Exact Solution Of Einstein's Equation To The Gravitational Field Of ...

Ozean Journal of Applied Sciences 1(1), 2008Ozean Journal of Applied Sciences 1(1), 2008ISSN 1943-2429© 2008 Ozean PublicationExact Solution Of Einstein’s Equation To The Gravitational Field Of Mass Point InPolar Coordinates And The Singularity Analysis Of The SolutionM. H. A. Biswas, S. S. Jahan, C. M. Asaduzzaman and M. KamruzzamanMathematis Discipline, Khulna University, Khulna-9208, Bangladesh.E-mail: mhabiswas@yahoo.com_____________________________________________________________________________________Abstract: In this paper, we obtained the exact solution of Einstein’s equation to the gravitational field ofmass point in polar coordinates considering the new transformation known as cumbersome transformation.We have also investigated the singularity of the new solution in compared to that of Schwarzschild solutionat R = a , where a represents the boundary of the isolated particle.Key words: Tensor; Euclidean geometry; Exact solution; Gravitational field; Mass point; Schwarzschildsingularity.___________________________________________________________________________________INTRODUCTIONThe exact solution of Einstein’s equation to the gravitational field of mass point is very much difficult dueto the highly nonlinearity of the problem. Einstein published his complete theory of relativity in 1915 andafter the publication of the paper he himself expected that the exact solution to the problem could not beformulated. K. Schwarzschild was first who found an exact solution in less than a year from the publicationof the theory 1916. It was the first physically significant exact solution to Einstein equations.The Einstein theory of gravitation as well as the general theory of relativity depends on the Einstein world1famous field equations in space-time manifold, as Ri j− gi jR = 8πT . It is clear by considering the2ijdefinition of the Ricci tensor that the Einstein equations depend on the derivatives of g up to the secondi jorder and that they are highly non-linear in gi js . It may be noted, however, that the Einstein equationsare linear in the second derivatives of g . In fact, the quantitiesi jR andi jR g are the only second-ranki jsymmetric tensor which are linear in the second derivatives of the metric and involve only up to secondderivatives of g . Actually, the Einstein equations are a coupled system of non-linear second order partiali jdifferential equations for g i js . This makes the task of solving them extremely difficult (Atwater, 1974).So, one generally needs to impose several symmetry assumptions on the space-time in order to work outthe metric components as a solution to the Einstein equations. Since, the Einstein equations form a highlynon-linear system of differential equations and due to their complexity, a completely general solution is notknown. Thus, the known exact solutions usually assume a rather high degree of symmetry such as thespherical or axial symmetry, and the existence of necessary killing vector fields on the space-time, and tothat extent represent on idealized situation (Bergmann, 1942). Thus, some of such solutions havesignificant physical applications. For example, the Minkowski space-time is both the geometry of thespecial theory of relativity and locally that of any general relativistic model. The non Euclidean nature ofspace in which there is present a gravitational field of the centrifugal carioles type, must be extended to allgravitational field. In the case of a field, such that which surrounds the earth. It will not possible to find aninertial frame of reference relative to which the field vanishes and for which the spatial geometry isEuclidean. In any gravitational field, it is always possible to define a frame relative to which the fieldvanishes over a restricted region and which behaves as an inertial frame for events occurring in this regionand extending over a small interval of time. Our aim is to find a solution of Einstein’s field equation of agravitating point particle, for example, the sun in polar coordinates. This will correspond to the field of an29

Ozean Journal of Applied Sciences 1(1), 2008isolated particle continually at rest at the origin. We will also investigate the singularity occurred in thesolution and thus will be compared to that of Schwarzschild solution (Biswas, 2008).EINSTEIN LAW OF GRAVITATIONWe look for a law of motion, which will be independent of the coordinates system used, describing thegravitational field of a single particle. In the special theory of relativity the line element for the space-timecoordinates is given by2 2 2 2 2 2ds = −dx− dy − dz + c dtIn the space of2222ds = −dr− r dθ− r sin θ dφ+ c dtx y,z,t, the αβ222g are constant and is flat (Euclidian) so that R = 0 .For a gravitational particle we postulate that the Ricci tensor Rijvanishes.SOLUTION FOR GRAVITATIONAL FIELD OF A MASS POINTLet a point moves according to the prescriptionδ ∫ ds = 0(1)Let s be a parameter defined on C such that, if s , s + ds are its values at the respective neighboringpoints p , p′ on C [ Fig. 1] then ds is the interval between these two point. Ifpoint p , p′ on C , then the curve will be defined by parametric equation (Lawden, 1968) as,( s)22ijklx are the coordinatexi = xi(2)dx iis termed as the unit tangent to the curve at P .dsSuppose P , P′are neighboring points on a geodesic having co-ordinates xi, xi+ dxirespectively. If theunit tangent at P is parallel displacement to p′ , then it will be identical with the actual unit tangent at thispoint now after parallel displacement from P , P′[Fig. 1].iFigure 1. Parallel displacement of a vector in curvilinear coordinates.The unit tangent has components30

