Short note on construction of gauge-invariant variables of linear metric   perturbations on an arbitrary background spacetime
  An outline of a proof of the decomposition of linear metric perturbations into gauge-invariant and gauge-variant parts on an arbitrary background spacetime which admits ADM decomposition is briefly discussed. We explicitly construct the gauge-invariant and gauge-variant parts of the linear metric perturbations based on some assumptions. This implies that we can develop the higher-order gauge-invariant perturbation theory on an arbitrary background spacetime. 
Kouji Nakamura
gr-qc astro-ph.CO hep-th math-ph math.MP