On the invariance properties of Vaidya-Bonner geodesics via symmetry operators

Davood Farrokhi†, Rohollah Bakhshandeh-Chamazkoti§ and Mehdi Nadjafikhah‡

†Department of Mathematics, Karaj Branch, Islamic Azad University, karaj,
Iran.

§Department of Mathematics, Faculty of Basic Sciences, Babol Noshirvani
University of Technology, Babol, Iran.

‡School of Mathematics, Iran University of Science and Technology Narmak,
Tehran, 16846-13114, Iran

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
In the present paper, we try to investigate the Noether symmetries and Lie
point symmetries of the Vaidya-Bonner geodesics. Classification of one-
dimensional subalgebras of Lie point symmetries are considered. In fact, the
collection of pairwise non-conjugate one-dimensional subalgebras that are
called the optimal system of subalgebras is determined. Moreover, as
illustrative examples, the symmetry analysis is implemented on two special
cases of the system.

##  1 Introduction

Noether’s theorem [11, 12] provides a method for finding conservation laws of
differential equations arising from a known Lagrangian and having a known Lie
symmetry. This theorem relies on the availability of a Lagrangian and the
corresponding Noether symmetries which leave invariant the action integral,
[7, 9].

Since the geodesic equations follow from the variation of the geodesic
Lagrangian defined by the metric and due to the fact that the Noether
symmetries are a subgroup of the Lie symmetries of these equations, one should
expect a relation of the Noether symmetries of this Lagrangian with the
projective collineations of the metric or with its degenerates, [1, 2, 6, 14].
In some works, [3, 4], the relation between the Noether symmetries and the Lie
point symmetries (Killing vectors) of some special spacetimes are discussed.

In [14], Tsamparlis and Paliathanasis have computed the Lie point symmetries
and the Noether symmetries explicitly together with the corresponding linear
and quadratic first integrals for the Schwarzschild spacetime and the Friedman
Robertson Walker (FRW) spacetime. These authors, in another paper [15], have
proved a theorem that relates the Lie symmetries of the geodesic equations in
a Riemannian space with the collineations of the metric. They applied the
results to Einstein spaces and spaces of constant curvature.

In, [9, 10], the authors present a complete analysis of symmetries of classes
of wave equations that arise as a consequence of some Vaidya metrics. Now in
this paper, we try to find Lie point symmetries and Noether symmetries for the
Vaidya-Bonner metric

$\displaystyle
ds^{2}=-\left(1-\frac{M(t)}{r}+\frac{Q(t)}{r^{2}}\right)dt^{2}-2dtdr+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\,,$
(1)

where $M(t)$ is dynamical mass of the black hole and $Q(t)$ is electric
charges that both of them depends to the advanced Eddington time coordinate
$t$, [8]. As examples we consider two cases with $M(t)=1,Q(t)=t$ and
$M(t)=t,Q(t)=t^{2}$.

In general relativity, the Vaidya metric describes the non-empty external
spacetime of a spherically symmetric and nonrotating star which is either
emitting or absorbing null dust. It is named after the Indian physicist
Prahalad Chunnilal Vaidya and constitutes the simplest non-static
generalization of the non-radiative Schwarzschild solution to Einstein’s field
equation, and therefore is also called the radiating (shining) Schwarzschild
metric.

##  2 Noether symmetries

Suppose $(M,g)$ is Riemannian manifold of dimension $n$. In a local space-time
coordinate like that ${\bf x}(s)=(x^{1}(s),\ldots,x^{n}(s))$, the geodesic
equations form a system of $n$ nonlinear, second order ordinary differential
equations

$\displaystyle\ddot{x}^{i}(t)+\Gamma^{i}_{jk}\dot{x}_{j}(t)\dot{x}_{k}(t)=0,\qquad
i=1,2,\ldots,n,$ (2)

where $\Gamma^{i}_{jk}$ are the Christoffel symbols and “.” represents
derivative in terms of $s$. Consider a second order system of $m$ ordinary
differential equations (2) with following form

$\displaystyle E_{i}(s,{\bf x}(s),{\bf x}^{(1)}(s),{\bf x}^{(2)}(s))=0,\quad
i=1,\ldots,n.$ (3)

Consider a vector field which is defined on a real parameter fibre bundle over
the manifold, [6],

$\displaystyle{\bf
X}=\overline{\xi}(s,x^{\mu})\partial_{s}+\overline{\eta}^{\nu}(s,x^{\mu})\partial_{x^{\nu}},$
(4)

where $\mu,\nu=1,2,3,4$. The first prolongation of the above vector field
defined on the real parameter fibre bundle over the tangent bundle to the
manifold, is expressed as follows:

