Birkhoffs Theorem and Lie Symmetry Analysis
aa r X i v : . [ g r- q c ] F e b Birkhoff ’s Theorem and Lie Symmetry Analysis
A. Mukherjee Relativity and Cosmology Research Centre, Department of PhysicsJadavpur University, India. andSubham B. Roy Department of PhysicsUniversity at Albany, State University of New YorkFormerly at Relativity and Cosmology Research Centre, Department ofPhysics, Jadavpur University, India
Abstract
Three dimensional space is said to be spherically symmetric if it admits SO(3) as the group ofisometries. Under this symmetry condition, the Einstein’s Field equations for vacuum, yieldsthe Schwarzschild Metric as the unique solution, which essentially is the statement of the wellknown Birkhoff’s Theorem. Geometrically speaking this theorem claims that the pseudo -Riemanian space-times provide more isometries than expected from the original metric holon-omy/ansatz. In this paper we use the method of Lie Symmetry Analysis to analyze the Ein-stein’s Vacuum Field Equations so as to obtain the Symmetry Generators of the correspondingDifferential Equation. Additionally, applying the Noether Point Symmetry method we haveobtained the conserved quantities corresponding to the generators of the Schwarzschild La-grangian and paving way to reformulate the Birkhoff’s Theorem from a different approach.
Keywords : Birkhoff’s theorem, Lie Symmetry Analysis, Noether Point Symmetry. [email protected] [email protected] 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 thegeometry of space-time. In a more qualitative manner, the field equations explains how themetric (of the space-time involved) respond to energy and momentum. Due to the non-linearityand 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 ofspherical symmetry, is known as the Schwarzschild solution.Lie Symmetry analysis of differential equations provides us a rudimentary yet very powerfulmachinery to derive the conservation laws of corresponding system represented by the (ordinaryor partial) differential equations. The invariance of the differential equations under transfor-mation of both dependent and independent variables involved essentially leads to the idea oftheir symmetry analysis. This transformation forms a local group of point transformation es-tablishing diffeomorphism on the space of independent and dependent variables, mapping thesolutions 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 involvedsymmetries and the conserved quantities of dynamical system to reiterate the well knownBirkhoff’s theorem.The paper is organized in the following manner. In Section 2, we quickly recapitulate theBirkhoff’s theorem and discuss about the Lie Algebra of the Killing vectors in a sphericallysymmetric space-time. Section 3, sheds some light upon the recent literatures that deals withBirkhoff’s theorem and it’s modern treatments to different dimensions. In the section followed2y we introduce the most important tool implemented for our work that is the Lie groups oftransformations and the Prolongation Theory of differential equations. In simple language,the term prolongation means becoming longer. Here the involved system certainly does notbecome longer but the space of the dependent variables does. Basically our requirement isof a differential equation not only representing the dependent and independent variables butalso the appearing partial derivatives. So, we prolong the space of dependent variables usingtheir partial derivatives. In section 5, we take the Einstein’s vacuum field equations and obtainthe maximal symmetry generator, using the method of Lie Symmetry analysis. Entire section6 comprises of Noether’s theorem and it’s modification for the first order prolongations. Thistheorem has been implemented for the case of Schwarzschild lagrangian. This section concludeswith recovering Birkhoff’s theorem from the infinitesimal symmetry generators obtained via theanalysis of Schwarzschild Lagrangian.
