Birkhoff’s Theorem and Lie Symmetry Analysis

A<EMAIL_ADDRESS>
Relativity and Cosmology Research Centre, Department of Physics
Jadavpur University, India.

and

Subham B<EMAIL_ADDRESS>
Department of Physics
University at Albany, State University of New York

Formerly at Relativity and Cosmology Research Centre, Department of Physics,
Jadavpur University, India

Abstract

Three dimensional space is said to be spherically symmetric if it admits SO(3)
as the group of isometries. Under this symmetry condition, the Einstein’s
Field equations for vacuum, yields the Schwarzschild Metric as the unique
solution, which essentially is the statement of the well known Birkhoff’s
Theorem. Geometrically speaking this theorem claims that the pseudo -
Riemanian space-times provide more isometries than expected from the original
metric holonomy/ansatz. In this paper we use the method of Lie Symmetry
Analysis to analyze the Einstein’s Vacuum Field Equations so as to obtain the
Symmetry Generators of the corresponding Differential Equation. Additionally,
applying the Noether Point Symmetry method we have obtained the conserved
quantities corresponding to the generators of the Schwarzschild Lagrangian and
paving way to reformulate the Birkhoff’s Theorem from a different approach.

Keywords : Birkhoff’s theorem, Lie Symmetry Analysis, Noether Point Symmetry.

##  1 INTRODUCTION

Einstein’s Field equations are the most fundamental equation in the realm of
General Relativity. This equation can be loosely summarized as the link up
between the matter content and the geometry of space-time. In a more
qualitative manner, the field equations explains how the metric (of the space-
time involved) respond to energy and momentum. Due to the non-linearity and
indeterminacy of the equations it is very hard to obtain the solution of the
field equations. All the solutions that are available in literature are
carried out under simplifying assumptions. The most well known solution of
Einstein’s equations is obtained under the assumption of spherical symmetry,
is known as the Schwarzschild solution.

Lie Symmetry analysis of differential equations provides us a rudimentary yet
very powerful machinery to derive the conservation laws of corresponding
system represented by the (ordinary or partial) differential equations. The
invariance of the differential equations under transformation of both
dependent and independent variables involved essentially leads to the idea of
their symmetry analysis. This transformation forms a local group of point
transformation establishing diffeomorphism on the space of independent and
dependent variables, mapping the solutions of the specific differential
equation to the other solutions.

In this paper we use the method of Lie Symmetry Analysis on Einstein’s field
equations. There after we use the Noether’s theorem, which reveals the inner
relation between the involved symmetries and the conserved quantities of
dynamical system to reiterate the well known Birkhoff’s theorem.

The paper is organized in the following manner. In Section 2, we quickly
recapitulate the Birkhoff’s theorem and discuss about the Lie Algebra of the
Killing vectors in a spherically symmetric space-time. Section 3, sheds some
light upon the recent literatures that deals with Birkhoff’s theorem and it’s
modern treatments to different dimensions. In the section followed by we
introduce the most important tool implemented for our work that is the Lie
groups of transformations and the Prolongation Theory of differential
equations. In simple language, the term prolongation means becoming longer.
Here the involved system certainly does not become longer but the space of the
dependent variables does. Basically our requirement is of a differential
equation not only representing the dependent and independent variables but
also the appearing partial derivatives. So, we prolong the space of dependent
variables using their partial derivatives. In section 5, we take the
Einstein’s vacuum field equations and obtain the maximal symmetry generator,
using the method of Lie Symmetry analysis. Entire section 6 comprises of
Noether’s theorem and it’s modification for the first order prolongations.
This theorem has been implemented for the case of Schwarzschild lagrangian.
This section concludes with recovering Birkhoff’s theorem from the
infinitesimal symmetry generators obtained via the analysis of Schwarzschild
Lagrangian.

##  2 BIRKHOFF’S THEOREM

A metric describing the space-time is obtained when Einstein’s field equations
are solved under simplifying assumptions. The Birkhoff’s theorem is a
fundamental theorem regarding the solution of the field equations. It states
that, any spherically symmetric solution of Einstein’s vacuum field equations
must be static and asymptotically flat. This statement leads to the fact that
Schwarzschild’s exterior (i.e. space-time outside of a spherically non-
rotating gravitational object) solution is the most general spherically
symmetric solution of Einstein’s vacuum field equations with zero cosmological
constant. The solution for non-zero cosmological constant is provided by the
Schwarzschild-deSitter metric. The Einstein’s field equations are given by,

$R_{ik}-\frac{1}{2}g_{ik}R=T_{ik}$

Now for of vacuum, the energy-momentum tensor vanishes i.e. $T_{ik}=0$

$R_{ik}-\frac{1}{2}g_{ik}R=0$

where $g_{ik}$ is the metric of the space-time involved and $R_{ik}$ and $R$
being the Ricci tensor and Ricci scalar respectively. The solution of this
equation for a space-time outside a spherical source or gravitating body is
given as,

$ds^{2}=\bigg{(}1-\frac{2GM}{c^{2}r}\bigg{)}dt^{2}-\bigg{(}1-\frac{2GM}{c^{2}r}\bigg{)}^{-1}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$

Where $(r,\theta,\phi)$ are the spherical coordinates used and $M$ is the mass
of the (spherical) gravitating object. Redefining $m=\frac{GM}{c^{2}}$ we get,

$ds^{2}=\bigg{(}1-\frac{2m}{r}\bigg{)}dt^{2}-\bigg{(}1-\frac{2m}{r}\bigg{)}^{-1}dr^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$

The theorem also conveys that the solutions of Einstein’s vacuum field
equations possess hypersurface orthogonal Killing vectors i.e. the metric
obtained as solution of field equations is static333The description of
Birkhoff theorem with Einstein’s vacuum solutions having static Schwarzschild
metric as solution can be a bit of misleading as well since in the region
$0<r<2M$, the time coordinate t fails to remain time like, see Hawking and
Ellis Appendix.. Additionally the theorem also tells us the metric is
independent of the changes in the matter distribution, which are the sources
of the gravitational field, provided the spherical symmetry is preserved.

