Recursive structure in the definitions of gauge-invariant variables for   any order perturbations
  The construction of gauge-invariant variables for any order perturbations is discussed. Explicit constructions of the gauge-invariant variables for perturbations to 4th order are shown. From these explicit construction, the recursive structure in the definitions of gauge-invariant variables for any order perturbations is found. Through this recursive structure, the correspondence with the fully non-linear exact perturbations is briefly discussed. 
Kouji Nakamura
gr-qc astro-ph.CO hep-th math-ph math.MP