The Einstein equation solution inside a ball with uniform density
DOI:
https://doi.org/10.21638/11701/spbu10.2024.101Abstract
A great number of solutions of the Einstein field equation are known. They describe the gravitational field in the empty space-time, in the space-time with electromagnetic field and for a ball filled with a liquid under pressure. The present work is devoted to gravitational field generated by some mass distribution. One of the simplest cases is considered, when mass is uniformly distributed inside a ball and is not moving. The boundary problem for the Einstein equation is formulated. Solution outside the ball is the Schwartzschild solution in vacuum. The coordinates at which the Schwartzschild solution is written are different from the coordinates used in equations for components of the metric tensor inside the ball. Relations between internal and external coordinates are found on the ball surface. They allow to use the Schwartzschild solution for formulation of boundary conditions for internal solution. The solution of the boundary problem is found for the case of weak field. This solution can be used as an example in the analysis of laws of conservation for the gravitational field, in which interaction of mass with field generated by the mass gives a contribution to momentum and energy of the gravitational field.
Keywords:
the Einstein equation, metric tensor, ball with uniform mass density
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References
Stephani H., Kramer D., MacCallum M. A. H., Hoenselaers C., Herlt E. Exact solutions of Einstein's field equations. Ed. 2. Cambridge, Cambridge University Press, 2003, 701 p.
Drivotin O. I. Rigorous definition of the reference frame. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2014, vol. 10, iss. 4, pp. 25–36.
Drivotin O. I. Ob opredelenii geometrii prostranstva-vremeni [On the determination of spacetime geometry]. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2022, vol. 18, iss. 3, pp. 316–327. https://www.doi.org/10.21638/11701/spbu10.2022.302 (In Russian)
Drivotin O. I. Covariant description of phase space distributions. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2016, vol. 12, iss. 3, pp. 39–52. https://www.doi.org/10.21638/11701/spbu10.2016.304
Einstein A. Die grundlage der allgemeinen relativitatsteorie. Ann. d. Phys., 1916, vol. 49, pp. 769–822.
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Wang C.-C. Mathematical principles of mechanics and electromagnetism. Pt B. Electromagnetism and gravitation. New York, Plenum Publ., 1979, 386 p.
Gron O., Hervik S. Einstein's general theory of relativity. New York, Springer Verlag Publ., 2007, 558 p.
Drivotin O. I. O plotnosti potoka impulsa gravitatsionnogo polya [On momentum flow density of the gravitational field]. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2021, vol. 17, iss. 2, pp. 137–147. https://www.doi.org/10.21638/11701/spbu10.2021.204 (In Russian)
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