The presence of SO(3) group as a group of isometries renders the metric to be
spherically symmetric. The existence of Killing vectors corresponding to
respective symmetries characterizes the symmetries of space-time or metric.
The existing Killing vectors of any given space-time are coordinate
independent quantities. None the less by labelling the space-time in specific
coordinates, we can formulate the Killing vectors in the chosen coordinate.
For example, consider a $S^{2}$ (having SO(3)) with the Killing vectors
labelled as X,Y,Z which in polar coordinates are given as follows,

$X=\partial_{\phi}$
$Y=(\cos\phi)\partial_{\theta}-(\cot\theta\sin\phi)\partial_{\phi}$
$Z=-(\sin\phi)\partial_{\theta}-(\cot\theta\cos\phi)\partial_{\phi}$

A quick evaluation of the closed commutators of these Killing vectors,

$[X,Y]=Z$ $[Y,Z]=X$ $[Z,X]=Y$

gives us the Lie algebra of SO(3) group.

Now it is observed that along with these three Killing vectors the
Schwarzschild solution does allow one extra time-like Killing vector; 3 for
spherical symmetry and 1 for time translation (only in the limit $r>2M$,
beyond the Schwarzschild radius the Killing vector remain hypersurface -
orthogonal i.e beyond the Schwarzschild radius the time-like coordinte becomes
space-like) [31]. Corresponding to every Killing vector, there will be a
constant of motion. For a free particle moving along a geodesic, having of
equation of motion

$\ddot{x}^{a}+\Gamma^{a}_{bc}\dot{x}^{b}\dot{x}^{c}=0$

if $K^{\nu}$ is a Killing vector then $K^{\nu}\frac{dx^{\nu}}{d\lambda}$
remains conserved. The explicit expressions for the four conserved quantities
associated with the Killing vectors can easily be written. Summarizing we can
conclude that any spherically symmetric solution of the vacuum field equations
allow us a fourth/extra Killing vector (we started with SO(3)).

However, this is one of the many aspects of Birkhoff theorem. There are many
different approaches available to this theorem. Such as, physically, it
implies that a spherically symmetric star undergoing strict radial pulsations
cannot propagate any disturbances into the surrounding space, indicating that
a radially oscillating star has a static gravitational field [29][30]. This
paper is based on the differential geometric aspect of Birkhoff’s theorem
which is stated as, every member of the family of the pseudo-Riemannian space-
time has more isometries than expected from the original metric
ansatz/holonomy (we have described briefly about this following the
pedagogical technique of using the Killing vectors).

##  3 GENERAL FORMULATIONS OF BIRKHOFF’S THEOREM

The Birkhoff’s theorem concerning Einstein’s General theory of Relativity was
first proposed and discussed by G.D.Birkhoff (1923) in his paper cited as
follows, The field outside the spherical distribution of matter is static
whether or not the matter is in a static or in a variable state…thus the
Schwarzschild solution is essentially the most general solution of the field
equations with spherical symmetry[27].

Over the years Birkhoff’s theorem has been addressed in many different ways.
Upon stating the fact that the theorem relies on the existence of a
3-parameter group of global isometries with 2-dimensional non-null orbits and
of an additional Killing vector associated with a $G_{4}$ group of motion, H.
Goenner [8] had put forward a generalized and geometric version of the
Birkhoff’s theorem. H.J.Schmidt on the other hand provided a complete
covariant proof of Birkhoff’s theorem by showing that the origin of Birkhoff’s
theorem rests on the property that compared to all other dimensions k, it
holds for k=2. Other different approaches to this theorem are very lucidly
described in his work [10][7].

[32][33] These two papers describe the 5-dimensional case related to
Birkhoff’s theorem. Other generalizations such as, extending this theorem to
fourth order gravity can be found in the works of P. Havas [34]. The relation
of Birkhoff’s theorem with 2-dimensional space-time can be found in [35][36].

The most evolved phrasing of this theorem in connection with the conformally
reducible metrics can be found in the works of Bona [37]. The author’s work
focuses on the different kinds of space-time to which the theorem is
applicable. It is based on the fact that these space-times are conformal to
the direct product of two 2-dimensional manifold. Other generalization of this
theorem for higher dimension was done by K.A. Bronnikov and V.N.Melnikov [38]
where they have discussed about the validity conditions for the extended
Birkhoff’s theorem in multidimensional gravity with no resctrictions on space-
time dimensionality. An elucidating discussion on the relation between
manifold dimensionality and the existance of Birkhoff like theorems has been
done by H.J.Schmidt [10][7]. G.F.R. Ellis and R. Goswami have investigated the
possibility of extending the Birkhoff’s theorem by analysing whether the
theorem holds approximately for an approximate spherical vacuum solution and
also for an almost vacuum like configuration [39] [24].

##  4 LIE GROUPS OF TRANSFORMATIONS

The method of change of variables involed while analysing differential
equations is a go-to tool used by physicists and mathematicians alike. Let us
illustrate this method by a 2 dimensional case. For example let $x$ and $y$ be
the set of variables for a given space. After transformation of the variables
we obtain,

$\hskip 28.45274ptx\rightarrow x^{\prime}=x^{\prime}(x,y)\hskip 28.45274pt$
$\hskip 28.45274pty\rightarrow y^{\prime}=y^{\prime}(x,y)\hskip 28.45274pt$

where $x^{\prime}$ and $y^{\prime}$ are new set of variables involved. This is
an example of a Point Transformation which maps points $(x,y)$ into
$(x^{\prime},y^{\prime})$.

As we are more delved into symmetry properties therefore we will be much more
interested in the transformation that also involves parameters.

###  4.1 Groups of Transformation

Let us consider a domain $D\subset\mathbb{R}^{N}$ (coordinate space) and
another space $S\subset\mathbb{R}^{1}$ ($S$ be our parameter space). Now let
us choose G to be the set of transformations (map) defined by,

$G:D\times S\rightarrow D$

Now within this set, choose a particular transformation $Z\in G$ defined by,

$\hskip 56.9055ptZ:x\times a\rightarrow x^{\prime}=Z(x;a)\hskip 71.13188pt(1)$

where $x$ and $x^{\prime}$ are old and new set of coordinates respectively
while $a$ being the parameter of transformation involved.

