  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
in January 1966
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V.V. Vasil'ev and L.V. Fedorov, "Stress State of an Elastic Ball in a Spherically Symmetric Gravitational Field," Mech. Solids. 49 (4), 370381 (2014) 
Year 
2014 
Volume 
49 
Number 
4 
Pages 
370381 
DOI 
10.3103/S0025654414040025 
Title 
Stress State of an Elastic Ball in a Spherically Symmetric Gravitational Field 
Author(s) 
V.V. Vasil'ev (Moscow State Aviation Technological University, ul. Orshanskaya 3, Moscow, 121552 Russia, vvvas@dol.ru)
L.V. Fedorov (Moscow State Aviation Technological University, ul. Orshanskaya 3, Moscow, 121552 Russia) 
Abstract 
We consider a spherically symmetric static problem of general relativity whose solution was obtained in 1916 by Schwarzschild for a metric form of a special type. This solution determines the metric coefficients of the exterior and interior Riemannian spaces generated by a gravitating solid ball of constant density and includes the socalled gravitational radius r_{g}. For a ball of outer radius R=r_{g}, the metric coefficients are singular, and hence the radius r_{g} is traditionally assumed to be the radius of the event horizon of an object called a black hole. The solution of the interior problem obtained for an incompressible ideal fluid shows that the pressure at the ball center increases without bound for R=9/8r_{g}, which is traditionally used for the physical justification of the existence of black holes. The discussion of Schwarzschild's traditional solution carried out in this paper shows that it should be generalized with respect to both the geometry of the Riemannian space and the elastic medium model. In this connection, we consider the general metric form of a spherically symmetric Riemannian space and prove that the solution of the corresponding static problem exists for a broad class of metric forms. A special metric form based on the assumption that the gravitation generating the Riemannian space inside a fluid ball or an elastic ball does not change the ball mass is singled out from this class. The solution obtained for the special metric form is singular with respect to neither the metric coefficients nor the pressure in the fluid ball and the stresses in the elastic ball. The obtained solution is compared with Schwarzschild's traditional solution. 
Keywords 
spherically symmetric solid, statics, gravitation, singular solution 
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Received 
24 May 2011 
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