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Einsteinian Gravitational Field of a Heterogeneous Fluid Sphere

Published online by Cambridge University Press:  20 January 2009

Jyotirmaya Ghosh
Affiliation:
Dacca University
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The field-equations of gravitation in Einstein's theory have been solved in the case of an empty space, giving rise to de Sitter's spherical world. In the case of homogeneous matter filling all space, the solution gives Einstein's cylindrical world. The field corresponding to an isolated particle has been obtained by Schwarzchild. He has also obtained a solution for a fluid sphere with uniform density, a problem treated also by Nordström and de Donder. A new solution of the gravitational equations has been obtained in this paper, which corresponds to the field of a heterogeneous fluid sphere, the density at any point being a certain function of the distance of the point from the centre. The law of density is quite simple and such as to give finite density at the centre and gradually diminishing values as the distance from the centre increases, as might be expected of a natural sphere of fluid of large radius. The general problem of the fluid sphere with any arbitrary law of density cannot be solved in exact terms. It will be seen, however, from a theorem obtained in this paper, that the solution depends on a linear differential equation of the second order with variable coefficients involving the density, and thus the laws of density for which the problem admits of exact solution are those for which the above coefficients satisfy the conditions of integrability of the differential equation. An approximate solution for any law of density may be obtained by the method of series.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1925

References

1 Eddington, , Mathematical Theory of Relativity, Art. 45.Google Scholar

2 Ibid, Art. 67.

3 Ibid, Art. 38. Über das Gravitationsfield eines Massenpunktes nach der Einsteinschen Theorie, Berlin-Sitzungsberichte, 1916, p. 189.Google Scholar

4 4Über das Gravitationsfield einer Kugel aus incompressibler Flüssigkeit, Berlin Sitz., 1916, p. 426.Google Scholar

5 Calculation of some special cases in Einstein's Theory of Gravitation, Proc. Amsterdam Acad., 21 (1919).Google Scholar

6 La Gravifique Einsteinniene.

* Cf. Combridge, Phil. Mag., Jan. 1920.