Second-Order Approximate Symmetries of the Geodesic Equations for the Reissner–Nordström Metric and

Ibrar Hussain ; Fazal M. Mahomed ; Asghar Qadir

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

Following the use of approximate symmetries for the Schwarzschild spacetime by A. H. Kara, F. M. Mahomed and A. Qadir (Nonlinear Dynam., to appear), we have investigated the exact and approximate symmetries of the system of geodesic equations for the Reissner–Nordström spacetime (RN). For this purpose we are forced to use second order approximate symmetries. It is shown that in the second-order approximation, energy must be rescaled for the RN metric. The implications of this rescaling are discussed.

Détail

Keywords: Reissner–Nordström metric; geodesic equations; second-order approximate symmetries.

Bibliographie

[1] Kara A. H., Mahomed F. M., Qadir A., “Approximate symmetries and conservation laws of the geodesic equations for the Schwarzschild metric”, Nonlinear Dynam., 51:1–2 (2008), 183–188 | MR | Zbl

[2] Kramer D., Stephani H., MacCullum M. A. H., Herlt E., Exact solutions of Einstein field equations, Cambridge University Press, Cambridge, 1980 | Zbl

[3] Landau L. D., Lifshitz E. M., The classical theory of fields, Pergamon Press, 1975 | MR

[4] Weber J., Wheeler J. A., “Reality of the cylindrical gravitational waves of Einstein and Rosen”, Rev. Modern Phys., 29 (1957), 509–515 | DOI | MR | Zbl

[5] Ehlers J., Kundt W., “Exact solutions of gravitational field equations”, Gravitation: An Introductoion to Current Research, ed. L. Witten, Wiley, New York, 1962, 49–101 | MR

[6] Komar A., “Asymtotic covariant conservation laws for gravitational radiation”, Phys. Rev., 127 (1962), 1411–1418 ; Komar A., “Positive-definite energy density and global consequences for general relativity”, Phys. Rev., 129 (1963), 1873–1876 | DOI | MR | Zbl | DOI | MR | Zbl

[7] Matzner R., “Almost symmetric spaces and gravitational radiation”, J. Math. Phys., 9 (1968), 1657–1668 | DOI | MR

[8] Isaacson R. A., “Gravitational radiation in the limit of high frequency. I. The linear approximation and geometric potics”, Phys. Rev., 166 (1968), 1263–1271 | DOI

[9] York J. W., “Covariant decomposition of symmetric tensor in the theory of gravitation”, Ann. Inst. H. Poincaré Sect. A (N.S.), 21 (1974), 319–332 | MR

[10] Taub A. H., “Empty spacetime admitting a three parameter groups of motions”, Ann. Math., 53 (1951), 472–490 | DOI | MR | Zbl

[11] Bona C., Carot J., Palenzueala-Luque C., “Almost-stationary motions and gauge conditions in general relativity”, Phys. Rev. D, 72 (2005), 124010, 5 pp., ages ; gr-qc/0509015 | DOI | MR

[12] Spero A., Baierlein R., “Approximate symmetry groups of inhomogeneous metrics”, J. Math. Phys., 18 (1977), 1330–1340 ; Spero A., Baierlein R., “Approximate symmetry groups of inhomogeneous metrics: examples”, J. Math. Phys., 19 (1978), 1324–1334 | DOI | Zbl | DOI | Zbl

[13] Aminova A. V., “Projective transformations and symmetries of differential equations”, Sb. Math., 186 (1995), 1711–1726 | DOI | MR | Zbl

[14] Feroze T., Mahomed F. M., Qadir A., “The connection between isometries and symmetries of the geodesic equations of the underlying spaces”, Nonlinear Dynam., 45 (2006), 65–74 | DOI | MR | Zbl

[15] Math. USSR Sb., 64 (1989), 427–441 | DOI | MR

[16] Ibragimov N. H., Elementary Lie group analysis and ordinary differential equations, Wiley, Chichester, 1999 | MR

[17] Transport Theory and Stat. Phys., 1 (1971), 186–207 | DOI | MR | Zbl

[18] Bluman G., “Connections between symmetries and conservation laws”, SIGMA, 1 (2005), 011, 16 pp., ages ; math-ph/0511035 | MR | Zbl

[19] Kara A. H., Mahomed F. M., “A basis of conservation laws for partial differential equations”, J. Nonlinear Math. Phys., 9 (2002), 60–72 | DOI | MR