$\displaystyle{\bf X}^{[1]}={\bf
X}+\left(\overline{\eta}^{\nu}_{,s}+\overline{\eta}^{\nu}_{,\mu}\dot{x}^{\mu}-\overline{\xi}_{,s}\dot{x}^{\nu}-\overline{\xi}_{,s}\dot{x}^{\mu}\dot{x}^{\nu}\right)\partial_{\dot{x}^{\nu}},$
(5)

then ${\bf X}$ is a Noether point symmetry of the Lagrangian, [6],

$\displaystyle{\mathcal{L}}(s,x^{\mu},\dot{x}^{\mu})=g_{\mu\nu}(x^{\sigma})\dot{x}^{\mu}\dot{x}^{\nu},$
(6)

if there exists a gauge function, $A(s,x^{\mu})$, such that

$\displaystyle{\bf X}^{[1]}{\mathcal{L}}+(D_{s}\xi){\mathcal{L}}=D_{s}A,$ (7)

where

$\displaystyle D_{s}=\partial_{s}+\dot{x}^{\mu}\partial_{x^{\mu}}.$ (8)

Since the corresponding Euler-Lagrange equations (geodesic equations) are
second order ordinary differential equations, one generally takes first order
Lagrangian. Particularly, we take ${\mathcal{L}}(s,x_{i},\dot{x}_{i})$, where
“dots” denotes differentiation in terms of $s$, which results a set of second
ODEs

$\displaystyle\ddot{x}^{\mu}=g(s,x^{\mu},\dot{x}^{\mu}).$ (9)

##  3 Lie point symmetries

Consider a second order system of $m$ ordinary differential equations (3) as
geodesic equations of our given Riemannian metric $g$. We consider a one-
parameter Lie group of transformations acting on $(s,{\bf x})$-space with
following forms

$\displaystyle\bar{s}=s+\epsilon\xi(s,{\bf
x}(s))+O(\epsilon^{2}),\qquad\bar{x}^{\alpha}(s)=x^{\alpha}(s)+\epsilon\eta^{\alpha}(s,{\bf
x}(s))+O(\epsilon^{2}),$ (10)

where $\alpha=1,\ldots,n$. The infinitesimal generator ${\bf X}$ associated
with the group of transformations (10) are

$\displaystyle{\bf X}=\xi(s,{\bf x})\partial_{s}+\eta^{\alpha}(s,{\bf
x})\partial_{x^{\alpha}},$ (11)

the second order prolongation of ${\bf X}$ is given by

$\displaystyle{\bf X}^{[2]}={\bf X}+\eta^{\alpha}_{,(1)}(s,{\bf x},{\bf
x}^{(1)})\partial_{x^{\alpha}_{,(1)}}+\eta^{\alpha}_{,(2)}(s,{\bf
x},\ldots,{\bf x}^{(2)})\partial_{x^{\alpha}_{,(2)}},$ (12)

in which the prolongation coefficients are

$\displaystyle\eta^{\alpha}_{,(1)}=D\eta^{\alpha}_{,(0)}-x^{\alpha}_{,(1)}D\xi,\qquad\eta^{\alpha}_{,(2)}=D\eta^{\alpha}_{,(1)}-x^{\alpha}_{,(2)}D\xi$
(13)

where $\eta^{\alpha}_{,(0)}=\eta^{\alpha}(s,{\bf x})$ and $D$ is the total
derivative operator.

The invariance of the system (3) under the one–parameter Lie group of
transformations (10) leads to the invariance criterions. So ${\bf X}$ is a
point symmetry generator of (3) if and only if

$\displaystyle{\bf X}^{[2]}E_{i}\big{|}_{E_{i}=0}=0.$ (14)

Using (14) we find a system of partial differential equations that is called
the determining equations. By solving the determining PDEs, the symmetry
operators of the considered geodesics will be found.

##  4 Symmetry computation of Vaidya-Bonner geodesics in general form

######  Theorem 1.

The Lie algebra of Noether symmetries associated to Vaidya-Bonner metric (1)
for the arbitrary functions $M(t)$ and $Q(t)$ is generated by following
infinitesimals:

$\displaystyle\begin{array}[]{lcl}&{\bf X}_{1}=\partial_{s},\qquad{\bf
X}_{2}=\partial_{t},\qquad X_{3}=\partial_{\varphi}\qquad{\bf
X}_{4}=-\cos\varphi\;\partial_{\theta}+\sin\varphi\cot\theta\;\partial_{\varphi},\\\\[8.53581pt]
&{\bf
X}_{5}=\sin\varphi\;\partial_{\theta}+\cos\varphi\csc\theta\;\partial_{\varphi}.\end{array}$
(17)

###### Proof.

Associated Lagrangian of the (1) metric for arbitrary functions $M(u)$ and
$Q(u)$ is

$\displaystyle{\mathcal{L}}=-\left(1-\frac{M(t)}{r}+\frac{Q(t)}{r^{2}}\right)\dot{t}^{2}-2\dot{t}\dot{r}+r^{2}(\dot{\theta}^{2}+\sin^{2}\theta\,\dot{\phi}^{2}),$
(18)

where the “dot” represents derivative in terms of $s$. Suppose that the vector
field

