Abstract
Third rank Killing tensors in (1 +1)-dimensional geometries are investigated andclassified. It is found that a necessary and sufficientcondition for such a geometry to admit a third rankKilling tensor can always be formulated as a quadratic PDE, oforder three or lower, in a Kahler type potential for themetric. This is in contrast to the case of first andsecond rank Killing tensors for which the integrability condition is a linear PDE. The motivation for studying higher rank Killing tensors in (1 +1)-geometries, is the fact that exact solutions of theEinstein equations are often associated with a first orsecond rank Killing tensor symmetry in the geodesicflow formulation of the dynamics. This is in particulartrue for the many models of interest for which thisformulation is (1 + 1)-dimensional, where just one additional constant of motion suffices forcomplete integrability. We show that new exact solutionscan be found by classifying geometries admitting higherrank Killing tensors.
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REFERENCES
Dietz, W., and Rüdiger, R. (1981). Proc. Roy. Soc. Lond. 375, 361.
Hietarinta, J. (1987). Phys. Rep. 147, 87.
Karlovini, M. (1998). Masters thesis, Stockholm University, http://vanosf.physto.se/exjobb/exjobb.html
Nakahara, M. (1990). Geometry, Topology and Physics (IOPP, Bristol and Philadelphia).
Penrose, R., and Walker, M. (1970). Commun. Math. Phys. 18, 265.
Rosquist, K. (1995). Class. Quantum Grav. 12, 1305.
Rosquist, K. (1996). In Proc. VII Marcel Grossmann Meeting (Stanford, 1994), R. T. Jantzen and M. Kaiser, ed. (World Scientific, Singapore), vol. 1, p.379.
Rosquist, K., and Goliath, M. (1998). Gen. Rel. Grav. 30, 1521.
Rosquist, K., and Uggla, C. (1991). J. Math. Phys. 32, 3412.
Rosquist, K., and Uggla, C. (1993). Mod. Phys. Lett. A 8, 2815.
Sklyanin, E. A. (1995). Prog. Theor. Phys. Suppl. 118, 35.
Sommers, P. (1973). J. Math. Phys. 14, 787.
Uggla, C., R. Jantzen R and Rosquist, K. (1995). Phys. Rev. D 51, 5522.
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Karlovini, M., Rosquist, K. Third Rank Killing Tensors in General Relativity. The (1 + 1)-dimensional Case. General Relativity and Gravitation 31, 1271–1294 (1999). https://doi.org/10.1023/A:1026724824465
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DOI: https://doi.org/10.1023/A:1026724824465