Einstein Equations in the Case of Static Cylindrical Symmetry and the Diagonal Stress-Energy Tensor with Mutually Proportional Components

  • S. B. Grigoryev Dnipropetrovsk National University
  • A. B. Leonov Dnipropetrovsk National University
Keywords: Einstein equations, cylindrical symmetry, stress-energy tensor

Abstract

The Einstein equations with the stress-energy tensor in the form of a diagonal matrix with mutually proportional components are studied in the static cylindrically symmetric case. Several known exact solutions fall into this case (static electric field, some perfect fluid solutions, and solution with the cosmological constant). Coefficients of proportionality in the stress-energy tensor serve as parameters that allow studying a more general case (as well as obtaining new solutions for particular values of these coefficients). The initial system of equations is simplified and transformed into a system of two first-order ordinary differential equations. An exact solution is found for a broad set of parameters. The equilibrium points of the system of equations are considered, and the qualitative behavior of the solutions near the hyperbolic equilibrium points is studied.

References



  1. H. Stephani et al., Exact Solutions of Einstein's Field Equations (CUP, Cambridge, 2003).
     https://doi.org/10.1017/CBO9780511535185

  2. D.M. Chitre, R. Giiven, and Y. Nutku, J. Math. Phys. 16, 475 (1975).
     https://doi.org/10.1063/1.522569

  3. N. Van den Bergh and P. Wils, J. Phys. A 16, 3843 (1983).
     https://doi.org/10.1088/0305-4470/16/16/022

  4. M.A.H. MacCallum, J. Phys. A 16, 3853 (1983).
     https://doi.org/10.1088/0305-4470/16/16/023

  5. A.K. Raychaudhuri, Ann. Phys. (USA) 11, 501 (1960).
     https://doi.org/10.1016/0003-4916(60)90009-9

  6. H. Stephani et al., Exact Solutions of Einstein's Field Equations (CUP, Cambridge, 2003), p. 345 (formula 22.14).

  7. A.B. Evans, J. Phys. A 10, 1303 (1977).
     https://doi.org/10.1088/0305-4470/10/8/009

  8. T.G. Philbin, Class. Quantum Grav. 13, 1217 (1996).
     https://doi.org/10.1088/0264-9381/13/5/032

  9. S. Haggag and F. Desokey, Class. Quantum Grav. 13, 3221 (1996).
     https://doi.org/10.1088/0264-9381/13/12/012

  10. S. Haggag, Gen. Relativ. Gravit. 31, 1169 (1999).
     https://doi.org/10.1023/A:1026756203992

  11. W. Davidson, Gen. Relativ. Gravit. 22, (1990).

  12. A. Krasinski, Rep. Math. Phys. 14, (1978).

  13. K.A. Bronnikov, J. Phys. A 12, 201 (1979).
     https://doi.org/10.1088/0305-4470/12/2/007

  14. A. Krasinski, Acta Phys. Polon. B 6, 223 (1975).

  15. M.P. Korkina, S.B. Grigoryev, Ukr. Fiz. Zh. 29, 1153 (1984).

  16. L. Herrera, G. Le Denmat, G. Marcilhacy, and N.O. Santos, Int. J. Mod. Phys. D 14, 657 (2005).
     https://doi.org/10.1142/S0218271805006626

  17. F.C. Mena, R. Tavakol, and R. Vera, Generalisations of the Einstein-Strauss model to cylindrically symmetric settings, arXiv:gr-qc/0405043v1.

  18. P. Tod and F.C. Mena, Phys. Rev. D 70, 104028 (2004).
     https://doi.org/10.1103/PhysRevD.70.104028

  19. J.M.M. Senovilla and Raul Vera, Class. Quant. Grav. 17, 2843 (2000).
     https://doi.org/10.1088/0264-9381/17/14/314

  20. Masafumi Seriu, Phys. Rev. D 69, 124030 (2004).
     https://doi.org/10.1103/PhysRevD.69.124030

  21. C. Chicone and B. Mashhoon, Phys. Rev. D 83, 064013 (2011).
     https://doi.org/10.1103/PhysRevD.83.064013

  22. A. Ashtekar and M. Varadarajan, Phys. Rev. D 50, 4944 (1994).
     https://doi.org/10.1103/PhysRevD.50.4944

  23. . S.B. Grigoryev, Proc. Int. Conf. on Gen. Relativ. Gravit. (GR-13) (1992).

  24. S.B. Grigoryev, Kinem. Fiz. Nebes. Tel 10, 25 (1994).

  25. N.N. Bautin and E.A. Leontovich, Methods and Procedures of the Qualitative Study of Dynamical Systems on a Plane (Nauka, Moscow, 1990) (in Russian).

  26. S.L. Parnovs'kyi and O.Z. Gaidamaka, J. of Phys. Stud. 8, No. 4, 308 (2004).

  27. Shawn J. Kolitch, Qualitative Analysis of Brans-Dicke Universes with a Cosmological Constant, e-print arXiv:grqc/9409002v2 (1995).

  28. H. Stephani et al., Exact Solutions of Einstein's Field Equations (CUP, Cambridge, 2003), p. 342 (formula 22.4a).

  29. H. Stephani et al., Exact Solutions of Einstein's Field Equations (CUP, Cambridge, 2003), p. 344 (formula 22.11).

  30. M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra (Academic Press, San Diego, 1974).

  31. H. Stephani et al., Exact Solutions of Einstein's Field Equations (CUP, Cambridge, 2003), p. 63.
     https://doi.org/10.1017/CBO9780511535185


Published
2018-10-11
How to Cite
Grigoryev, S., & Leonov, A. (2018). Einstein Equations in the Case of Static Cylindrical Symmetry and the Diagonal Stress-Energy Tensor with Mutually Proportional Components. Ukrainian Journal of Physics, 58(9), 894. https://doi.org/10.15407/ujpe58.09.0894
Section
Astrophysics and cosmology