${\bf
X}=\xi\;\partial_{s}+\eta^{1}\;\partial_{t}+\eta^{2}\;\partial_{r}+\eta^{3}\;\partial_{\theta}+\eta^{4}\;\partial_{\varphi},$

is the Noether symmetry operator of (18). Assuming $A=0$ in (7) formula, we
have

$\displaystyle{\bf X}^{[1]}{\mathcal{L}}+(D_{s}\xi){\mathcal{L}}=0,$ (19)

where

$\displaystyle
D=\partial_{s}+\dot{t}\;\partial_{t}+\dot{r}\;\partial_{r}+\dot{\theta}\;\partial_{\theta}+\dot{\varphi}\;\partial_{\varphi}.$
(20)

The equation (19) leads to some differential equations that their solutions
are the Noether symmetries (17). ∎

######  Theorem 2.

Lie algebra of point symmetries associated to Vaidya-Bonner metric (1), for
arbitrary functions $M(u)$ and $Q(u)$ is generated by:

$\displaystyle\begin{array}[]{lcl}&{\bf X}_{1}=\partial_{s},\quad{\bf
X}_{2}=\partial_{t},\quad{\bf X}_{3}=\partial_{\varphi},\quad{\bf
X}_{4}=\cot(\theta)\sin(\varphi)\partial_{\varphi}-\cos(\varphi)\partial_{\theta},\\\\[8.53581pt]
&{\bf
X}_{5}=\cot(\theta)\cos(\varphi)\partial_{\varphi}+\sin(\varphi)\partial_{\theta}.\end{array}$
(23)

###### Proof.

The Euler-Lagrange geodesic equations associated with the Lagrangian (18) are

$\displaystyle\left\\{\begin{array}[]{lcl}E_{1}:3\ddot{t}+\frac{2Q(t)-rM(t)}{4r^{3}}\dot{t}^{2}+\frac{r\sin^{2}\theta}{2}\dot{\phi}^{2}+\frac{r}{2}\ddot{\theta}^{2}=0,\\\\[5.69054pt]
E_{2}:5\ddot{r}+\frac{-2(rM^{{}^{\prime}}(t)+4Q^{{}^{\prime}}(t))(rM(t)-2Q(t))(rM(t)-r^{2}-Q(t))}{8r^{5}}\dot{t}^{2}\\\\[5.69054pt]
\hskip
19.91692pt+\frac{rM(t)-2Q(t)}{4r^{3}}\dot{t}\dot{r}+\frac{M(t)-r^{2}-Q(t)}{4r}\dot{\theta}^{2}+\frac{(M(t)r-r^{2}-Q(t))(sin^{2}\theta)}{4r}\ddot{\phi}^{2}=0,\\\\[5.69054pt]
E_{3}:3\ddot{\theta}+\frac{2}{r}\dot{r}\dot{\theta}-\sin(\theta)\cos(\theta)\dot{\phi}^{2}=0,\\\\[5.69054pt]
E_{4}:4\ddot{\phi}+\frac{1}{r}\dot{r}\dot{\phi}+2\cot(\theta)\dot{\theta}\dot{\phi}=0\\\
\end{array}\right.$ (29)

where the $M^{\prime}(t),Q^{\prime}(t)$ represents derivative in terms of $t$.
Now the second prolongation of the operator ${\bf X}$ is computed. ${\bf X}$.
Applying the invariance criteria (14) on equations (29) we have some
determining equations. By inserting (11) to invariance condition (14) and then
comparing of the powers $\dot{t},\dot{r},\dot{\theta},\dot{\phi}$ , we obtain
complete set of determining equations. By solving this system of PDEs we find
that

$\displaystyle\xi=c_{1},\qquad\eta^{1}=c_{2},\qquad\eta^{2}=0,\eta^{3}=-c_{4}\cos(\varphi)+c_{5}\sin(\varphi),$
$\displaystyle\eta^{4}=c_{3}+\cot(\theta)(c_{4}\sin(\varphi)+c_{5}\cos(\varphi)).$

∎

##  5 Algebraic structure of the operators (23)

The algebra ${\mathfrak{g}}=\langle{\bf X}_{1},{\bf X}_{2},\cdots,{\bf
X}_{5}\rangle$ is non-solvable because we have
${\mathfrak{g}^{(1)}}=[{\mathfrak{g}},{\mathfrak{g}}]=\langle{\bf X}_{2},{\bf
X}_{3},{\bf X}_{4}\rangle$ and we have the following chain of ideals
${\mathfrak{g}}\supset{\mathfrak{g}^{(1)}}\supset{\mathfrak{g}^{(2)}}\dots\neq
0$ which shows that ${\mathfrak{g}}$ is non–salvable . Also ${\mathfrak{g}}$
is not semi–simple, because its Killing form

