Abstract
A gauge-invariant treatment of the monopole (l = 0) and dipole (l = 1) modes in linear perturbations of the Schwarzschild background spacetime is proposed. Through this gauge-invariant treatment, we derived the solutions to the linearized Einstein equation for these modes with a generic matter field. In the vacuum case, these solutions include the Kerr parameter perturbations in the l = 1 odd modes and the additional mass parameter perturbations of the Schwarzschild mass in the l = 0 even modes. The linearized version of Birkhoff’s theorem is also confirmed in a gauge-invariant manner. In this sense, our proposal is reasonable.
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