(a) For each value of parameter $a\hskip 2.84526pt\in\hskip 2.84526ptS$, the
transformations are bijective.

(b) S with the law of composition $\mu$ is a group with identity e.

(c) $Z(x\hskip 1.42262pt;\hskip 1.42262pte)={x}\hskip 5.69054pt;\hskip
5.69054pt\forall\hskip 2.84526ptx\hskip 2.84526pt\in\hskip 2.84526ptD.$

(d) $Z\big{(}Z(x\hskip 1.42262pt;\hskip 1.42262pta)\hskip 1.42262pt;\hskip
1.42262ptb\big{)}=Z\big{(}x;\hskip 1.42262pt\mu(a\hskip 1.42262pt,\hskip
1.42262ptb)\big{)}\hskip 2.84526pt;\hskip 2.84526pt\forall\hskip
2.84526ptx\hskip 2.84526pt\in\hskip 2.84526ptD\hskip 2.84526pt$ and
$\forall\hskip 2.84526pta,b\hskip 2.84526pt\in\hskip 2.84526ptS.$

###  4.2 Lie Groups of Transformation

Furthermore satisfying the axioms that are mentioned above if the following
properties,

(a) involved parameter $a$ is continuous.

(b) Z is $C^{\infty}$ ; $\forall\hskip 2.84526ptx\in D$ and an analytic
function of $a\in S$.

are also satisfied then these axioms together with the ones stated in section
4.1 elevates the group of transformation into the class of Lie groups of
transformations.

Expanding equation (1) in Taylor series about a=0,

$\hskip 14.22636ptx^{\prime}=x+a\frac{\partial Z(x\hskip 1.42262pt;\hskip
1.42262pta)}{\partial a}\bigg{|}_{a=0}+O(a^{2})\hskip 28.45274pt(2)$

and defining444often referred as first fundamental theorem of Lie.,

$\hskip 28.45274pt\xi(x)=\frac{\partial Z(x\hskip 1.42262pt;\hskip
1.42262pta)}{\partial a}\bigg{|}_{a=0}\hskip 28.45274pt$

we found,

$\hskip 28.45274ptx^{\prime}=x+a\xi(x)\hskip 14.22636pt(3)$

where $\xi(x)$ is often termed as auxilliary function of the transformation
involved.

To shed some more light on the physical aspect of the theory we use the
following one parameter transformation group.

$\hskip 28.45274ptx\rightarrow x^{\prime}=f(x\hskip 1.42262pt,\hskip
1.42262pt\mu\hskip 1.42262pt,\hskip 1.42262pta)\hskip 28.45274pt(4.1)$ $\hskip
28.45274pt\mu\rightarrow\mu^{\prime}=\phi(x\hskip 1.42262pt,\hskip
1.42262pt\mu\hskip 1.42262pt,\hskip 1.42262pta)\hskip 28.45274pt(4.2)$

where $f$ and $\phi$ being transformation maps.

With our definition,

$\hskip 28.45274ptx^{\prime}|_{a=0}=f(x\hskip 1.42262pt,\hskip
1.42262pt\mu\hskip 1.42262pt,\hskip 1.42262pta)=x\hskip 28.45274pt(5.1)$
$\hskip 28.45274pt\mu^{\prime}|_{a=0}=\phi(x\hskip 1.42262pt,\hskip
1.42262pt\mu\hskip 1.42262pt,\hskip 1.42262pta)=\mu\hskip 28.45274pt(5.2)$

These properties ensures the above being a one parameter group of point
transformation[4]. A simple example of a one parameter group is given by the
rotations

$x^{\prime}=x(\cos a)-\mu(\sin a)$ $\mu^{\prime}=x(\sin a)+\mu(\cos a)$

where a is the involved parameter. On contrary, the reflection

$x^{\prime}=-x$ $\mu^{\prime}=-\mu$

is an useful point transformation which does not constitute a one parameter
group.

The one parameter group and its action are best pictorially observed as motion
in the $x-\mu$ plane.

Consider an arbitrary starting point $(x_{o},\mu_{o})$ in that plane (with the
involved parameter $a$ being zero). Varying this parameter shifts the starting
point along some curve. Again repeating this procedure for different values of
$(x_{o},\mu_{o})$ we can obtain a bunch of curves where under the action of a
group, each curve can be transformed into one another and are collectively
referred to as orbits of the group [6].

###  4.3 Invariants

Let us assume, an $C^{\infty}$ fucntion $F(x)$, which is said to be an
Invariant of the Lie group of Transformation if and only if for any group of
transformation the condition,

$\hskip 28.45274ptF(x^{\prime})=F(x)\hskip 28.45274pt$

(where $x$ and $x^{\prime}$ are old and new set of coordinates respectively)
holds.

Infinitesimal generator of the group can be easily exploited to characterize
the invariance of a function as is illustrated by the following theorem.

F(x) is invariant under a coordinate transformation, $x^{\prime}=Z(x\hskip
1.42262pt;\hskip 1.42262pta)$, if and only if,

$\hskip 28.45274ptXF(x)=0\hskip 28.45274pt$

where X is called the infinitesimal generator of the transformation. It is
given as,

$\hskip 28.45274ptX=\xi_{1}(x)\frac{\partial}{\partial
x_{1}}+\xi_{2}(x)\frac{\partial}{\partial
x_{2}}+.....+\xi_{N}(x)\frac{\partial}{\partial x_{N}}\hskip 14.22636pt$

where $\xi_{1}$, $\xi_{2}$,…..,$\xi_{N}$ are set of generalized auxilliary
functions for the N dimensional case.

###  4.4 Lie Groups of differential equations : Prolongation Theory

Consider Lie groups of transformations associated to any given type of
differential equation (say F) involving n independent variables
$(x^{1},x^{2},.....,x^{n})$ $\in$ $\mathbb{R}^{n}$ and m dependent variables
$(u^{1},u^{2},.....,u^{m})$ $\in$ $\mathbb{R}^{m}$ and $\mathbb{R}^{m+n}$ be
the space of all variables $(x,u)$.

