Gauge-invariant variables in general-relativistic perturbations:   globalization and zero-mode problem
  An outline of a proof of the local decomposition of linear metric perturbations into gauge-invariant and gauge-variant parts on an arbitrary background spacetime is briefly explained. We explicitly construct the gauge-invariant and gauge-variant parts of the linear metric perturbations based on some assumptions. We also point out the zero-mode problem is an essential problem to globalize of this decomposition of linear metric perturbations. The resolution of this zero-mode problem implies the possibility of the development of the higher-order gauge-invariant perturbation theory on an arbitrary background spacetime in a global sense. 
Kouji Nakamura
gr-qc astro-ph.CO hep-th math-ph math.MP