Journal of Applied Science Studies - Ozean Publications

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Ozean Journal of Applied Sciences 1(1), 2008 Now introduce these values of the functions f in the expression (12) of the line element and get the exact solution of Einstein’s problem in polar coordinates 2 2 ⎛ a ⎞ 2 ⎛ 1 1 ⎞ 2 2 dx2 2 2 2 ds = ⎜1 − ⎟dx0 − ⎜ dx1 − R − R dx3 ( ) ( 1− x2 ) 4 ⎟ 2 ⎝ R ⎠ ⎝ R 1− aR ⎠ 1− x2 2 ⎛ a ⎞ 2 ⎛ 1 1 ⎞ 4 2 2 sinθdθ = 2 2 ⎜1 − ⎟dt − ⎜ ⎟R dR − R − R dφ ( sinθ ) R 4 ⎝ ⎠ ⎝ R 1− aR ⎠ sinθ 2 ⎛ a ⎞ 2 ⎛ 1 ⎞ 2 2 2 2 ds = ⎜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 of Schwarzschild 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 isolated particle whose radius must be longer than a . On the other hand, two singularities one is intrinsic singularity at r = 0 and another is coordinate singularity at r = 2M were occurred in Schwarzschild solution which was extensively investigated by Biswas (Biswas, 2008). Another investigation was made on curvature invariant by Biswas et al. (Biswas, Mallik, Parvin and Islam, 2008). UNIQUENESS OF THE SOLUTION The uniqueness of the solution resulted spontaneously through the present calculation. Without the continuity condition it would have 4 4 − 3 − ( 3x1 + c) 3 ( r + c) 3 f 1 = = 1 1 − 1 ( 3 ) 3 3 − − a x1 + c 1− a( r + c) 3 1 = ( ) 4 3 3 3 r + c − a( r + c) 1 1 = 4 4⎛ c ⎞ 3 ⎛ ⎞ ( ) ⎜ + ⎟ − 3 c r 1 ar 3 ⎜1 + 3 ⎟ ⎝ r ⎠ ⎝ r ⎠ 1 = ⎛ 2 ⎞ 4⎜ 4 c 4 ⎛ 4 ⎞⎛ c ⎞ 1 ⎟ ⎛ ⎞ ⎟⎜ ⎟ + − 3 c r 1+ + ⎜ −1 LLL ar ⎜1 + ⎟ ⎜ 3 3 3 3 ⎟ 3 ⎝ r ⎝ 3 ⎠⎝ r ⎠ 2! ⎠ ⎝ r ⎠ 1 = ⎧ ⎫ ⎪ ⎛ 2 3 4 c 2 1 ⎞ ⎪ ⎨ ⎜ c + + ⎟ ac r r + LLL. a − 2 5 3 ⎬ 3 r 3 3 ⎪⎩ ⎝ r ⎠ r ⎪⎭ Now neglecting more than 2 nd order then we get 37
Ozean Journal of Applied Sciences 1(1), 2008 1 4⎛ 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 same accuracy all the conditions of the problem. Within this expression the condition of continuity does not introduce anything new, since discontinuous occur spontaneously only in the origin. Then the two constants a and c appear to remain arbitrary, hence the problem would be physically undetermined. The exact solution reaches that in reality, by extending the approximation, the discontinuity does not occur at the 1 3 3 origin, but at r = ( a − c)3 , and that one must set just c = a For the discontinuity to go in the origin with the approximation in powers of a and c should survey very closely the law of the coefficient in order to recognize the necessity of this link between a and c . CONCLUSION This solution is of great importance on account of the fact that it provides a treatment of gravitational field surrounding the sun. To find out the Gravitational field of a mass point we use the Schwarzschild line element, the Einstein Law of Gravitation and Ricci theorem in the empty space. Here we get a singularity at r = a , whereas there were two singularities in Schwarzschild another solution. The occurrence of a singularity in the field equation is a draw back of field theory. Einstein attributed this singularity to the electromagnetic field associated with the interior structure of the particle. He expected to have singularity free solutions of the field equations of the total field. By total field, we mean a field which is not only concerned with gravitational aspects but it is also concerned with electromagnetic field as well as meson field. Thus we may conclude in such a way that the singularity is nothing but the artifact of transformation of coordinates. REFERENCES Atwater, 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 (not published) Department of Mathematics, University of Rajshahi, Rajshahi, Bangladesh. Biswas, M. H. A., Mallik, U. K., Parvin, S. & Islam, M. A. (2008). Curvature Invariants of Spherically Symmetric Schwarzschild Solution without Cosmological Constant. Journal of Applied Sciences Research, 4(1), 16-31. Lawden. D. F. (1968). An Introduction to Tensor Calculus and Relativity. Second edition. Chapman and 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
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