Let us consider the transformations,

$\hskip 28.45274ptx^{\prime i}=Z^{i}(x,u,a)\hskip 28.45274pt(6.1)$ $\hskip
28.45274ptu^{\prime k}=U^{k}(x,u,a)\hskip 28.45274pt(6.2)$

This transformation acts on the space $\mathbb{R}^{n+m}$ of variables ($x,u$).

Let $p^{k}_{i}$ denote the derivative of dependent variable $u$ with respect
to the independent variable $x$,

$p^{k}_{i}=\frac{\partial u^{k}}{\partial x^{i}}\hskip
5.69054pt;(i=1,2,.....,n\hskip 1.42262pt;\hskip 1.42262ptk=1,2,.....,m)$

The transformation of derivatives with the formula (8) leads us to an
extension of the Lie group of transformation which are called Prolongations.
This extended group acts on the space with variables ($x^{i}$, $u^{k}$,
$p^{k}_{i}$) rather than ($x^{i}$, $u^{k}$), so the $k^{th}$ prolonged space
will have ($x^{i}$ , $u^{k}$, $p^{k}_{i}$, ….., $p^{k}_{j}$) as variables,
often called the jet space.

####  4.4.1 First Prolongation:

The infinitesimal generator for the extended group $X^{(1)}$, after first
prolongation is given by[2][4][6],

$\hskip
28.45274ptX^{(1)}=X+\zeta^{k}_{i}\frac{\partial}{\partial\phi^{k}_{i}}\hskip
28.45274pt(7)$

Where X is the generator of the Lie group of transformation of variable
($x^{i}$, $u^{k}$). Given as,

$\hskip 28.45274ptX=\xi^{i}\frac{\partial}{\partial
x^{i}}+\eta^{k}\frac{\partial}{\partial u^{k}}\hskip 28.45274pt(8)$

where,

$\zeta^{i}_{k}=D_{i}(\eta^{k})-p^{k}_{j}D_{i}(\xi^{j})\hskip
5.69054pt;(i=1,2,.....,n\hskip 1.42262pt;k=1,2,.....,m)\hskip 5.69054pt(9)$

and

$\hskip 28.45274ptD_{i}=\frac{\partial}{\partial
x^{i}}+p^{k}_{i}\frac{\partial}{\partial u^{k}}\hskip 85.35826pt(10)$

The operator $D_{i}$ is the Lie derivative operator.

####  4.4.2 Second Prolongation:

Following the same methods the generator of the extended group $X^{(2)}$, can
also be obtained as,

$X^{(2)}=X^{(1)}+\sigma^{k}_{ij}\frac{\partial}{\partial r^{k}_{ij}}\hskip
14.22636pt(11)$

where $r^{k}_{ij}$ is defined as,

$r^{k}_{ij}=\frac{\partial p^{k}_{i}}{\partial u_{j}}\hskip
5.69054pt;(k=1,2,.....,m\hskip 1.42262pt;i,j=1,2,.....,n)\hskip 5.69054pt(12)$

and

$\hskip
34.1433pt\sigma^{k}_{ij}=\bar{D}_{i}(\zeta^{k}_{i})-r^{k}_{ij}\bar{D}_{i}(\xi^{t})\hskip
56.9055pt(13)$ $\hskip 28.45274pt\bar{D}_{i}=\frac{\partial}{\partial
x^{i}}+p^{k}_{i}\frac{\partial}{\partial
u^{k}}+r^{k}_{ij}\frac{\partial}{\partial p^{k}_{j}}\hskip 42.67912pt(14)$

Proceeding in the aforesaid manner, the $k^{th}$ order prolongation can also
be found. Taking into account the cumbersome calculations and our limited need
upto second order prolongations for our purpose, we leave the calculations of
$k^{th}$ order prolongation for the interested readers.

We are predisposed towards the symmetry properties of the differential
equations, indicating that any transformation T $\in$ G, when applied to any
solution of the associated differential equation, maps itself into another
solution of the same differential equation, that is to say the $i^{th}$ order
differential equation

$\hskip 28.45274ptF^{(i)}(x,u,.....)=0\hskip 42.67912pt(15)$

remains invarient under any transformation belonging to group G. If
$X=\xi^{i}\frac{\partial}{\partial x^{i}}+\eta^{k}\frac{\partial}{\partial
u^{k}}$ be the infinitesimal generator corresponding to equation (6) and if
the operator of $k^{th}$ prolongation is denoted by $X^{(k)}$ then equation
(6) holds for equation (15), the system of PDE, if and only if

$\hskip 28.45274ptX^{(k)}F^{(i)}\bigg{|}_{F=0}=0\hskip 62.59605pt(16)$

Equation (16) is often referred as the Symmetry Condition or Condition of
Invariance.

##  5 LIE SYMMETRY ANALYSIS OF EINSTEIN’S VACUUM FIELD EQUATIONS:

The Einstein’s vacuum field equations as we know,

$\hskip 28.45274ptR_{ik}-\frac{1}{2}g_{ik}R=0\hskip 42.67912pt(17.1)$ $\hskip
28.45274ptR_{ik}\bigg{|}_{vacuum}=0\hskip 42.67912pt(17.2)$

The general covariance of this equation can be interpreted as the notion of a
group of generalized motion.

The application of Lie symmetry analysis of Einstein’s empty space field
equations can give rise to the maximal group of the involved point
transformations.

To extend the theory of Lie symmetry analysis to Einstein’s equation, we
consider a n-dimensional manifold M and along with it, we define a metric
tensor $g_{ij}$ for the space-time involved (Riemannian). It is convenient to
consider the transformation of the coordinates of space-time rather than the
function $g_{ij}$ itself. It is pretty clear that the transformations of the
type,

$\hskip 28.45274ptx^{i}\rightarrow x^{\prime i}=f^{i}(x,a)\hskip
5.69054pt;\hskip 5.69054pt(i=1,2,.....,n)\hskip 28.45274pt$

where $a$ is the involved parameter. We can write,

$\hskip 14.22636ptg_{ij}(x)\rightarrow
g^{\prime}_{ij}(x^{\prime})=g_{kl}\frac{\partial f^{k}(x,a)}{\partial
x^{i}}\frac{\partial f^{l}(x,a)}{\partial x^{j}}\hskip 28.45274pt(18)$

The generator of the group G,

$\hskip 56.9055ptX=\xi^{i}(x)\frac{\partial}{\partial x^{i}}\hskip
71.13188pt(19)$

where $\xi^{i}=\frac{\partial f^{i}}{\partial a}\bigg{|}_{a=0}\hskip
5.69054pt;\hskip 5.69054pt(i=1,2,.....,n)$ is the auxilliary function
involved.

