Construction of gauge-invariant variables of linear metric perturbations on an arbitrary background spacetime

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Abstract
An outline of a proof of the decomposition of linear metric perturbations into gauge-invariant and gauge-variant parts on an arbitrary background spacetime that admits ADM decomposition is discussed. We explicitly construct the gauge-invariant and gauge-variant parts of the linear metric perturbations through the assumption of the existence of some Green functions. We also confirm the result through another approach. This implies that we can develop a higher-order gauge-invariant perturbation theory on an arbitrary background spacetime. Remaining issues to complete the general framework of the general-relativistic higher-order gauge-invariant perturbation theories are also discussed.

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