Abstract
This paper investigates the relationship between Killing Tensors and separable systems for the geodesic Hamilton-Jacobi equation in Riemannian and Lorentzian manifolds: locally, a separable system consists of the vector and covector associated with a separable coordinate. It is shown that there are only two types of separable system, those associated with local symmetry groups and those which can be obtained by a simple transformation from orthogonal systems. Some sufficient conditions for existence are given and some global problems are enumerated. The results are illustrated with a demonstration that the existence of separable systems in a certain class of {2, 2} space-times is a consequence of the algebraic properties of the Weyl tensor.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abraham, R., Marsden, J.E.: Foundations of mechanics. Reading, Mass.: Benjamin 1967
Bardeen, J.M.: Timelike and null geodesics in the Kerr metric, 1972. Les Houches Summer School, pp. 215–239. London-New-York-Paris: Gordon and Breach 1973
Boyer, R.H., Lindquist, R.W.: J. Math. Phys.8, 265 (1967)
Burgatti, P.: R. Acad. dei Lincei (Roma)20, 108 (1911)
Carter, B.: Phys. Rev.174, 1559 (1968); and: Black Hole Equilibrium States. 1972, Les Houches Summer School, pp. 57–214. London-New-York-Paris: Gordon and Breach 1973
Dall'Acqua, F.A.: Math. Ann.66, 398 (1908); and: Rend. di Palermo33, 341 (1912)
Eisenhart, L.P.: Ann. Math.35, 284 (1934)
Galgani, L., Scotti, A.: Nuovo Cimento2, 189 (1972)
Geroch, R.P.: J. Math. Phys.11, 1955 (1970)
Geroch, R.P.: Special topics in particle physics: mimeographed lecture notes: Chicago 1971
Hauser, I., Malhiot, R.J.: J. Math. Phys. (to appear) (1975)
Herman, R.: Differential geometry and the calculus of variations. New York: Academic Press 1968
Hughston, L.P., Penrose, R., Sommers, P., Walker, M.: Commun. math. Phys.27, 303 (1972); Hughston, L.P., Sommers, P.: Commun. math. Phys.32, 147 (1973); and: Commun. math. Phys.33, 129 (1973)
Iarov-Iarovoi, M.S.: J. Appl. Math. Mech.27, 1499 (1964)
Kerr, R.P.: Phys. Rev. Lett.11, 237 (1963)
Kostant, B.: Symplectic spinors. Symposia Math. to appear
Levi-Civita, T.: Math. Ann.49, 383 (1904)
Liouville, J.: J. de Math.11, 345 (1846)
Loomis, L.H., Sternberg, S.: Advanced calculus. Reading, Mass.: Addison Wesley 1968
Nijenhuis, A.: Nederl. Akad. Wetensch. Proc. Ser. A58, 390 (1955)
Onofri, E., Pauri, M.: J. Math. Phys.14, 1106 (1973)
Penrose, R.: Structure of space-time. In: De Witt, C., Wheeler, J.A. (Eds.): Battelle rencontres, pp. 121–235. New York: Benjamin 1968
Pirani, F.A.E.: Capt. 8 of A. Trautman's lectures. 1964 Brandeis Summer Institute, pp. 201–227. Englewood Cliffs, N. J.: Prentice-Hall 1965
Ricci, G., Levi-Civita, T.: Math. Ann.54, 178 (1901)
Stäckel, P.: Math. Ann.42, 537 (1893); and: C. R. Acad. Sci. Paris121, 489 (1895)
Serre, J.-P.: Lie groups and Lie algebras. New-York-Amsterdam: Benjamin 1965
Sommers, P.: J. Math. Phys.14, 787 (1973)
Sommers, P.: Killing tensors and type {2, 2} space-times. Ph. D. Austin 1973
Stewart, J.M., Walker, M.: Perturbations of space-times in general relativity. Preprint, Max-Planck-Institut, Munich 1973
Teukolsky, S.A.: Perturbations of rotating black holes, I. Press, W.A., Teukolsky, S.A.: Perturbations of rotating black holes, II. Preprint, California Institute of Technology 1973
Walker, M., Penrose, R.: Commun. Math. Phys.18, 265 (1970)
Walker, M.: Private communication
Author information
Authors and Affiliations
Additional information
Communicated by J. Ehlers
Research supported by the Science Research Council.
Rights and permissions
About this article
Cite this article
Woodhouse, N.M.J. Killing tensors and the separation of the Hamilton-Jacobi equation. Commun.Math. Phys. 44, 9–38 (1975). https://doi.org/10.1007/BF01609055
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01609055