$k=\begin{bmatrix}0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&-2&0&0\\\ 0&0&0&-2&0\\\
0&0&0&0&-2\end{bmatrix}$

is degenerate. ${\mathfrak{g}}$ has a Levi–decomposition of the form
${\mathfrak{g}}={\mathfrak{r}}\oplus_{{\mathfrak{h}}}{\mathfrak{h}}$ where
${\mathfrak{r}}=\langle{\bf X}_{1},{\bf X}_{2}\rangle$ is the radical of
${\mathfrak{g}}$ and ${\mathfrak{h}}=\langle{\bf X}_{3},{\bf X}_{4},{\bf
X}_{5}\rangle$ is a semi–simple and non–solvable subalgebra of
${\mathfrak{g}}$.

###  5.1 Classification of subalgebras

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. There is an inner
automorphism $\tau_{a}\longrightarrow\tau\tau_{a}\tau^{-1}$ of the group $G$
for every arbitrary element $\tau\in G$. Every automorphism of the group $G$
induces an automorphism of $\mathfrak{g}$. The set of all these automorphism
forms a Lie group called the adjoint group $G^{A}$. For arbitrary
infinitesimal generators ${\bf X}$ and ${\bf Y}$ in $\mathfrak{g}$, the linear
mapping ${\rm Ad}~{}{\bf X}({\bf Y}):{\bf Y}\longrightarrow[{\bf X},{\bf Y}]$
is an automorphism of $\mathfrak{g}$, called the inner derivation of
$\mathfrak{g}$. The set of all these inner derivations equipped by the Lie
bracket $[{\rm Ad}{\bf X},{\rm Ad}{\bf Y}]={\rm Ad}[{\bf X},{\bf Y}]$ is a Lie
algebra $\mathfrak{g}^{A}$ called the adjoint algebra of $\mathfrak{g}$. Two
subalgebras in $\mathfrak{g}$ are conjugate if there is a transformation of
$G^{A}$ which takes one subalgebra into the other. The collection of pairwise
non-conjugate $p$-dimensional subalgebras is called the optimal system of
subalgebras of order $p$ which is introduced by Ovsiannikov [13]. Actually
solving the optimal system problem is to determine the conjugacy inequivalent
subalgebras with the property that any other subalgebra is equivalent to a
unique member of the list under some element of the adjoint representation
i.e. $\overline{\mathfrak{h}}\,{\rm Ad(\tau)}\,\mathfrak{h}$ for some $\tau$
in a given Lie group.

The adjoint action is given by the Lie series

$\displaystyle{\rm Ad}(\exp(q\,{\bf X}_{i})){\bf X}_{j}={\bf X}_{j}-q\,[{\bf
X}_{i},{\bf X}_{j}]+\frac{q^{2}}{2}\,[{\bf X}_{i},[{\bf X}_{i},{\bf
X}_{j}]]-\cdots,$ (30)

where $q$ is a parameter and $i,j=1,\cdots,n$. Suppose

$\displaystyle{\bf X}=\sum_{i=1}^{5}a_{i}{\bf X}_{i},$ (31)

is an arbitrary member of the Lie algebra $\mathfrak{g}=\langle{\bf
X}_{1},\cdots,{\bf X}_{5}\rangle$. Note that the elements of $\mathfrak{g}$
can be represented by vectors $(a_{1},\ldots,a_{5})\in{\mathbb{R}}^{5}$ since
each of them can be written in the form (31) for some constants
$a_{1},\ldots,a_{5}$. Hence, the adjoint action can be regarded as a group of
linear transformations of the vectors $(a_{1},\ldots,a_{5})$.

######  Theorem 3.

An optimal system of one-dimension Lie subalgebras associated to the Lie point
symmetries algebra of (23) is generated by

$\displaystyle\begin{array}[]{lcl}&1)~{}{\bf X}_{1}+a_{2}{\bf X}_{2}+a_{5}{\bf
X}_{5}\qquad 2)~{}{\bf X}_{1}+a_{2}{\bf X}_{2}+a_{3}{\bf X}_{3}\qquad
3)~{}{\bf X}_{1}+a_{2}{\bf X}_{2}+a_{4}{\bf X}_{4}\\\\[8.53581pt]
&~{}~{}~{}4)~{}{\bf X}_{2}+a_{5}{\bf X}_{5}\qquad 5)~{}{\bf X}_{2}+a_{3}{\bf
X}_{3}6)~{}{\bf X}_{2}+a_{4}{\bf X}_{4}\qquad 7)~{}{\bf X}_{3}\qquad 8)~{}{\bf
X}_{4}\qquad 9)~{}{\bf X}_{5}\end{array}$ (34)

