×

zbMATH — the first resource for mathematics

Kerr-Schild symmetries. (English) Zbl 0982.83015
Summary: We study continuous groups of generalized Kerr-Schild transformations and the vector fields that generate them in any \(n\)-dimensional manifold with a Lorentzian metric. We prove that all these vector fields can be intrinsically characterized and that they constitute a Lie algebra if the null deformation direction is fixed. The properties of these Lie algebras are briefly analyzed and we show that they are generically finite-dimensional but that they may have infinite dimension in some relevant situations. The most general vector fields of the above type are explicitly constructed for the following cases: any two-dimensional metric, the general spherically symmetric metric and deformation direction, and the flat metric with parallel or cylindrical deformation directions.

MSC:
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
53C80 Applications of global differential geometry to the sciences
17B81 Applications of Lie (super)algebras to physics, etc.
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Kerr, R. P., and Schild, A. (1965). In Proceeding of the Galileo Galilei Centenary Meeting on General Relativity, Problems of Energy and Gravitational Waves, G. Barbera, ed., Comiato Nazionale per le Manifestazione Celebrative, Florence, pp. 222–233.
[2] Trautman A. (1962). In Recent Developments on General Relativity (Pergamon Press, New York).
[3] Kerr, R. P. · Zbl 0112.21904
[4] Kerr, R. P., and Schild A. (1969). Proc. Symp. Appl. Math. 17, 199; Debney, G. C., Kerr, R. P., and Schild A
[5] Vaidya, P
[6] Vaidya,
[7] Mas, L. (1969)
[8] Kramer, D., Stephani, H., Herlt, E., and MacCallum, M. A. H. (1980). Exact Solutions of Einstein’s Field Equations (Cambridge University Press, Cambridge). · Zbl 0449.53018
[9] Thompson, A
[10] Taub, A. H. (1981). Ann. Phys. (NY) 134, 326–372. · Zbl 0467.53036
[11] Bilge, A. K., and Gürses, M. (1982). In ”XI International Colloquiam on Group Theoretical Methods in Physics”, M. Cerdaroglu and E. Inönü, (Springer Verlag, Istanbul), pp. 252–255
[12] Nahmad-Achar, E · Zbl 0651.53052
[13] Coll, B., (1999). In Relativity and Gravitation in General. Proceedings of the Spanish Relativity Meeting in Honour of the 65th Birthday of L. Bel, J. Martin, E. Ruiz, F. Atrio, and A. Molina, eds. (World Scientific, Singapore), pp. 91–98.
[14] Talbot, C. J. (19 · Zbl 0186.58701
[15] Martín, J., and Senovilla, J. M. M. · Zbl 0608.76127
[16] Magli, · Zbl 0842.73017
[17] Gergely, L. Å., and Perjés,
[18] Xanthopoulos, B. C · Zbl 0978.83509
[19] Katzin, G. H., Levine, J., and Davis, W. R · Zbl 0176.19402
[20] Ehlers, J., and Kundt, W. (1962). in Gravitation: An Introduction to Current Research, ed. L. Witten (Wiley, New York.)
[21] Vaidya, P. C. (1951). P
[22] Defrise-Carter L. (19 · Zbl 0322.53008
[23] Hildebrandt, S. R., (2000). Preprint.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.