Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime: Part III—Realization of Exact Solutions

This is the Part III paper of our series of papers on a gauge-invariant perturbation theory on the Schwarzschild background spacetime. After reviewing our general framework for the gauge-invariant perturbation theory and the proposal for gauge-invariant treatments of l=0,1 mode perturbations on the Schwarzschild background spacetime in the Part I paper, we examine the problem of whether the l=0,1 even-mode solutions derived in the Part II paper are physically reasonable. We consider the linearized versions of the Lemaître–Tolman–Bondi solution and the non-rotating C-metric. As a result, we show that our derived even-mode solutions to the linearized Einstein equations realize these two linearized solutions. This supports the conclusion that our derived solutions are physically reasonable, which implies that our proposal for gauge-invariant treatments of l=0,1 mode perturbations is also physically reasonable. We also briefly summarize the conclusions of our series of papers.