Gauge-invariant perturbation theory on the Schwarzschild background spacetime Part I : -- Formulation and odd-mode perturbations

This is the Part I paper of our series of full papers on a gauge-invariant {\it linear} perturbation theory on the Schwarzschild background spacetime which was briefly reported in our short papers [K.~Nakamura, Class. Quantum Grav. {\bf 38} (2021), 145010; K.~Nakamura, Letters in High Energy Physics {\bf 2021} (2021), 215.]. We first review our general framework of the gauge-invariant perturbation theory, which can be easily extended to the {\it higher-order} perturbation theory. When we apply this general framework to perturbations on the Schwarzschild background spacetime, a gauge-invariant treatments of $l=0,1$ mode perturbations are required. On the other hand, in the current consensus on the perturbations of the Schwarzschild spacetime, gauge-invariant treatments for $l=0,1$ modes are difficult if we keep the reconstruction of the original metric perturbations in our mind. Based on this situation, we propose a strategy of a gauge-invariant treatments of $l=0,1$ mode perturbations through the decomposition of the metric perturbations by singular harmonic functions at once and the regularization of this singularity through the imposition of the boundary conditions to the Einstein equations. Following this proposal, we derive the linearized Einstein equations for any modes of $l\geq 0$ in a gauge-invariant manner. We discuss the solutions to the odd-mode perturbation equations in the linearized Einstein equations and show that these perturbations include the Kerr parameter perturbation in these odd-mode perturbation, which is physically reasonable.