Now the generator of the extended group $\bar{G}$,

$\bar{X}=\xi^{i}\frac{\partial}{\partial
x^{i}}+\eta_{ij}\frac{\partial}{\partial g_{ij}}\hskip 1.42262pt;\hskip
1.42262pt\eta_{ij}=\frac{\partial g^{\prime}_{ij}}{\partial
a}\bigg{|}_{a=0}\hskip 28.45274pt(20)$

The preceeding equation along with the fact that when a=0,
$g^{\prime}_{ij}(x)=g_{ij}(x)$ gives us,

$\eta_{ij}=-\bigg{(}g_{ik}\frac{\partial\xi^{k}}{\partial
x^{j}}+g_{kj}\frac{\partial\xi^{k}}{\partial x^{i}}\bigg{)}\hskip
5.69054pt;(i,j=1,2,3,.....n)\hskip 14.22636pt(21)$

The generator of the extended group $\bar{G}$ is

$\bar{X}=\xi^{i}(x)\frac{\partial}{\partial
x^{i}}-\bigg{(}g_{ik}\frac{\partial\xi^{k}}{\partial
x^{j}}+g_{kj}\frac{\partial\xi^{k}}{\partial
x^{i}}\bigg{)}\frac{\partial}{\partial g_{i}j}\hskip 14.22636pt(22)$

As it is known very well that the Einstein’s vacuum field equations allow all
possible transformations of coordinates i.e. a infinite dimensional Lie
algebra is obtained from equation (22). However the operator satisfied by
equation (22) does not admit the maximal algebra. 555The Einstein’s equation
allows the transformation, $x^{\prime i}\rightarrow x^{i}$ and
$g^{\prime}_{ij}\rightarrow ag_{ij}$ which is indeed a simple transformation
with a generator,  $X=g_{ij}\frac{\partial}{\partial g_{ij}}\hskip
28.45274pt(23)$ which certainly do not belong to the class of generators
specified in equation (22)

So assuming a generator of the form,

$X=\xi^{i}(x,g)\frac{\partial}{\partial
x^{i}}+\eta_{ij}(x,g)\frac{\partial}{\partial g_{ij}}\hskip 34.1433pt(24)$

along with the invariance condition,

$\hskip 56.9055ptX^{(2)}R_{ik}=\Omega^{jl}_{ik}R_{jl}\hskip 56.9055pt(25)$

where $\Omega^{jl}_{ik}$ are undeterminded coefficients. Segregating the terms
with second derivatives of $g_{ij}$ we obtain the Lie determining equations
for Einstein’s equations resulting in,

$\hskip 71.13188pt\xi^{i}=\xi^{i}(x)\hskip 156.49014pt(26.1)$ $\hskip
28.45274pt\eta_{ij}=-\bigg{(}\frac{\partial\xi^{k}}{\partial
x^{i}}g_{kj}+\frac{\partial\xi^{k}}{\partial
x^{j}}g_{ki}\bigg{)}+ag_{ij}\hskip 99.58464pt(26.2)$ $\hskip
28.45274ptX=\xi^{i}(x)\frac{\partial}{\partial
x^{i}}-\bigg{(}\xi^{k}\frac{\partial\xi^{k}}{\partial
x^{i}}g_{kj}+\frac{\partial\xi^{k}}{\partial
x^{j}}g_{ki}\bigg{)}+ag_{ij}\frac{\partial}{\partial g_{ij}}\hskip
42.67912pt(26.3)$

Thus it can be seen that the operators given in equation (23) and equation
(22) span the maximal Lie algebra admitted by Einstein’s empty space field
equations.

The generator obtained here is immoderately general for our liking because it
produces different sets of infinitesimal generator and different Lie algebras
depending upon which space-time or the metric we choose to work on. The
generator might give us the gravity group which evidently cannot depend on the
metric of choice (just like the Lie algebra of spherically symmetric metric is
given as SO(3), independent of the choice of metric.). So drifting away from
the far too generalized analysis of Einstein’s equation we intend to analyse
the geodesic equation of any particular metric (second order is less
cumbersome) using the same algorithm.

###  5.1 Geodesics of Schwarzschild metric:

The geodesics of Schwarzschild metric can be obtained by calculating the
Christoffel symbol and using the geodesic equation

$\hskip
56.9055pt\ddot{x}^{\beta}+\Gamma^{\beta}_{\gamma\mu}\dot{x}^{\gamma}\dot{x}^{\mu}=0\hskip
56.9055pt(27)$

$\hskip 28.45274pt\Gamma^{\alpha}_{\mu\nu}=\frac{1}{2}g^{\alpha
l}\big{(}g_{\mu l,\nu}+g_{\nu l,\mu}-g_{\mu\nu,l}\big{)}\hskip 28.45274pt(28)$

Using equation (27) and (28) for Schwarzschild metric

$ds^{2}=\bigg{(}1-\frac{2m}{r}\bigg{)}dt^{2}-\bigg{(}1-\frac{2m}{r}\bigg{)}^{-1}dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta
d\phi^{2}\hskip 28.45274pt(29)$

one obtains the geodesic equation for the above metric. The equations are
given as,

