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The flux integral for axisymmetric perturbations of static space-times

The axisymmetric perturbations of static space-times with prevailing sources (a Maxwell field or a perfect fluid) are considered; and it is shown how a flux integral can be derived directly from the relevant linearized equations. The flux integral ensures the conservation of energy in the attendant scattering of radiation and the sometimes accompanying transformation of one kind of radiation into another. The flux integral derived for perturbed Einstein-Maxwell space-times will be particularly useful in this latter context (as in the scattering of radiation by two extreme Reissner-Nordström black-holes) and in the setting up of a scattering matrix. And the flux integral derived for a space-time with a perfect-fluid source will be directly applicable to the problem of the non-radial oscillations of a star with accompanying emission of gravitational radiation and enable its reformulation as a problem in scattering theory.