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The decomposition of the linear-order metric perturbation is discussed in thecontext of the higher-order gauge-invariant perturbation theory. We show thatthe linear order metric perturbation is decomposed into gauge-invariant andgauge-variant parts on the general background spacetime which admits ADMdecomposition. This decomposition was an important premise of the generalframework of the higher order gauge-invariant perturbation theory proposed inthe papers [K. Nakamura, Prog. Theor. Phys. vol.110 (2003), 723; ibid. vol.113(2005), 481]. This implies that we can develop the higher-order gauge-invariantperturbation theory on generic background spacetime. Remaining issues tocomplete the general-framework of the higher-order gauge-invariant perturbationtheories are also discussed.

Decomposition of linear metric perturbations on generic background  spacetime -- Toward higher-order general-relativistic gauge-invariant  perturbation theory