###### Proof.

The function $F_{i}^{s_{i}}:{\mathfrak{g}}\mapsto{\mathfrak{g}}$ defined by
${\bf X}\mapsto{\rm Ad}(\exp(s_{i}{\bf X}_{i}).{\bf X})$ is a linear map for
$i=1,\dots,5$. The matrices $M_{i}^{s_{i}}$ of $F_{i}^{s_{i}}$ with respect to
basis $\\{{\bf X}_{1},\cdots,{\bf X}_{5}\\}$ are

$M_{1}^{s_{1}}=\begin{bmatrix}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&1&0\\\ 0&0&0&0&1\end{bmatrix}\qquad
M_{2}^{S_{2}}=\begin{bmatrix}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&1&0\\\ 0&0&0&0&1\end{bmatrix}\qquad
M_{3}^{s_{3}}=\begin{bmatrix}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&\cos(s_{3})&-\sin(s_{3})\\\ 0&0&0&\sin(s_{3})&\cos(s_{3})\end{bmatrix}$
$M_{4}^{s_{4}}=\begin{bmatrix}1&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&\cos(s_{4})&0&\sin(s_{4})\\\ 0&0&0&1&0\\\
0&0&-\sin(s_{4})&0&\cos(s_{4})\end{bmatrix}\qquad
M_{5}^{s_{5}}=\begin{bmatrix}1&0&0&0&0\\\ 0&1&0&0&0\\\
0&0&\cos(s_{5})&-\sin(s_{5})&0\\\ 0&0&\sin(s_{5})&\cos(s_{5})&0\\\
0&0&0&0&1\end{bmatrix},$

therefore

$\displaystyle F_{5}^{s_{5}}\circ F_{4}^{s_{4}}\circ F_{3}^{s_{3}}\circ
F_{2}^{s_{2}}\circ F_{1}^{s_{1}}:{\bf X}\mapsto a_{1}{\bf X}_{1}+a_{2}{\bf
X}_{2}+[(\cos(s_{4})\cos(s_{5}))a_{3}+$
$\displaystyle(\cos(s_{5})\sin(s_{3})\sin(s_{4})+\cos(s_{3})\sin(s_{5}))a_{4}+(\sin(s_{3})\sin(s_{5})-\cos(s_{3})\sin(s_{4})\cos(s_{5}))a_{5}]{\bf
X}_{3}+$
$\displaystyle[(-\sin(s_{5})\cos(s_{4}))a_{3}+(\cos(s_{3})\cos(s_{5})-\sin(s_{3})\sin(s_{4})\sin(s_{5}))a_{4}+(\cos(s_{3})\sin(s_{4})\sin(s_{5})$
$\displaystyle+\sin(s_{3})\cos(s_{5}))a_{5}]{\bf
X}_{4}+[(-\sin(s_{3})\cos(s_{4}))a_{4}+(\cos(s_{3})\cos(s_{4}))a_{5}]{\bf
X}_{5}.$

The one-dimensional optimal system of subalgebras associated to (23) is
determined as follows. The operator ${\bf X}$ is simplified with the following
cases:

  * •

If $a_{1}\neq 0$ we have three cases:

1) The coefficients ${\bf X}_{3},{\bf X}_{4}$ could be vanished by
$F_{3}^{s_{3}},F_{4}^{s_{4}}$ by setting $s_{3}=\arctan(\frac{a_{4}}{a_{5}})$
and $s_{4}=-\arctan(\frac{a_{3}}{a_{5}})$ respectively. By scaling ${\bf X}$
we can assume that $a_{1}=1$ so ${\bf X}$ is reduced to case (1).

2) The coefficients ${\bf X}_{4},{\bf X}_{5}$ could be vanished by
$F_{3}^{s_{3}},F_{4}^{s_{4}}$ by setting
$s_{3}=\arctan(\frac{a_{4}}{a_{5}}),s_{4}=\arctan(\frac{a_{5}}{a_{3}})$
respectively. By scaling ${\bf X}$ we can assume that $a_{1}=1$ so ${\bf X}$
is reduced to case (2).

3) The coefficients ${\bf X}_{3},{\bf X}_{5}$ could be vanished by
$F_{3}^{s_{3}},F_{5}^{s_{5}}$, by setting
$s_{3}=-\arctan(\frac{a_{5}}{a_{4}}),s_{5}=\arctan(\frac{a_{3}}{a_{4}})$
respectively. By scaling ${\bf X}$ we assume that $a_{1}=1$ so ${\bf X}$ is
reduced to case (3).

  * •

Now If $a_{1}=0,a_{2}\neq 0$,we have three cases:

1) The coefficients ${\bf X}_{3},{\bf X}_{4}$ would be vanished by
$F_{3}^{s_{3}},F_{4}^{s_{4}}$ by setting
$s_{3}=\arctan(\frac{a_{4}}{a_{5}}),s_{4}=\arctan(\frac{a_{3}}{a_{5}})$
respectively. By scaling ${\bf X}$, we can put $a_{2}=1$ and then ${\bf X}$ is
reduced to case (4).