[20] Bessel-Hagen E., “Über die Erhaltungssätze der Elektrodynamik”, Math. Ann., 84 (1921), 258–276 | DOI | MR | Zbl

[21] Ibragimov N. H., Transformation groups applied to mathematical physics, Reidel, Boston, 1985 | MR | Zbl

[22] Olver P. J., Applications of Lie groups to differential equations, Springer-Verlag, New York, 1993 | MR

[23] Bluman G., Kumei S., Symmetries and differential equations, Springer-Verlarg, New York, 1989 | MR | Zbl

[24] Lie S., Sophus Lie's 1880 transformation group paper, Math Sci. Press, Brookline MA, 1975, translated by Michael Ackerman. Comments by Robert Hermann | MR

[25] Stephani H., Differential equations: their solutions using symmetry, Cambridge University Press, New York, 1989 | MR | Zbl

[26] Hawking S. W., Ellis G. F. R., The large scale structure of spacetime, Cambridge University Press, Cambridge, 1973 | MR | Zbl

[27] Caviglia G., “Dynamical symmetries: an approach to Jacobi fields and to constants of geodesic motion”, J. Math. Phys., 24 (1983), 2065–2069 | DOI | MR | Zbl

[28] Katzin G. H., Levine J., “Geodesic first integral with explicit path-parameter dependence in Riemannian spacetime”, J. Math. Phys., 22 (1981), 1878–1891 | DOI | MR | Zbl

[29] Baikov V., Gazizov R. K., Ibragimov N. H., “Differential equations with a small parameter: exact and approximate symmetries”, CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 3, ed. N. H. Ibragimov, CRC Press, Boca Raton, Florida, 1996, 217–282

[30] Kara A. H., Mahomed F. M., Ünal G., “Approximate symmetries and conservation laws with applications”, Int. J. Theor. Phys., 38 (1999), 2389–2399 | DOI | MR | Zbl

[31] Gazizov R. K., “Lie algebras of approximate symmetries”, J. Nonlinear Math. Phys., 3 (1996), 96–101 | DOI | MR | Zbl

[32] Baikov V., Gazizov R. K., Ibragimov N. H., Mahomed F. M., “Closed orbits and their stable symmetries”, J. Math. Phys., 35 (1994), 6525–6535 | DOI | MR | Zbl

[33] Fushchich W. I., Shtelen W. M., “On approximate symmetry and approximate solutions of the non-linear wave equation with a small parameter”, J. Phys. A: Math. Gen., 22 (1989), L887–L890 | DOI | MR | Zbl

[34] Pakdemirli M., Yürüsoy M., Dolapci I., “Comparison of approximate symmetry methods for differential equatons”, Acta Appl Math., 80 (2004), 243–271 | DOI | MR | Zbl

[35] Valenti A., “Approximate symmetries for a model describing dissipative media”, Proceedings of 10th International Conference in Modern Group Analysis (MOGRAN X) (October 24–31, 2004, Larnaca, Cyprus), ed. N. H. Ibrahimov, C. Sophocleous and P. A. Damianou, 2005, 236–243

[36] Hall G. S., Symmetries and curvature structure in general relativity, World Scientific, Singapore, 2004 | MR

[37] Misner C. W., Thorne K. S., Wheele J. A., Gravitation, W. H. Freeman and Company, San Francisco, 1973 | MR

[38] Inverno D. R., Introducing Einstein's relativity, Oxford University Press, New York, 1992 | Zbl

[39] Qadir A., “General relativity in terms of forces”, Proceedings of Third Regional Conference on Mathematical Physics, ed. F. Hussain and A. Qadir, World Scientific, 1990, 481–490 | MR

[40] Qadir A., “The gravitational force in general relativity”, M.A.B. Beg Memorial Volume, eds. A. Ali and P. Hoodbhoy, World Scientific, 1991, 159–178

[41] Qadir A., Sharif M., “The relativistic generalization of the gravitational force for arbitary spacetimes”, Nuovo Cimento B, 117 (1992), 1071–1083 | DOI | MR

[42] Qadir A., Zafarullah I., “The pseudo-Newtonian force in time-varying spacetimes”, Nuovo Cimento B, 111 (1996), 79–84 | DOI

[43] Qadir A., Sharif M., “General formula for the momentum imparted to test particles in arbitrary spacetimes”, Phys. Lett. A, 167 (1992), 331–334 ; gr-qc/0701098 | DOI | MR