Conservation equations and the gravitational symplectic form

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Abstract
By considering space–times whose metric is given by a perturbation expansion away from a background which admits a Killing vector, conservation equations, based on the energy–momentum tensor, are derived to first and second orders in the perturbation expansion. To the first order, equations are derived independently of the Einstein field equations, and describe secular changes in the energy–momentum distribution of the matter fields. To the second order, a gravitational energy–momentum contribution arises from the conservation equations which may be constructed from the symplectic inner product on the solution space of the linearized Einstein field equations. Considering a similar scheme based on the Bel–Robinson tensor, it is shown that whilst first order conservation equations can be formulated, the lack of a symplectic form for the perturbed Bel–Robinson tensor implies the nonexistence of second order conservation equations, except when the background is flat. The results are applied to perturbations of a stationary black hole, and simple expressions are found for the mass and angular momentum fluxes, through the event horizon, due to a gravitational perturbation. By considering a monochromatic wave, it is seen that the conservation of the gravitational symplectic form reduces, in suitable coordinates, to the Wronskian condition of Teukolsky and Press.

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