Abstract
An outline of a proof of the decomposition of the linear metric perturbation into gauge-invariant and gauge-variant parts on an arbitrary background spacetime is discussed through an exlicit construction of gauge-invariant and gauge-variant parts. Although this outline is incomplete, yet, due to our assumptions, we propose a conjecture which states that the linear metric perturbation is always decomposed into its gauge-invariant and gageu-variant parts. If this conjecture is true, we can develop the higher-order gauge-invariant perturbation theory on an arbitrary background spacetime.
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