A metric describing the space-time is obtained when Einstein’s field equations are solved un-der simplifying assumptions. The Birkhoff’s theorem is a fundamental theorem regarding thesolution of the field equations. It states that, any spherically symmetric solution of Einstein’svacuum field equations must be static and asymptotically flat. This statement leads to the factthat Schwarzschild’s exterior (i.e. space-time outside of a spherically non-rotating gravitationalobject) solution is the most general spherically symmetric solution of Einstein’s vacuum fieldequations with zero cosmological constant. The solution for non-zero cosmological constant isprovided by the Schwarzschild-deSitter metric. The Einstein’s field equations are given by, R ik − g ik R = T ik Now for of vacuum, the energy-momentum tensor vanishes i.e. T ik = 03 ik − g ik R = 0where g ik is the metric of the space-time involved and R ik and R being the Ricci tensorand Ricci scalar respectively. The solution of this equation for a space-time outside a sphericalsource or gravitating body is given as, ds = (cid:18) − GMc r (cid:19) dt − (cid:18) − GMc r (cid:19) − dr − r ( dθ + sin θdφ )Where ( r, θ, φ ) are the spherical coordinates used and M is the mass of the (spherical)gravitating object. Redefining m = GMc we get, ds = (cid:18) − mr (cid:19) dt − (cid:18) − mr (cid:19) − dr − r ( dθ + sin θdφ )The theorem also conveys that the solutions of Einstein’s vacuum field equations possesshypersurface orthogonal Killing vectors i.e. the metric obtained as solution of field equationsis static . Additionally the theorem also tells us the metric is independent of the changes inthe matter distribution, which are the sources of the gravitational field, provided the sphericalsymmetry is preserved.The presence of SO(3) group as a group of isometries renders the metric to be sphericallysymmetric. The existence of Killing vectors corresponding to respective symmetries character-izes the symmetries of space-time or metric. The existing Killing vectors of any given space-timeare coordinate independent quantities. None the less by labelling the space-time in specific co-ordinates, we can formulate the Killing vectors in the chosen coordinate. For example, considera S (having SO(3)) with the Killing vectors labelled as X,Y,Z which in polar coordinates aregiven as follows, X = ∂ φ Y = (cos φ ) ∂ θ − (cot θ sin φ ) ∂ φ The description of Birkhoff theorem with Einstein’s vacuum solutions having static Schwarzschild metric assolution can be a bit of misleading as well since in the region 0 < r < M , the time coordinate t fails to remaintime like, see Hawking and Ellis Appendix. = − (sin φ ) ∂ θ − (cot θ cos φ ) ∂ φ A quick evaluation of the closed commutators of these Killing vectors,5
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 doesallow one extra time-like Killing vector; 3 for spherical symmetry and 1 for time translation (onlyin the limit r > M , 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 particlemoving along a geodesic, having of equation of motion¨ x a + Γ abc ˙ x b ˙ x c = 0if K ν is a Killing vector then K ν dx ν dλ remains conserved. The explicit expressions for the fourconserved quantities associated with the Killing vectors can easily be written. Summarizing wecan conclude that any spherically symmetric solution of the vacuum field equations allow us afourth/extra Killing vector (we started with SO(3)).However, this is one of the many aspects of Birkhoff theorem. There are many differentapproaches available to this theorem. Such as, physically, it implies that a spherically symmetricstar undergoing strict radial pulsations cannot propagate any disturbances into the surroundingspace, indicating that a radially oscillating star has a static gravitational field [29][30]. Thispaper is based on the differential geometric aspect of Birkhoff’s theorem which is stated as, everymember of the family of the pseudo-Riemannian space-time has more isometries than expectedfrom the original metric ansatz/holonomy (we have described briefly about this following thepedagogical technique of using the Killing vectors).6 GENERAL FORMULATIONS OF BIRKHOFF’S THE-OREM