$\hskip
28.45274ptt^{\prime\prime}-\frac{a}{r(a-r)}t^{\prime}(\tau)r^{\prime}(\tau)=0\hskip
85.35826pt(30.1)$ $r^{\prime\prime}(\tau)+\frac{a}{2r(a-r)}r^{\prime
2}(\tau)+(a-r)\bigg{(}\theta^{\prime 2}(\tau)+\sin^{2}\theta\phi^{\prime
2}(\tau)-\frac{a}{2r^{3}}t^{\prime 2}(\tau)\bigg{)}=0\hskip 28.45274pt(30.2)$
$\theta^{\prime\prime}(\tau)+\frac{2}{r}r^{\prime}(\tau)\theta^{\prime}(\tau)-\frac{1}{2}\sin(2\theta)\phi^{\prime
2}(\tau)=0\hskip 42.67912pt(30.3)$
$\phi^{\prime\prime}(\tau)+\frac{2}{r}r^{\prime}(\tau)\phi^{\prime}(\tau)+2\cot\theta\theta^{\prime}(\tau)\phi^{\prime}(\tau)=0\hskip
14.22636pt(30.4)$

Considering infinitesimal transformation of the form

$\hskip 28.45274pt\bar{\tau}=\tau+\epsilon\sigma^{\tau}\hskip 28.45274pt$
$\hskip 28.45274pt\bar{t}=t+\epsilon T^{t}\hskip 28.45274pt$ $\hskip
28.45274pt\bar{r}=r+\epsilon R^{r}\hskip 28.45274pt$ $\hskip
28.45274pt\bar{\theta}=\theta+\epsilon J^{\theta}\hskip 28.45274pt$ $\hskip
28.45274pt\bar{\phi}=\phi+\epsilon F^{\phi}\hskip 28.45274pt$

and the involved generator X given as,

$\hskip
28.45274ptX=\sigma\frac{\partial}{\partial\tau}+T\frac{\partial}{\partial
t}+R\frac{\partial}{\partial
r}+J\frac{\partial}{\partial\theta}+F\frac{\partial}{\partial\phi}\hskip
28.45274pt$

Running through the usual algorithm of the Lie symmetry analysis of
differential equation (reference last chapter) we obtain the vector fields
that span the Lie algebra of the space as,

$\hskip 85.35826ptX_{1}=\frac{\partial}{\partial\tau}\hskip 71.13188pt(31.1)$
$\hskip 85.35826ptX_{2}=\frac{\partial}{\partial t}\hskip 71.13188pt(31.2)$
$\hskip 85.35826ptX_{3}=\frac{\partial}{\partial\phi}\hskip 71.13188pt(31.3)$
$\hskip 85.35826ptX_{4}=\tau\frac{\partial}{\partial\tau}\hskip
62.59605pt(31.4)$ $\hskip
28.45274ptX_{5}=\sin\phi\frac{\partial}{\partial\theta}+\cot\theta\cos\phi\frac{\partial}{\partial\phi}\hskip
28.45274pt(31.5)$ $\hskip
28.45274ptX_{6}=-\cos\phi\frac{\partial}{\partial\theta}+\cot\theta\sin\phi\frac{\partial}{\partial\phi}\hskip
28.45274pt(31.6)$

##  6 NOETHER POINT SYMMETRIES

Emmy Noether[14] proved a very important result that every conservation law
pertaining to a system must originate from a corresponding symmetry property
of the system. In a more complicated manner, according to Noether’s theorem
there is an algorithm which relates the constants of the Lagrangian of any
given system to its symmetry transformation.

While dealing with the Lie point symmetry of geodesic equations of a space-
time yielding conserved quantities, we realize that they are non-Noether type
conserved quantities and hence are redundant to us. The symmetries of the
Lagrangian on the other hand being directly connected to the symmetry
transformation of a system, gives us conserved quantities of our interest.

###  6.1 Pedagogical Noether’s theorem and Prolongations

Consider a dynamical system with Lagrangian L, where $L=L(q,\dot{q},t)$. Using
the variational principle to the action we can obtain the Euler-Lagrangian
equations written as,

$\hskip 56.9055pt\frac{\partial L}{\partial
q^{i}}-\frac{d}{dt}\bigg{(}\frac{\partial
L}{\partial\dot{q}^{i}}\bigg{)}=0\hskip 28.45274pt(32)$

The variational principle involving the action functional can be implemented
to formulate the Noether theorem. Here the key constituent is its variations
under infinitesimal transformations of the generalized coordinates and time.

$\hskip
56.9055ptq^{i}\rightarrow\tilde{q}^{i}=q^{i}+\epsilon\xi^{i}(q,\dot{q},t)\hskip
28.45274pt(33.1)$ $\hskip
56.9055ptt\rightarrow\tilde{t}=t+\epsilon\eta(q,\dot{q},t)\hskip
28.45274pt(33.2)$

$\epsilon$ is an infinitesimal parameter and $\xi^{i}$, $\eta$ are analytic
functions. The generator of the transformation is given by,

$\hskip 56.9055ptX=\eta\frac{\partial}{\partial
t}+\xi^{i}\frac{\partial}{\partial q^{i}}\hskip 28.45274pt(34)$

The transformation maps the velocities (terms involving first order derivating
of the generalized coordinates),

$\hskip
56.9055pt\frac{d{\tilde{q}}^{i}}{d\tilde{t}}=\dot{q}^{i}+\epsilon(\dot{\xi}^{i}-\dot{q}^{i}\dot{\eta})\hskip
28.45274pt(35)$

Now any function like the lagrangian, depending on the velocities transform
like,

$\hskip 14.22636ptL(q,\dot{q},t)\rightarrow
L(\tilde{q},\dot{\tilde{q}},t)=L(q,\dot{q},t)+\epsilon
X^{(1)}(L(q,\dot{q},t))+O(\epsilon^{2})\hskip 56.9055pt(36)$

$X^{(1)}$ being the first prolongation of the generator X. Given as,

$\hskip
28.45274ptX^{(1)}=X+\big{(}\dot{\xi}^{i}-\dot{q}^{i}\dot{\eta}\big{)}\frac{\partial}{\partial\dot{q}^{i}}\hskip
28.45274pt(37)$

Due to the infinitesimal shift of the coordinate, the change in action can be
put forward as,

$\delta
A=\int_{\bar{t}_{1}}^{\bar{t}_{2}}L\bigg{(}\tilde{q},\frac{d\tilde{q}}{d\tilde{t}},\tilde{t}\bigg{)}d\tilde{t}-\int_{t_{1}}^{t_{2}}L\bigg{(}q,\frac{dq}{dt},t\bigg{)}dt\hskip
28.45274pt(38)$ $\hskip 56.9055pt\delta
A=\epsilon\int(X^{(1)}L+\dot{\eta}L)dt\hskip 42.67912pt(39)$

We have neglected the higher order terms throughout. We can conclude that the
variation of action is invariant upto divergence term f, if the integrand of
equation (39) is total time derivative of some function f(q, $\dot{q}$,t),

$\hskip 28.45274ptX^{(1)}L+\dot{\eta}L=\dot{f}\hskip 28.45274pt(40)$

This is the Rund-Trautman identity which can be used to unearth the carefully
hidden symmetries that the Lagrangian fails to exhibit.