2) The coefficients ${\bf X}_{4},{\bf X}_{5}$ could be vanished by
$F_{3}^{s_{3}},F_{4}^{s_{4}}$,by setting
$s_{3}=\arctan(\frac{a_{4}}{a_{5}}),s_{4}=\arctan(\frac{a_{5}}{a_{3}})$
respectively. By scaling ${\bf X}$ we assume that $a_{2}=1$ so ${\bf X}$ is
reduced to case (5)

3) The coefficients ${\bf X}_{3},{\bf X}_{5}$ could be vanished by
$F_{3}^{s_{3}},F_{5}^{s_{5}}$, by setting
$s_{3}=-\arctan(\frac{a_{5}}{a_{4}}),s_{5}=\arctan(\frac{a_{3}}{a_{4}})$
respectively. By scaling ${\bf X}$, we can put $a_{2}=1$ and then ${\bf X}$ is
reduced to case (6).

  * •

If $a_{1}=0,a_{2}=0,a_{3}\neq 0$, the coefficients ${\bf X}_{4},{\bf X}_{5}$
could be vanished by $F_{3}^{s_{3}},F_{4}^{s_{4}}$, by setting
$s_{3}=\arctan(\frac{a_{4}}{a_{5}}),s_{4}=\arctan(\frac{a_{5}}{a_{3}})$
respectively.By scaling ${\bf X}$ we can put $a_{3}=1$ so ${\bf X}$ is reduced
to case (7).

  * •

If $a_{1}=0,a_{2}=0,a_{3}=0,a_{4}\neq 0$, the coefficient ${\bf X}_{5}$ could
be vanished by $F_{3}^{s_{3}}$,by setting
$s_{3}=-\arctan(\frac{a_{5}}{a_{4}})$. By scaling ${\bf X}$, we can put
$a_{4}=1$ and then ${\bf X}$ is reduced to case (8).

  * •

If $a_{1}=0,a_{2}=0,a_{3}=0,a_{4}=0,a_{5}\neq 0$, by scaling ${\bf X}$ we can
put $a_{5}=1$ so it reduces to case (9).

∎

##  6 Some illustrative examples

###  6.1 Symmetry analysis of the metric (1) for $M(t)=1$ and $Q(t)=t$

In this case the associated Lagrangian of the metric (1) is

$\mathcal{L}=-(1-\frac{1}{r}+\frac{t}{r^{2}})\dot{t}^{2}-2\dot{t}\dot{r}+r^{2}(\dot{\theta}^{2}+\sin^{2}(\theta)\dot{\phi}^{2}).$
(35)

Suppose that the vector field ${\bf
X}=\xi\;\partial_{s}+\eta^{1}\;\partial_{t}+\eta^{2}\;\partial_{r}+\eta^{3}\;\partial_{\theta}+\eta^{4}\partial_{\varphi}$
is the Noether symmetry operator of Lagrangian (35). By solving the equation
${\bf X}^{[1]}{\mathcal{L}}+(D_{s}\xi){\mathcal{L}}=0$ we can find the
following Noether symmetries:

$\displaystyle{\bf X}_{1}=\partial_{s},\quad{\bf
X}_{2}=\partial_{\varphi},\quad{\bf
X}_{3}=\cot(\theta)\sin(\varphi)\partial_{\varphi}-\cos(\varphi)\partial_{\theta},\quad{\bf
X}_{4}=\cot(\theta)\cos(\varphi)\partial_{\varphi}+\sin(\varphi)\partial_{\theta}.$
(36)

The Euler-Lagrange geodesic equations associated with the Lagrangian (35) are

$\displaystyle\left\\{\begin{array}[]{lcl}E_{1}:5\ddot{t}+\frac{2t-r}{4r^{3}}\ddot{t}^{2}+\frac{r}{2}\dot{\theta}^{2}+\frac{r\sin^{2}\theta}{2}\dot{\phi}^{2}=0\\\\[5.69054pt]
E_{2}:3\ddot{r}+\frac{2r^{3}+(r^{2}+t-r)(r-2t)}{8r^{5}}\dot{t}^{2}+\frac{r-2t}{2r^{3}}\dot{t}\dot{r}+\frac{r-r^{2}-t}{4r}(\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2})=0\\\\[5.69054pt]
E_{3}:3\ddot{\theta}+\frac{2}{r}\dot{r}\dot{\theta}-\sin(\theta)\cos(\theta)\dot{\phi}^{2}=0\\\\[5.69054pt]
E_{4}:4\ddot{\phi}+\frac{2}{r}\dot{r}\dot{\phi}+2\cot(\theta)\dot{\theta}\dot{\phi}=0.\end{array}\right.$
(41)