The Birkhoff’s theorem concerning Einstein’s General theory of Relativity was first proposedand discussed by G.D.Birkhoff (1923) in his paper cited as follows,
The field outside the sphericaldistribution of matter is static whether or not the matter is in a static or in a variable state...thusthe Schwarzschild solution is essentially the most general solution of the field equations withspherical symmetry [27].Over the years Birkhoff’s theorem has been addressed in many different ways. Upon statingthe fact that the theorem relies on the existence of a 3-parameter group of global isometries with2-dimensional non-null orbits and of an additional Killing vector associated with a G group ofmotion, H. Goenner [8] had put forward a generalized and geometric version of the Birkhoff’stheorem. H.J.Schmidt on the other hand provided a complete covariant proof of Birkhoff’stheorem by showing that the origin of Birkhoff’s theorem rests on the property that comparedto all other dimensions k, it holds for k=2. Other different approaches to this theorem are verylucidly 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 inthe works of P. Havas [34]. The relation of Birkhoff’s theorem with 2-dimensional space-timecan be found in [35][36].The most evolved phrasing of this theorem in connection with the conformally reduciblemetrics can be found in the works of Bona [37]. The author’s work focuses on the differentkinds 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 generalizationof 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 theoremin multidimensional gravity with no resctrictions on space-time dimensionality. An elucidatingdiscussion on the relation between manifold dimensionality and the existance of Birkhoff liketheorems has been done by H.J.Schmidt [10][7]. G.F.R. Ellis and R. Goswami have investigatedthe possibility of extending the Birkhoff’s theorem by analysing whether the theorem holds7pproximately for an approximate spherical vacuum solution and also for an almost vacuumlike configuration [39] [24]. The method of change of variables involed while analysing differential equations is a go-to toolused by physicists and mathematicians alike. Let us illustrate this method by a 2 dimensionalcase. For example let x and y be the set of variables for a given space. After transformation ofthe variables we obtain, x → x ′ = x ′ ( x, y ) y → y ′ = y ′ ( x, y )where x ′ and y ′ are new set of variables involved. This is an example of a Point Transfor-mation which maps points ( x, y ) into ( x ′ , y ′ ).As we are more delved into symmetry properties therefore we will be much more interestedin the transformation that also involves parameters. Let us consider a domain D ⊂ R N (coordinate space) and another space S ⊂ R ( S be ourparameter space). Now let us choose G to be the set of transformations (map) defined by, G : D × S → D Now within this set, choose a particular transformation Z ∈ G defined by, Z : x × a → x ′ = Z ( x ; a ) (1)8here x and x ′ are old and new set of coordinates respectively while a being the parameterof transformation involved.(a) For each value of parameter a ∈ S , the transformations are bijective.(b) S with the law of composition µ is a group with identity e.(c) Z ( x ; e ) = x ; ∀ x ∈ D. (d) Z (cid:0) Z ( x ; a ) ; b (cid:1) = Z (cid:0) x ; µ ( a , b ) (cid:1) ; ∀ x ∈ D and ∀ a, b ∈ S. Furthermore satisfying the axioms that are mentioned above if the following properties,(a) involved parameter a is continuous.