Now the Rund-Trautman identity is valid for all type of paths $t\rightarrow
q(t)$ so it is more logical to replace the dots with total time derivative
operator,

$\hskip 28.45274ptD=\frac{\partial}{\partial t}+\frac{\partial}{\partial
q^{i}}\dot{q}^{i}+\frac{\partial}{\dot{q}^{i}}\ddot{q}^{i}+....\hskip
28.45274pt$

So rewritting the equation (40) as,

$\hskip 28.45274ptX^{(1)}L+(D\eta)L=(Df)\hskip 28.45274pt(41)$

where the first prolongation operator is given as,

$\hskip
28.45274ptX^{(1)}=X+\big{(}D\xi-\dot{q}(D\eta)\big{)}\frac{\partial}{\partial\dot{q}}\hskip
28.45274pt$

We can solve unknown generator defined in equation (34) using the Rund-
Trautman identity, if the Lagrangian of a dynamical system is specified.

If the action functional is invariant under the infinitesimal change of
generalized coordinates upto the divergence term f, then the quantity,

$\hskip 28.45274ptI=f-L\eta-\frac{\partial
L}{\partial\dot{q}^{i}}\big{(}\xi^{i}-\dot{q}^{i}\eta\big{)}\hskip
28.45274pt(42)$

is a first integral of the system involved.

The Lagrangian associated with geodesic equations can be written as,

$\hskip
14.22636ptL(q^{i},\dot{q}^{i})=g_{\alpha\beta}\dot{q}^{\alpha}\dot{q}^{\beta}=g_{\alpha\beta}\frac{dq^{\alpha}}{d\tau}\frac{dq^{\beta}}{d\tau}\hskip
28.45274pt(43)$

###  6.2 Schwarzschild case

For Schwarzschild metric the Lagrangian takes the form,

$L=\bigg{(}1-\frac{2m}{r}\bigg{)}\dot{t}^{2}-\bigg{(}1-\frac{2m}{r}\bigg{)}^{-1}\dot{r}^{2}-r^{2}\dot{\theta}^{2}-r^{2}\sin^{2}\theta\dot{\phi}^{2}\hskip
56.9055pt(44)$

(bringing back the conventional terminology instead of using q and t).

Taking the symmetry generator to be of the standard form,

$\hskip
11.38109ptX=\sigma\frac{\partial}{\partial\tau}+T\frac{\partial}{\partial
t}+R\frac{\partial}{\partial
r}+J\frac{\partial}{\partial\theta}+F\frac{\partial}{\partial\phi}\hskip
28.45274pt(45)$

Substituting this into equation (40) and using the Schwarzschild Lagrangian,
we obtain huge cumbersome equations which upon seperation of monomials gives
us 16 coupled PDE as listed below.

$\hskip 56.9055pt\sigma=\sigma(s)\hskip 85.35826pt(46.1)$
$R\bigg{(}\frac{2m}{r^{2}}\bigg{)}+2\bigg{(}1-\frac{2m}{r}\bigg{)}T_{t}-\bigg{(}1-\frac{2m}{r}\bigg{)}\sigma_{\tau}=0\hskip
42.67912pt(46.2)$
$R\bigg{(}\frac{2m}{(r-2m)^{2}}\bigg{)}-R_{r}\bigg{(}\frac{2r}{(r-2m)}\bigg{)}+\bigg{(}\frac{r}{(r-2m)}\bigg{)}\sigma_{\tau}=0\hskip
28.45274pt(46.3)$ $\hskip
36.98857pt-2Rr-2r^{2}J_{\theta}+r^{2}\sigma_{\tau}=0\hskip 56.9055pt(46.4)$
$R(-2r^{2}\sin^{2}\theta)-2Jr^{2}\sin\theta\cos\theta-2r^{2}\sin^{2}\theta
F\phi+r^{2}\sin^{2}\theta\sigma_{\tau}\hskip 14.22636pt=0\hskip
28.45274pt(46.5)$ $\hskip
28.45274pt2\bigg{(}1-\frac{2m}{r}\bigg{)}T_{r}-\bigg{(}\frac{2r}{r-2m}\bigg{)}R_{t}=0\hskip
28.45274pt(46.6)$ $\hskip
28.45274pt2\bigg{(}1-\frac{2m}{r}\bigg{)}T_{\theta}-2r^{2}J_{t}=0\hskip
56.9055pt(46.7)$ $\hskip
28.45274pt2\bigg{(}1-\frac{2m}{r}\bigg{)}T_{\phi}-2r^{2}\sin^{2}\theta
F_{t}=0\hskip 28.45274pt(46.8)$ $\hskip
14.22636pt-\bigg{(}\frac{2r}{r-2m}\bigg{)}R_{\phi}-2r^{2}\sin^{2}\theta
F_{r}=0\hskip 34.1433pt(46.9)$ $\hskip
28.45274pt-2r^{2}J_{\phi}-2r^{2}\sin^{2}\theta F_{\theta}=0\hskip
48.36967pt(46.10)$ $\hskip
28.45274pt-\bigg{(}\frac{2r}{r-2m}\bigg{)}R_{\theta}-2r^{2}J_{r}=0\hskip
48.36967pt(46.11)$ $\hskip 71.13188ptf_{s}=0\hskip 76.82243pt(46.12)$ $\hskip
42.67912pt2\bigg{(}1-\frac{2m}{r}\bigg{)}T_{\tau}=f_{t}\hskip
42.67912pt(46.13)$ $\hskip
28.45274pt-\bigg{(}\frac{2r}{r-2m}\bigg{)}R_{s}=f_{r}\hskip 28.45274pt(46.14)$
$\hskip 42.67912pt-2r^{2}J_{s}=f_{\theta}\hskip 42.67912pt(46.15)$ $\hskip
28.45274pt-2r^{2}\sin^{2}\theta F_{s}=f_{\phi}\hskip 28.45274pt(46.16)$

Cumulating the above equations we can obtain the infinitesimal generators of
the Noether point symmetry which happens to span the involved Lie algebra as
well.

