, resource
handling, throughput, reliability, control management, and energy efficiency.
It has been found that BC comes with issues such as scalability, high
transaction cost, energy inefficiency, and fee fairness when it is used in
IoT. It has also been found that IOTA has more potential to address those
problems using its DAG consensus mechanism.
Along with scalability problems, resource handling, latency, energy
efficiency, data accumulation, backup, monitoring of devices, and control
management are significant issues in EC that have been overlooked in recent
years. Energy efficiency and resource handling in EC is a hot research area
requiring intelligent solutions to use BC or IOTA. Similarly, there is no
progress being made to cope with backup, monitoring, and control management
problems in EC. These areas are also requiring considerable motivation toward
the solutions to such problems. Another important direction for future
research in EC while using distributed ledgers is to provide solutions for the
detection and avoidance of potential attacks on IoT devices.

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