Approximate Noether Symmetries of the Geodesic Equations for the Charged-Kerr Spacetime and Rescaling of Energy

Details Approximate Noether Symmetries of the Geodesic Equations for the Charged-Kerr Spacetime and Rescaling of Energy Approximate Noether Symmetries of the Geodesic Equations for the Charged-Kerr Spacetime and Rescaling of Energy

2009-03-07

arxiv.org/abs/0903.1342v1

Gen.Rel.Grav.41:2399-2414,2009

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

17 Pages

Scientific paper

10.1007/s10714-009-0772-3

Using approximate symmetry methods for differential equations we have investigated the exact and approximate symmetries of a Lagrangian for the geodesic equations in the Kerr spacetime. Taking Minkowski spacetime as the exact case, it is shown that the symmetry algebra of the Lagrangian is 17 dimensional. This algebra is related to the 15 dimensional Lie algebra of conformal isometries of Minkowski spacetime. First introducing spin angular momentum per unit mass as a small parameter we consider first-order approximate symmetries of the Kerr metric as a first perturbation of the Schwarzschild metric. We then consider the second-order approximate symmetries of the Kerr metric as a second perturbation of the Minkowski metric. The approximate symmetries are recovered for these spacetimes and there are no non-trivial approximate symmetries. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the charged-Kerr metric has to be rescaled and the rescaling factor is $r$-dependent. This rescaling factor is compared with that for the Reissner-Nordstr\"{o}m metric.

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