Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime Part II: Even-Mode Perturbations

This is the Part II paper of our series of papers on a gauge-invariant perturbation theory on the Schwarzschild background spacetime. After reviewing our general framework of the gauge-invariant perturbation theory and the proposal of gauge-invariant treatments for l=0,1-mode perturbations on the Schwarzschild background spacetime in the Part I paper, we examine the linearized Einstein equations for even-mode perturbations. We discuss the strategy to solve the linearized Einstein equations for these even-mode perturbations including l=0,1 modes. Furthermore, we explicitly derive the l=0,1-mode solutions to the linearized Einstein equations in both the vacuum and the non-vacuum cases. We show that the solutions for l=0-mode perturbations includes the additional Schwarzschild mass parameter perturbation, which is physically reasonable. Then, we conclude that our proposal of the resolution of the l=0,1-mode problem is physically reasonable due to the realization of the additional Schwarzschild mass parameter perturbation and the Kerr parameter perturbation in the Part I paper.