Ozean Journal of Applied Sciences 1(1), 200833)(1,)(1,,2233322222111000 xfgxfgfgfg−= −−= −= −=Also, the reciprocal metric tensors are given as,)(11,)(1,1,12233322222111000xfgfxgfgfg−= −−= −= −=Now we get the non-zero components of Christofell symbols of 2 nd Kind,⎟⎠⎞∂∂−∂∂+⎜⎝⎛∂∂=Γkkkkxgxgxgg010110000121100001121xff∂∂=Γ∴⎟⎠⎞⎜⎝⎛∂∂−∂∂+∂∂=Γkkkkxgxgxgg000000110021101100121xff ∂∂=Γ∴⎟⎠⎞⎜⎝⎛∂∂−∂∂+∂∂=Γkkkkxgxgxgg111111111121111111121xff ∂∂= −Γ∴⎟⎠⎞⎜⎝⎛∂∂−∂∂+∂∂=Γkkkkxgxgxgg222222112221( )2212112211121xxff−∂∂= −Γ∴⎟⎠⎞⎜⎝⎛∂∂−∂∂+∂∂=Γkkkkxgxgxgg212112222121)(11)(12121222221xxffx−∂∂−=Γ122221121xff∂∂=Γ∴⎟⎠⎞⎜⎝⎛∂∂−∂∂+∂∂=Γkkkkxgxgxgg222222222221222222222211121fxxfx⎟⎟⎠⎞⎜⎜⎝⎛−∂∂−=Γ)(1222222xx−= −Γ∴⎟⎠⎞⎜⎝⎛∂∂−∂∂+∂∂=Γkkkkxgxgxgg333333223321

Ozean Journal of Applied Sciences 1(1), 200822 1 ⎛ 1−x ⎞2 ∂2∴ Γ33 = −2 2x22⎜−f⎟⎝ 2 ⎠ ∂x3 1 3k⎛ ∂g3k∂gk1∂g31⎞Γ31= g ⎜ + − ⎟21 3 k⎝ ∂x∂x∂x⎠3 1 1 ∂2Γ 31 =( f 2 ((1− x 2 ))2 2 1f2(1 − x2) ∂x3 1 1 ∂f2∴ Γ31= −2 f12 ∂x3 1 3k⎛ ∂g3k∂gk2∂g32⎞Γ32= g ⎜ + − ⎟2 2 3 k⎝ ∂x∂x∂x⎠3 1 1 ∂2Γ 32 = −( − f (((12 ))223− xf2(1− x2) ∂x23 x21∴ Γ32=21−x2Also, we get the non-zero components of Ricci tensor by applying Einstein law,0 1 2 3R = R + R + R +(13)11 110 111 112R1132( − f (1 − ))1 ⎛ 1 ∂f0⎞ ∂ ⎛ 1 ∂f1⎞ 1 ⎛ 1 ∂f1⎞ ⎛ 1 ∂f2⎞∴ R11=2⎜⎟ −⎜⎟ ++0 11 1 12⎜⎟⎜⎟(14)⎝ f ∂x⎠ ∂x⎝ f ∂x⎠ ⎝ f1∂x1⎠ ⎝ f2∂x1⎠0 1 2 3R 22 = R220+ R221+ R222+ R223(15)∴ ∂ ⎛ 1 1 ∂f2 ⎞ ⎛ 1 1 ∂f2 ⎞⎛1 1 ∂f2 ⎞22 =⎜−+ 21 2⎟ +⎜1 1 2⎟⎜1 1 2⎟∂x⎝ f ∂x⎠ ⎝ f ∂x⎠⎝f2∂x1⎠(16)R 33 = 0(17)0 1 2 300 = R000+ R001+ R002R003(18)R +∂ ⎛ 1 1 ∂f0⎞ 1 1 ∂f0 1 1 ∂f0∴ R00 = −x⎜12 f1x⎟ +(19)∂ ⎝ ∂1 ⎠ 2 f0∂x12 f1∂x1Due to rotational symmetry around the origin it is sufficient to write the field equations only for the equator( x 2 = 0); therefore, since they will be differentiated only once,Now we get the field equations from (14), (16), (17) and (18) by applying R = 0∂∂x1∂∂x∂∂x11⎛ 1 ∂f⎜⎝ f1∂x11⎛ 1 ∂f⎜⎝ f2∂x⎛ 1 ∂f⎜⎝ f1∂x01⎞ 1 ⎛ 1 ∂f⎟ =2⎜⎠ ⎝ f0∂x21⎞⎟ = 2 +⎠⎞ 1⎟ =⎠ f1f01f f1021⎛ ∂f⎜⎝ ∂x01⎞⎟⎠2⎛ ∂f⎜⎝ ∂x⎞⎟⎠21 ⎛ 1 ∂f1⎞+2⎜⎟⎝ f1∂x1⎠212⎞⎟⎠22⎛ 1 ∂f+⎜⎝ f2∂x212⎞⎟⎠αβ2(20a)(20b)(20c)34

