Comparing a Gauge-Invariant Formulation and a “Conventional Complete Gauge-Fixing Approach” for l=0,1-Mode Perturbations on the Schwarzschild Background Spacetime

This article provides a comparison of the gauge-invariant formulation for l=0,1-mode perturbations on the Schwarzschild background spacetime, proposed by the same author in 2021, and a “conventional complete gauge-fixing approach” where the spherical harmonic functions Ylm as the scalar harmonics are used from the starting point. Although it is often stated that “gauge-invariant formulations in general-relativistic perturbations are equivalent to complete gauge-fixing approaches”, we conclude that, as a result of this comparison, the derived solutions through the proposed gauge-invariant formulation and those through a “conventional complete gauge-fixing approach” are different. It is pointed out that there is a case where the boundary conditions and initial conditions are restricted in a conventional complete gauge-fixing approach.