Now suppose ${\bf X}$ is a Lie point symmetry generator of (35) thus ${\bf
X}^{[2]}E_{i}\big{|}_{E_{i}=0}=0$ for $i=1,2,3,4$. By solving this equation we
obtain

$\displaystyle\xi=c_{1},\;\eta^{1}=0,\;\eta^{2}=0\quad\eta^{3}=-c_{3}\cos(\varphi)+c_{4}\sin(\varphi),\;\eta^{4}=c_{2}+\cot(\theta)(c_{3}\sin(\varphi)+c_{4}\cos(\varphi)).$
(42)

Therefore Lie symmetry generators associated to the metric (1) for $M(t)=1$
and $Q(t)=t$ are:

$\displaystyle{\bf X}_{1}=\partial_{s}\quad{\bf
X}_{2}=\partial_{\varphi}\quad{\bf
X}_{3}=\cot(\theta)\sin(\varphi)\partial_{\varphi}-\cos(\varphi)\partial_{\theta}\quad{\bf
X}_{4}=\cot(\theta)\cos(\varphi)\partial_{\varphi}+\sin(\phi)\partial_{\theta},$
(43)

and commutator table of these symmetry generators is given in following table:

Table 1: commutator table (43) [ , ] | ${\bf X}_{1}$ | ${\bf X}_{2}$ | ${\bf X}_{3}$ | ${\bf X}_{4}$
---|---|---|---|---
${\bf X}_{1}$ | 0 | 0 | 0 | 0
${\bf X}_{2}$ | 0 | 0 | ${\bf X}_{4}$ | -${\bf X}_{3}$
${\bf X}_{3}$ | 0 | -${\bf X}_{4}$ | 0 | ${\bf X}_{2}$
${\bf X}_{4}$ | 0 | ${\bf X}_{3}$ | $-{\bf X}_{2}$ | 0

The algebra ${\mathfrak{g}}=\langle{\bf X}_{1},{\bf X}_{2},\cdots,{\bf
X}_{4}\rangle$ is non-solvable because
${\mathfrak{g}}\supset{\mathfrak{g}^{(1)}}\neq 0$ where
${\mathfrak{g}^{(1)}}=[{\mathfrak{g}},{\mathfrak{g}}]=\langle{\bf X}_{2},{\bf
X}_{3},{\bf X}_{4}\rangle$. Also ${\mathfrak{g}}$ is not semi-simple, because
its Killing form

$k=\begin{bmatrix}0&0&0&0\\\ 0&-2&0&0\\\ 0&0&-2&0\\\ 0&0&0&-2\end{bmatrix}$

is degenerate. ${\mathfrak{g}}$ has a Levi-decomposition of the form
${\mathfrak{g}}={\mathfrak{r}}\oplus_{{\mathfrak{h}}}{\mathfrak{h}}$ where
${\mathfrak{r}}=\langle{\bf X}_{1}\rangle$ is the radical of ${\mathfrak{g}}$
and ${\mathfrak{h}}=\langle{\bf X}_{2},{\bf X}_{3},{\bf X}_{4}\rangle$ is a
semi-simple and non-solvable subalgebra of ${\mathfrak{g}}$.

###  6.2 Symmetry analysis of the metric (1) for $M(t)=t,Q(t)=t^{2}$

The Vaidya metric in special case, $M(t)=t$, is known as the Papapetrou model
[5]. Therefore in this case we consider the Vaidya-Bonner metric (1) for
$M(t)=t,Q(t)=t^{2}$.

The Lie algebra of Noether symmetries associated to metric (1) for
$M(t)=t,Q(t)=t^{2}$ is a 5-dimensional algebra spanned by the following vector
fields:

$\displaystyle{\bf
X}_{1}=s\;\partial_{s}+\frac{t}{2}\;\partial_{t}+\frac{r}{2}\;\partial_{r}\quad{\bf
X}_{2}=\partial_{s}\quad{\bf X}_{3}=\partial_{\varphi}\qquad{\bf
X}_{4}=\cot(\theta)\sin(\varphi)\partial_{\varphi}-\cos(\varphi)\partial_{\theta}$
$\displaystyle{\bf
X}_{5}=\cot(\theta)\cos(\varphi)\partial_{\varphi}+\sin(\varphi)\partial_{\theta}.$