Ozean Journal of Applied Sciences 1(1), 2008Beside these three equations the functions f 1 , f2,f3must fulfill also the equation of the determinant2f 0 f1f2= 1(20d)1 ∂f01 ∂f11 ∂f2i.e. + + = 0.f0∂x1f1∂x1f2∂x1From (20c) we can get∂ ⎛ 1 ∂f0 ⎞ 1 ∂f1∂f0x⎜1 f0x⎟ =(20e)∂ ⎝ ∂ 1 ⎠ f1f0∂x1∂x1Now integrating on both sides we get∂ ⎛ 1 ∂f⎞ ⎛⎜0 1⎟ = ∫⎜∂x1⎝ f0∂x1⎠ ⎝ f1f1 ∂f01 ∂f0∂=f ∂xf f ∂x∫ ∂ x∫011∂f10110∂f∂x11( f )1∂f∂x0∴ = af1, where a is integration constantf0∂x1Adding (20a) and (20e) and with help of (20d), we get∂∂x1∂⇒∂x⎛ 1 ∂f⎜⎝ f1∂x1∂⇒ − 2∂x11⎛ 1 ∂f⎜⎝ f1∂x11 ∂f+f ∂x11⎛ 1⎜⎝ f201+f001⎞ 1⎟ =⎠ f1f001∂f∂x⎟ ⎞⎠11∂f0⎞ ⎛ 1 ∂f⎟ =⎜∂x1⎠ ⎝ f2∂x∂f2⎞ ⎛ 1 ∂f⎟ = 3⎜∂x1⎠ ⎝ f2∂xBy integrating on both sides, we get21⎞⎟⎠2∂f∂x∂ ⎛ 1 ∂f⎞ ⎛ 1 ∂f⎞2 dx∫ ⎜ 2⎟ = ∫ ⎜ 2dx13∂x⎟ 1 ⎝ f2∂x1⎠ ⎝ f2∂x1⎠− 1⎛ 1 ∂f2⎞ ⎛ 1 ∂f2⎞− 2 ⎜⎟ = 3⎜⎟∫ dx⎝ f2∂x1⎠ ⎝ f2∂x1⎠1 ∂f2f2∂x1⇒ − 2 = 3x+ c2 1⎛ 1 ∂f⎞2⎜f x⎟⎝ 2∂1 ⎠1 3 c⇒ = x1+⎛ 1 ∂f2 ⎞ 2 2⎜f2x⎟⎝ ∂ 1 ⎠Where c is integration constantBy integrating once more⇒1221⎞⎟⎠01221 ⎛ 1 ∂f1⎞+2⎜⎟⎝ f1∂x1⎠1 ⎛ 1 ∂f+2⎜⎝ f1∂x112⎛ 1 ∂f+⎜⎝ f2∂x1 ∂f+f ∂x001⎞⎟⎠221⎞⎟⎠21 ⎛+2⎜⎝1f0∂f∂x01⎞⎟⎠235