(b) Z is C ∞ ; ∀ x ∈ D and an analytic function of a ∈ S .are also satisfied then these axioms together with the ones stated in section 4.1 elevates thegroup of transformation into the class of Lie groups of transformations.Expanding equation (1) in Taylor series about a=0, x ′ = x + a ∂Z ( x ; a ) ∂a (cid:12)(cid:12)(cid:12)(cid:12) a =0 + O ( a ) (2)and defining , ξ ( x ) = ∂Z ( x ; a ) ∂a (cid:12)(cid:12)(cid:12)(cid:12) a =0 we found, x ′ = x + aξ ( x ) (3)where ξ ( 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 often referred as first fundamental theorem of Lie. x → x ′ = f ( x , µ , a ) (4 . µ → µ ′ = φ ( x , µ , a ) (4 . f and φ being transformation maps.With our definition, x ′ | a =0 = f ( x , µ , a ) = x (5 . µ ′ | a =0 = φ ( x , µ , a ) = µ (5 . x ′ = x (cos a ) − µ (sin a ) µ ′ = x (sin a ) + µ (cos a )where a is the involved parameter. On contrary, the reflection x ′ = − xµ ′ = − µ 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 − µ plane.Consider an arbitrary starting point ( x o , µ o ) in that plane (with the involved parameter a being zero). Varying this parameter shifts the starting point along some curve. Again repeatingthis procedure for different values of ( x o , µ o ) we can obtain a bunch of curves where under theaction of a group, each curve can be transformed into one another and are collectively referred10o as orbits of the group [6]. Let us assume, an C ∞ fucntion F ( x ), which is said to be an Invariant of the Lie group ofTransformation if and only if for any group of transformation the condition, F ( x ′ ) = F ( x )(where x and x ′ are old and new set of coordinates respectively) holds.Infinitesimal generator of the group can be easily exploited to characterize the invarianceof a function as is illustrated by the following theorem.F(x) is invariant under a coordinate transformation, x ′ = Z ( x ; a ), if and only if, XF ( x ) = 0where X is called the infinitesimal generator of the transformation. It is given as, X = ξ ( x ) ∂∂x + ξ ( x ) ∂∂x + ..... + ξ N ( x ) ∂∂x N where ξ , ξ ,....., ξ N are set of generalized auxilliary functions for the N dimensional case. Consider Lie groups of transformations associated to any given type of differential equation(say F) involving n independent variables ( x , x , ....., x n ) ∈ R n and m dependent variables( u , u , ....., u m ) ∈ R m and R m + n be the space of all variables ( x, u ).Let us consider the transformations, x ′ i = Z i ( x, u, a ) (6 . ′ k = U k ( x, u, a ) (6 . R n + m of variables ( x, u ).Let p ki denote the derivative of dependent variable u with respect to the independent variable x , p ki = ∂u k ∂x i ; ( i = 1 , , ....., n ; k = 1 , , ....., m )The transformation of derivatives with the formula (8) leads us to an extension of the Liegroup of transformation which are called Prolongations . This extended group acts on thespace with variables ( x i , u k , p ki ) rather than ( x i , u k ), so the k th prolonged space will have ( x i , u k , p ki , ....., p kj ) as variables, often called the jet space. The infinitesimal generator for the extended group X (1) , after first prolongation is given by[2][4][6], X (1) = X + ζ ki ∂∂φ ki (7)Where X is the generator of the Lie group of transformation of variable ( x i , u k ). Given as, X = ξ i ∂∂x i + η k ∂∂u k (8)where, ζ ik = D i ( η k ) − p kj D i ( ξ j ) ; ( i = 1 , , ....., n ; k = 1 , , ....., m ) (9)and D i = ∂∂x i + p ki ∂∂u k (10)The operator D i is the Lie derivative operator.12 .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) + σ kij ∂∂r kij (11)where r kij is defined as, r kij = ∂p ki ∂u j ; ( k = 1 , , ....., m ; i, j = 1 , , ....., n ) (12)and σ kij = ¯ D i ( ζ ki ) − r kij ¯ D i ( ξ t ) (13)¯ D i = ∂∂x i + p ki ∂∂u k + r kij ∂∂p kj (14)Proceeding in the aforesaid manner, the k th order prolongation can also be found. Takinginto account the cumbersome calculations and our limited need upto second order prolongationsfor 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, indicatingthat any transformation T ∈ G, when applied to any solution of the associated differentialequation, maps itself into another solution of the same differential equation, that is to say the i th order differential equation F ( i ) ( x, u, ..... ) = 0 (15)remains invarient under any transformation belonging to group G. If X = ξ i ∂∂x i + η k ∂∂u k bethe infinitesimal generator corresponding to equation (6) and if the operator of k th prolongationis denoted by X ( k ) then equation (6) holds for equation (15), the system of PDE, if and only if X ( k ) F ( i ) (cid:12)(cid:12)(cid:12)(cid:12) F =0 = 0 (16)Equation (16) is often referred as the Symmetry Condition or Condition of Invariance.13 LIE SYMMETRY ANALYSIS OF EINSTEIN’S VAC-UUM FIELD EQUATIONS:
The Einstein’s vacuum field equations as we know, R ik − g ik R = 0 (17 . R ik (cid:12)(cid:12)(cid:12)(cid:12) vacuum = 0 (17 . g ij for the space-timeinvolved (Riemannian). It is convenient to consider the transformation of the coordinates ofspace-time rather than the function g ij itself. It is pretty clear that the transformations of thetype, x i → x ′ i = f i ( x, a ) ; ( i = 1 , , ....., n )where a is the involved parameter. We can write, g ij ( x ) → g ′ ij ( x ′ ) = g kl ∂f k ( x, a ) ∂x i ∂f l ( x, a ) ∂x j (18)The generator of the group G, X = ξ i ( x ) ∂∂x i (19)where ξ i = ∂f i ∂a (cid:12)(cid:12)(cid:12)(cid:12) a =0 ; ( i = 1 , , ....., n ) is the auxilliary function involved.Now the generator of the extended group ¯ G ,14 X = ξ i ∂∂x i + η ij ∂∂g ij ; η ij = ∂g ′ ij ∂a (cid:12)(cid:12)(cid:12)(cid:12) a =0 (20)The preceeding equation along with the fact that when a=0, g ′ ij ( x ) = g ij ( x ) gives us, η ij = − (cid:18) g ik ∂ξ k ∂x j + g kj ∂ξ k ∂x i (cid:19) ; ( i, j = 1 , , , .....n ) (21)The generator of the extended group ¯ G is¯ X = ξ i ( x ) ∂∂x i − (cid:18) g ik ∂ξ k ∂x j + g kj ∂ξ k ∂x i (cid:19) ∂∂g i j (22)As it is known very well that the Einstein’s vacuum field equations allow all possible trans-formations 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. So assuming a generator of the form, X = ξ i ( x, g ) ∂∂x i + η ij ( x, g ) ∂∂g ij (24)along with the invariance condition, X (2) R ik = Ω jlik R jl (25)where Ω jlik are undeterminded coefficients. Segregating the terms with second derivatives of g ij we obtain the Lie determining equations for Einstein’s equations resulting in, ξ i = ξ i ( x ) (26 . η ij = − (cid:18) ∂ξ k ∂x i g kj + ∂ξ k ∂x j g ki (cid:19) + ag ij (26 . X = ξ i ( x ) ∂∂x i − (cid:18) ξ k ∂ξ k ∂x i g kj + ∂ξ k ∂x j g ki (cid:19) + ag ij ∂∂g ij (26 . The Einstein’s equation allows the transformation, x ′ i → x i and g ′ ij → ag ij which is indeed a simpletransformation with a generator, X = g ij ∂∂g ij (23)which certainly do not belong to the class of generators specified in equation (22) The geodesics of Schwarzschild metric can be obtained by calculating the Christoffel symboland using the geodesic equation ¨ x β + Γ βγµ ˙ x γ ˙ x µ = 0 (27)Γ αµν = 12 g αl (cid:0) g µl,ν + g νl,µ − g µν,l (cid:1) (28)Using equation (27) and (28) for Schwarzschild metric ds = (cid:18) − mr (cid:19) dt − (cid:18) − mr (cid:19) − dr − r dθ − r sin θdφ (29)one obtains the geodesic equation for the above metric. The equations are given as, t ′′ − ar ( a − r ) t ′ ( τ ) r ′ ( τ ) = 0 (30 . r ′′ ( τ ) + a r ( a − r ) r ′ ( τ ) + ( a − r ) (cid:18) θ ′ ( τ ) + sin θφ ′ ( τ ) − a r t ′ ( τ ) (cid:19) = 0 (30 . θ ′′ ( τ ) + 2 r r ′ ( τ ) θ ′ ( τ ) −
12 sin(2 θ ) φ ′ ( τ ) = 0 (30 . ′′ ( τ ) + 2 r r ′ ( τ ) φ ′ ( τ ) + 2 cot θθ ′ ( τ ) φ ′ ( τ ) = 0 (30 . τ = τ + ǫσ τ ¯ t = t + ǫT t ¯ r = r + ǫR r ¯ θ = θ + ǫJ θ ¯ φ = φ + ǫF φ and the involved generator X given as, X = σ ∂∂τ + T ∂∂t + R ∂∂r + J ∂∂θ + F ∂∂φ
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, X = ∂∂τ (31 . X = ∂∂t (31 . X = ∂∂φ (31 . X = τ ∂∂τ (31 . X = sin φ ∂∂θ + cot θ cos φ ∂∂φ (31 . X = − cos φ ∂∂θ + cot θ sin φ ∂∂φ (31 . NOETHER POINT SYMMETRIES
Emmy Noether[14] proved a very important result that every conservation law pertaining toa system must originate from a corresponding symmetry property of the system. In a morecomplicated manner, according to Noether’s theorem there is an algorithm which relates theconstants 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 yieldingconserved quantities, we realize that they are non-Noether type conserved quantities and henceare redundant to us. The symmetries of the Lagrangian on the other hand being directlyconnected to the symmetry transformation of a system, gives us conserved quantities of ourinterest.
