 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
|
| In English (Mech. Solids): | | 4760 |
|
| << Previous article | Volume 49, Issue 4 / 2014 | Next article >> |
| V.V. Vasil'ev and L.V. Fedorov, "Stress State of an Elastic Ball in a Spherically Symmetric Gravitational Field," Mech. Solids. 49 (4), 370-381 (2014) |
| Year |
2014 |
Volume |
49 |
Number |
4 |
Pages |
370-381 |
| 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 so-called gravitational radius rg. For a ball of outer radius R=rg, the metric coefficients are singular, and hence the radius rg 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=9/8rg, 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 |
| References |
| 1. | J. L. Synge,
Relativity: The General Theory
(Interscience, New York, 1960; Izd-vo Inostr. Lit., Moscow, 1963). |
| 2. | N. A. Kilchevskii,
Foundations of Tensor Calculus with Applications to Mechanics
(Naukova Dumka, Kiev, 1972)
[in Russian]. |
| 3. | L. D. Landau and E. M. Lifshits,
Field Theory
(Fizmatizd, Moscow, 1988)
[in Russian]. |
| 4. | K. Torn,
Black Holes and Folds of Time
(Fizmatgiz, Moscow, 2009)
[in Russian]. |
| 5. | V. V. Vasiliev and L. V. Fedorov,
"On the Gravitational Constant and Radius in Spherically Symmetric Problem
in General Relativity,"
Appl. Phys. Res.
5 (3), 116-122 (2013). |
| 6. | A. E. H. Love,
A Treatise on the Mathematical Theory of Elasticity
(Cambridge Univ. Press, Cambridge, 1927; ONTI HKGiP SSSR, Moscow-Leningrad, 1935). |
| 7. | R. P. Feynman, F. Morinigo, and W. Wagner,
Feynman Lectures on Gravitation
(Addison-Wesley, Massachusetts, 1995; Yanus-K, Moscow, 2000). |
| 8. | V. Z. Vlasov,
Strain Continuity Equations in Curvilinear Coordinates,
Selected Works, Vol. 1
(Izd-vo AN SSSR, Moscow, 1962)
[in Russian]. |
| 9. | V. V. Vasil'ev and L. V. Fedorov,
"An Elasticity Problem for a Gravitating Ball and Some Geometrical Effects,"
Izv. Akad. Nauk.
Mekh. Tverd. Tela,
No. 1, 84-92 (2003).
[Mech. Solids (Engl. Transl.)
38 (1), 67-74 (2003)]. |
| 10. | V. V. Vasiliev and L. V. Fedorov,
"On Singular Solutions in Spherically Symmetric Static Problem of General Relativity,"
Appl. Phys. Res.
4 (2), 188-174 (2012). |
| 11. | V. V. Vasiliev and S. A. Lurie,
"On the Solution Singularity in the Plane Elasticity Problem for a Cantilever Strip,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 4, 40-49 (2013).
[Mech. Solids (Engl. Transl.)
48 (4), 388-396 (2013)]. |
| 12. | Yu. S. Vladimirov,
Classical Theory of Gravitation
(LIBROKOM, Moscow, 2009)
[in Russian]. |
| 13. | V. A. Fock,
The Theory of Space Time and Gravitation
(Pergamon Press, Oxford, London, New York, Paris, 1964; LKI, Moscow, 2007). |
| 14. | V. V. Vasiliev and L. V. Fedorov,
"On the Solution of Spherically Symmetric Static Problem
for a Fluid Sphere in General Relativity,"
Appl. Phys. Res.
6 (3), 40-49 (2014). |
|
| Received |
24 May 2011 |
| Link to Fulltext |
|
| << Previous article | Volume 49, Issue 4 / 2014 | Next article >> |
|
 If you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter
|
|
Russian 
English