Ozean Journal of Applied Sciences 1(1), 2008Now introduce these values of the functions f in the expression (12) of the line element and get the exactsolution of Einstein’s problem in polar coordinates22 ⎛ a ⎞ 2 ⎛ 1 1 ⎞ 2 2 dx22 2 2ds = ⎜1 − ⎟dx0− ⎜ dx1− R − R dx3( )( 1−x2)4⎟2⎝ R ⎠ ⎝ R 1−aR ⎠ 1−x22⎛ a ⎞ 2 ⎛ 1 1 ⎞ 4 2 2 sinθdθ=2 2⎜1− ⎟dt− ⎜ ⎟RdR − R − R dφ( sinθ)R4⎝ ⎠ ⎝ R 1−aR ⎠sinθ2 ⎛ a ⎞ 2 ⎛ 1 ⎞ 2 2 22ds = ⎜1 − ⎟dt− ⎜ ⎟dR− R ( dθ + sinθdφ)(22)⎝ R ⎠ ⎝1−aR ⎠Hence equation (22) is the required solution of Einstein equation which is also called the another form ofSchwarzschild solution.⎛ R − a ⎞This expression has a singularity at distance given by ⎜ ⎟ = 0.Thus R = a represents the⎝ R ⎠boundary of the isolated particle. The solution (22) holds in empty space outside the spherical isolatedparticle whose radius must be longer than a . On the other hand, two singularities one is intrinsicsingularity at r = 0 and another is coordinate singularity at r = 2Mwere occurred in Schwarzschildsolution which was extensively investigated by Biswas (Biswas, 2008). Another investigation was madeon curvature invariant by Biswas et al. (Biswas, Mallik, Parvin and Islam, 2008).UNIQUENESS OF THE SOLUTIONThe uniqueness of the solution resulted spontaneously through the present calculation. Without thecontinuity condition it would have44−3 −( 3x1+ c)3 ( r + c)3f 1 ==11−1 ( 3 ) 33 −− a x1+ c 1−a( r + c) 31=( ) 4333r + c − a( r + c) 11=44⎛ c ⎞ 3 ⎛ ⎞( ) ⎜ + ⎟ −3 cr 1 ar3⎜1+3⎟⎝ r ⎠ ⎝ r ⎠1=⎛2 ⎞4⎜4 c 4 ⎛ 4 ⎞⎛c ⎞ 1 ⎟ ⎛ ⎞⎟⎜⎟ + −3 cr 1++ ⎜ −1LLLar ⎜1+ ⎟⎜ 3333⎟3⎝ r ⎝ 3 ⎠⎝r ⎠ 2!⎠ ⎝ r ⎠1=⎧⎫⎪⎛ 23 4 c 2 1 ⎞⎪⎨⎜c+ + ⎟acr r+ LLL.a −253⎬3 r 3 3⎪⎩⎝ r ⎠r ⎪⎭Now neglecting more than 2 nd order then we get37

Ozean Journal of Applied Sciences 1(1), 200814⎛4 c a ⎞r ⎜1+− ⎟3⎝ 3 r r ⎠This expression, together with the corresponding expansions of f 0 , f 2,f3, satisfies up to the sameaccuracy all the conditions of the problem. Within this expression the condition of continuity does notintroduce anything new, since discontinuous occur spontaneously only in the origin. Then the two constantsa and c appear to remain arbitrary, hence the problem would be physically undetermined. The exactsolution reaches that in reality, by extending the approximation, the discontinuity does not occur at the133origin, but at r = ( a − c)3, and that one must set just c = aFor the discontinuity to go in the origin with the approximation in powers of a and c should survey veryclosely the law of the coefficient in order to recognize the necessity of this link between a and c .CONCLUSIONThis solution is of great importance on account of the fact that it provides a treatment of gravitational fieldsurrounding the sun. To find out the Gravitational field of a mass point we use the Schwarzschild lineelement, the Einstein Law of Gravitation and Ricci theorem in the empty space. Here we get a singularity atr = a , whereas there were two singularities in Schwarzschild another solution. The occurrence of asingularity in the field equation is a draw back of field theory. Einstein attributed this singularity to theelectromagnetic field associated with the interior structure of the particle. He expected to have singularityfree solutions of the field equations of the total field. By total field, we mean a field which is not onlyconcerned with gravitational aspects but it is also concerned with electromagnetic field as well as mesonfield. Thus we may conclude in such a way that the singularity is nothing but the artifact of transformationof coordinates.REFERENCESAtwater, H. A. (1974). Introduction to General Relativity. Pergamon Press, New York, USA.Bergman, P. G. (1942). Introduction to the Theory of Relativity. Perentese Hall, Englewood, New Jersey.Biswas, M. H. A. (2008). Studies on Mathematics and Physics of Collapsing Stars. M. Phil Thesis (notpublished) Department of Mathematics, University of Rajshahi, Rajshahi, Bangladesh.Biswas, M. H. A., Mallik, U. K., Parvin, S. & Islam, M. A. (2008). Curvature Invariants ofSpherically Symmetric Schwarzschild Solution without Cosmological Constant. Journal ofApplied Sciences Research, 4(1), 16-31.Lawden. D. F. (1968). An Introduction to Tensor Calculus and Relativity. Second edition. Chapmanand Hall Ltd. London.Schwarzschild, K. (1916). On the Gravitational Field of a Mass Point According to Einstein’s Theory.Sitzungsberichte der K¨oniglich Preussischen Akademie der Wissenschaften zu Berlin, Phys.Math. Klasse, 189-196 (Germany).38