Consider a dynamical system with Lagrangian L, where L = L ( q, ˙ q, t ). Using the variationalprinciple to the action we can obtain the Euler-Lagrangian equations written as, ∂L∂q i − ddt (cid:18) ∂L∂ ˙ q i (cid:19) = 0 (32)The variational principle involving the action functional can be implemented to formulatethe Noether theorem. Here the key constituent is its variations under infinitesimal transforma-tions of the generalized coordinates and time. q i → ˜ q i = q i + ǫξ i ( q, ˙ q, t ) (33 . t → ˜ t = t + ǫη ( q, ˙ q, t ) (33 . ǫ is an infinitesimal parameter and ξ i , η are analytic functions. The generator of thetransformation is given by, X = η ∂∂t + ξ i ∂∂q i (34)18he transformation maps the velocities (terms involving first order derivating of the gener-alized coordinates), d ˜ q i d ˜ t = ˙ q i + ǫ ( ˙ ξ i − ˙ q i ˙ η ) (35)Now any function like the lagrangian, depending on the velocities transform like, L ( q, ˙ q, t ) → L (˜ q, ˙˜ q, t ) = L ( q, ˙ q, t ) + ǫX (1) ( L ( q, ˙ q, t )) + O ( ǫ ) (36) X (1) being the first prolongation of the generator X. Given as, X (1) = X + (cid:0) ˙ ξ i − ˙ q i ˙ η (cid:1) ∂∂ ˙ q i (37)Due to the infinitesimal shift of the coordinate, the change in action can be put forward as, δA = Z ¯ t ¯ t L (cid:18) ˜ q, d ˜ qd ˜ t , ˜ t (cid:19) d ˜ t − Z t t L (cid:18) q, dqdt , t (cid:19) dt (38) δA = ǫ Z ( X (1) L + ˙ ηL ) dt (39)We have neglected the higher order terms throughout. We can conclude that the variationof action is invariant upto divergence term f, if the integrand of equation (39) is total timederivative of some function f(q, ˙ q ,t), X (1) L + ˙ ηL = ˙ f (40)This is the Rund-Trautman identity which can be used to unearth the carefully hiddensymmetries that the Lagrangian fails to exhibit.Now the Rund-Trautman identity is valid for all type of paths t → q ( t ) so it is more logicalto replace the dots with total time derivative operator,19 = ∂∂t + ∂∂q i ˙ q i + ∂ ˙ q i ¨ q i + .... So rewritting the equation (40) as, X (1) L + ( Dη ) L = ( Df ) (41)where the first prolongation operator is given as, X (1) = X + (cid:0) Dξ − ˙ q ( Dη ) (cid:1) ∂∂ ˙ q We can solve unknown generator defined in equation (34) using the Rund-Trautman iden-tity, if the Lagrangian of a dynamical system is specified.If the action functional is invariant under the infinitesimal change of generalized coordinatesupto the divergence term f, then the quantity, I = f − Lη − ∂L∂ ˙ q i (cid:0) ξ i − ˙ q i η (cid:1) (42)is a first integral of the system involved.The Lagrangian associated with geodesic equations can be written as, L ( q i , ˙ q i ) = g αβ ˙ q α ˙ q β = g αβ dq α dτ dq β dτ (43)20 .2 Schwarzschild case For Schwarzschild metric the Lagrangian takes the form, L = (cid:18) − mr (cid:19) ˙ t − (cid:18) − mr (cid:19) − ˙ r − r ˙ θ − r sin θ ˙ φ (44)(bringing back the conventional terminology instead of using q and t).Taking the symmetry generator to be of the standard form, X = σ ∂∂τ + T ∂∂t + R ∂∂r + J ∂∂θ + F ∂∂φ (45)Substituting this into equation (40) and using the Schwarzschild Lagrangian, we obtainhuge cumbersome equations which upon seperation of monomials gives us 16 coupled PDE aslisted below. σ = σ ( s ) (46 . R (cid:18) mr (cid:19) + 2 (cid:18) − mr (cid:19) T t − (cid:18) − mr (cid:19) σ τ = 0 (46 . R (cid:18) m ( r − m ) (cid:19) − R r (cid:18) r ( r − m ) (cid:19) + (cid:18) r ( r − m ) (cid:19) σ τ = 0 (46 . − Rr − r J θ + r σ τ = 0 (46 . R ( − r sin θ ) − J r sin θ cos θ − r sin θF φ + r sin θσ τ = 0 (46 . (cid:18) − mr (cid:19) T r − (cid:18) rr − m (cid:19) R t = 0 (46 . (cid:18) − mr (cid:19) T θ − r J t = 0 (46 . (cid:18) − mr (cid:19) T φ − r sin θF t = 0 (46 . − (cid:18) rr − m (cid:19) R φ − r sin θF r = 0 (46 . − r J φ − r sin θF θ = 0 (46 . − (cid:18) rr − m (cid:19) R θ − r J r = 0 (46 . s = 0 (46 . (cid:18) − mr (cid:19) T τ = f t (46 . − (cid:18) rr − m (cid:19) R s = f r (46 . − r J s = f θ (46 . − r sin θF s = f φ (46 . X = ∂∂τ (47 . X = ∂∂t (47 . X = cos φ ∂∂θ − cot θ sin φ ∂∂φ (47 . X = sin φ ∂∂θ + cot θ cos φ ∂∂φ (47 . X = ∂∂φ (47 . X , X ] = X [ X , X ] = X [ X , X ] = X So the generators spanning the Lie algebra follow the SO(3) structure. This is quite antic-ipated since the Schwarzschild solution describes a spherically symmetric space-time. A quick22omparison between the Killing vectors involved in SO(3) space-time and the infinitesimal gen-erators obtained using Noether Point Symmetry will tell us that the generators are nothing butKilling 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 primaryfocus will be pertaining to the generator X . The presence of conserved charges corresponding to the respective Noether Point Symmetriesis the next obvious thing ensured by the Noether theorem. We can calculate the conservedquantities using equation (42). But for our purpose, out of the five conserved quantities foundwe are mostly interested in the conserved charge that corresponds to the particular generator X = ∂∂t which is very much intertwined with time translation symmetry.The conserved quantity corresponding to this generator is, I = (cid:18) − mr (cid:19) ˙ t (48)This quantity so obtained is not only a constant of motion but also consistent with thequantity that remains conserved corresponding to a time-like Killing vector. Hence we canconclude that the time translation invariance generator X behaves like a timelike Killingvector 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 withthe infinitesimal generators given in equation (43). Now we can also see that the vectors X , X , X form SO(3) algebra. [ X , X ] = X (49 . X , X ] = X (49 . X , X ] = X (49 . S which weare expected to obtain since we are working with a spherically symmetric metric, see equation(47). The vectors obtained as X , X , X are nothing but Killing vectors of SO(3).Now along with this group structure of the obtained generators. The conserved quantitiescorresponding to ∂∂t is given by (1 − mr ) ˙ 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 µ is a Killing vector then, K µ dx µ dλ = constant (50)Hence from equation (48) and (50) we can conclude that the generator X we obtained bysolving the partial differential equations is actually a Killing vector (time-like as we are in region r > m ). The conserved quantity given in the equation (48) is the corresponding constant ofthe Killing vector.We initiated our work with the spherically symmetric metric ansatz exhibitingSO(3) algebra. While calculating the Noether point symmetries of the Schwarzschild La-grangian we observe the Lie Algebra of SO(3) among the infinitesimal symmetry generators wehave obtained which is an expected result as we had started with the spherically symmetricmetric (these generators recognised as nothing but the Killing vectors of the SO(3)). In ad-dition to these generators we have also obtained another generator which was identified withthe 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 symmetricvacuum solutions of Einstein’s equations allow us a fourth or additional timelike Killing vector(which is what we have obtained). 24 cknowledgement: One of the author of this paper, Subham Basanta Roy, would like to thank University GrantsCommission (UGC) India, for financial support (JRF Registration Id : 521612) during thecourse of this work and also The Relativity and Cosmology Centre, Department of Physics, Ja-davpur University for allowing to rightfully conduct the research work without any disruptions.25 eferences [1] N. H. Ibragimov : “Elementary Lie Group Analysis and Ordinary Differential Equation”.(Wiley Series in Mathematical Method in practice)[2] Peter J. 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