Proposal of a gauge-invariant treatment of l=0,1-mode perturbations on
  Schwarzschild Background Spacetime
  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.
