Recursive structure in the definitions of gauge-invariant variables for any order perturbations

Abstract

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.

arXiv comment21 pages, no figure: (v2) one section and additional explanations are added. Accepted for publication to Classical and Quantum Gravity

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• Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part III: -- Realization of exact solutions
• Comparing a gauge-invariant formulation and a "conventional complete gauge-fixing approach" for $l=0,1$ mode perturbations on the Schwarzschild background spacetime
• Construction and applications of the manifestly gauge invariant expressions of the solutions of the cosmological perturbation theory
• Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part I : -- Formulation and odd-mode perturbations