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Approximate Noether symmetries of the geodesic equations for the charged-Kerr spacetime and rescaling of energy

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Abstract

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 re-scaling factor is compared with that for the Reissner–Nordström metric.

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Authors and Affiliations

  1. Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus of the College of Electrical and Mechanical Engineering, Peshawar Road, Rawalpindi, Pakistan

    Ibrar Hussain & Asghar Qadir

  2. Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa

    F. M. Mahomed

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  1. Ibrar Hussain
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  2. F. M. Mahomed
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  3. Asghar Qadir
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Correspondence to Ibrar Hussain.

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Hussain, I., Mahomed, F.M. & Qadir, A. Approximate Noether symmetries of the geodesic equations for the charged-Kerr spacetime and rescaling of energy. Gen Relativ Gravit 41, 2399–2414 (2009). https://doi.org/10.1007/s10714-009-0772-3

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