The Euler-Lagrange geodesic equations are

$\displaystyle\left\\{\begin{array}[]{lcl}E_{1}:3\ddot{t}+(\frac{2t^{2}-rt}{4r^{3}})\dot{t}^{2}+\frac{r}{2}\dot{\theta}^{2}+\frac{r\sin^{2}(\theta)}{2}\dot{\phi}^{2}=0\\\\[5.69054pt]
E_{2}:5\ddot{r}+\frac{2r^{3}(2t-r)+(r^{2}-rt+t^{2})(rt-2t^{2})}{8r^{5}}\dot{t}^{2}+\frac{tr-2t^{2}}{2r^{3}}\dot{r}\dot{t}+\frac{r^{2}-rt+t^{2}}{4r}(\sin^{2}(\theta)\dot{\phi}^{2}-\dot{\theta}^{2})=0\\\\[5.69054pt]
E_{3}:3\ddot{\theta}+\frac{1}{r}\dot{r}\dot{\theta}-\sin(\theta)\cos(\theta)\dot{\phi}^{2}=0\\\\[5.69054pt]
E_{4}:4\ddot{\phi}+\frac{2}{r}\dot{r}\dot{\phi}+2\cot(\theta)\dot{\theta}\dot{\phi}=0,\end{array}\right.$
(48)

by applying the symmetry criteria we have

$\displaystyle\xi=c_{1}s+c_{2}\quad\eta^{1}=c_{1}t,\quad\eta^{2}=c_{1}r,\quad\eta^{3}=-c_{4}\cos(\varphi)+c_{5}\sin(\varphi)$
$\displaystyle\eta^{4}=c_{3}+\cot(\theta)(c_{4}\sin(\varphi)+c_{5}\cos(\varphi)).$

Thus the Lie point symmetries are as follows:

$\displaystyle{\bf
X}_{1}=s\;\partial_{s}+t\;\partial_{u}+r\;\partial_{r}\quad{\bf
X}_{2}=\partial_{s},\quad{\bf X}_{3}=\partial_{\varphi}\qquad{\bf
X}_{4}=\cot(\theta)\sin(\varphi)\partial_{\varphi}-\cos(\varphi)\partial_{\theta}$
$\displaystyle{\bf
X}_{5}=\cot(\theta)\cos(\varphi)\partial_{\varphi}+\sin(\varphi)\partial_{\theta}.$
(49)

The commutator table of symmetry generators is given in following table:

Table 2: commutator table (49) [ , ] | ${\bf X}_{1}$ | ${\bf X}_{2}$ | ${\bf X}_{3}$ | ${\bf X}_{4}$ | ${\bf X}_{5}$
---|---|---|---|---|---
${\bf X}_{1}$ | 0 | $-{\bf X}_{2}$ | 0 | 0 | 0
${\bf X}_{2}$ | ${\bf X}_{2}$ | 0 | 0 | 0 | 0
${\bf X}_{3}$ | 0 | 0 | 0 | ${\bf X}_{5}$ | $-{\bf X}_{4}$
${\bf X}_{4}$ | 0 | 0 | $-{\bf X}_{5}$ | 0 | ${\bf X}_{3}$
${\bf X}_{5}$ | 0 | 0 | ${\bf X}_{4}$ | $-{\bf X}_{3}$ | 0

${\mathfrak{g}}$ is non-solvable because we have
${\mathfrak{g}^{(1)}}=[{\mathfrak{g}},{\mathfrak{g}}]=$ $\langle{\bf
X}_{2},{\bf X}_{3},{\bf X}_{4},{\bf X}_{5}\rangle$ ,
${\mathfrak{g}^{(2)}}=[{\mathfrak{g}^{(1)}},{\mathfrak{g}^{(1)}}]=$
$\langle{\bf X}_{3},{\bf X}_{4},{\bf X}_{5}\rangle$,
${\mathfrak{g}^{(3)}}=[{\mathfrak{g}^{(2)}},{\mathfrak{g}^{(2)}}]=$
$\langle{\bf X}_{3},{\bf X}_{4},{\bf X}_{5}\rangle$, thus we have the
following chain of ideals
${\mathfrak{g}}\supset{\mathfrak{g}^{(1)}}\supset{\mathfrak{g}^{(2)}}\dots\neq
0$ which shows that ${\mathfrak{g}}$ is non-salvable . Also ${\mathfrak{g}}$
is not semi-simple, because its Killing form

$k=\begin{bmatrix}1&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&-2&0&0\\\ 0&0&0&-2&0\\\
0&0&0&0&-2\end{bmatrix}$

is degenerate. ${\mathfrak{g}}$ has a Levi decomposition of the form
${\mathfrak{g}}={\mathfrak{r}}\oplus_{{\mathfrak{h}}}{\mathfrak{h}}$ where
${\mathfrak{r}}=\langle{\bf X}_{1},{\bf X}_{2}\rangle$ is the radical of
${\mathfrak{g}}$ and ${\mathfrak{h}}=\langle{\bf X}_{3},{\bf X}_{4},{\bf
X}_{5}\rangle$ is a semi-simple and non-solvable subalgebra of
${\mathfrak{g}}$.

## Acknowledgement

The second author (Rohollah Bakhshandeh-Chamazkoti) acknowledges the funding
support of Babol Noshirvani University of Technology under Grant No.
BNUT/391024/99.

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