$\hskip 71.13188ptX_{1}=\frac{\partial}{\partial\tau}\hskip 71.13188pt(47.1)$
$\hskip 71.13188ptX_{2}=\frac{\partial}{\partial t}\hskip 71.13188pt(47.2)$
$\hskip
28.45274ptX_{3}=\cos\phi\frac{\partial}{\partial\theta}-\cot\theta\sin\phi\frac{\partial}{\partial\phi}\hskip
28.45274pt(47.3)$ $\hskip
28.45274ptX_{4}=\sin\phi\frac{\partial}{\partial\theta}+\cot\theta\cos\phi\frac{\partial}{\partial\phi}\hskip
28.45274pt(47.4)$ $\hskip 71.13188ptX_{5}=\frac{\partial}{\partial\phi}\hskip
71.13188pt(47.5)$

So the obtained infinitesimal Noether Point Symmetry generators involved for
the Schwarzschild Lagrangian are nothing but the vectors that span associated
Lie Algebra. As we can see from the expression of the infinitesimal symmetry
generators, a quick calculation of the closed commutators lead us to,

$[X_{3},X_{5}]=X_{4}$ $[X_{5},X_{4}]=X_{3}$ $[X_{4},X_{3}]=X_{5}$

So the generators spanning the Lie algebra follow the SO(3) structure. This is
quite anticipated since the Schwarzschild solution describes a spherically
symmetric space-time. A quick comparison between the Killing vectors involved
in SO(3) space-time and the infinitesimal generators obtained using Noether
Point Symmetry will tell us that the generators are nothing but Killing
vectors of the spherically symmetric space-time. But apart from the three
generators, satisfying the SO(3) algebra we have also obtained other
generators out of which our primary focus will be pertaining to the generator
$X_{2}$.

###  6.3 Conserved charge and Birkhoff theorem

The presence of conserved charges corresponding to the respective Noether
Point Symmetries is the next obvious thing ensured by the Noether theorem. We
can calculate the conserved quantities using equation (42). But for our
purpose, out of the five conserved quantities found we are mostly interested
in the conserved charge that corresponds to the particular generator
$X_{2}=\frac{\partial}{\partial t}$ which is very much intertwined with time
translation symmetry.

The conserved quantity corresponding to this generator is,

$\hskip 28.45274ptI=\bigg{(}1-\frac{2m}{r}\bigg{)}\dot{t}\hskip
28.45274pt(48)$

This quantity so obtained is not only a constant of motion but also consistent
with the quantity that remains conserved corresponding to a time-like Killing
vector. Hence we can conclude that the time translation invariance generator
$X_{2}$ behaves like a timelike Killing vector giving us the extra symmetry
that we were looking for since the onset of the problem.

We obtained the Noether point symmetries of Schwarzschild geodesic equations
along with the infinitesimal generators given in equation (43). Now we can
also see that the vectors $X_{3}$, $X_{4}$, $X_{5}$ form SO(3) algebra.

$\hskip 28.45274pt[X_{3},X_{5}]=X_{4}\hskip 28.45274pt(49.1)$ $\hskip
28.45274pt[X_{5},X_{4}]=X_{3}\hskip 28.45274pt(49.2)$ $\hskip
28.45274pt[X_{4},X_{3}]=X_{5}\hskip 28.45274pt(49.3)$

The Schwarzschild metric which does offer SO(3) algebra and the vector so
obtained span the Lie algebra of Noether point symmetries are nothing but the
Killing vectors of $S^{2}$ which we are expected to obtain since we are
working with a spherically symmetric metric, see equation (47). The vectors
obtained as $X_{3}$ , $X_{4}$, $X_{5}$ are nothing but Killing vectors of
SO(3).

Now along with this group structure of the obtained generators. The conserved
quantities corresponding to $\frac{\partial}{\partial t}$ is given by
$(1-\frac{2m}{r})\dot{t}$ is in tune with the above mentioned proposition.

We do know that the Killing vector lead us to a constant of motion (for a free
particle)[18]. If $K^{\mu}$ is a Killing vector then,

$\hskip 28.45274ptK_{\mu}\frac{dx^{\mu}}{d\lambda}=constant\hskip
28.45274pt(50)$

Hence from equation (48) and (50) we can conclude that the generator $X_{2}$
we obtained by solving the partial differential equations is actually a
Killing vector (time-like as we are in region $r>2m$). The conserved quantity
given in the equation (48) is the corresponding constant of the Killing
vector.

We initiated our work with the spherically symmetric metric ansatz exhibiting
SO(3) algebra. While calculating the Noether point symmetries of the
Schwarzschild Lagrangian we observe the Lie Algebra of SO(3) among the
infinitesimal symmetry generators we have obtained which is an expected result
as we had started with the spherically symmetric metric (these generators
recognised as nothing but the Killing vectors of the SO(3)). In addition to
these generators we have also obtained another generator which was identified
with the extra Killing vector that shows an additional symmetry than what we
had started with. So essentially we have regained the Birkhoff’s theorem which
states that spherically symmetric vacuum solutions of Einstein’s equations
allow us a fourth or additional timelike Killing vector (which is what we have
obtained).

Acknowledgement:
One of the author of this paper, Subham Basanta Roy, would like to thank
University Grants Commission (UGC) India, for financial support (JRF
Registration Id : 521612) during the course of this work and also The
Relativity and Cosmology Centre, Department of Physics, Jadavpur University
for allowing to rightfully conduct the research work without any disruptions.

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