  We present a formalism to study the metric perturbations of the Schwarzschild
spacetime. The formalism is gauge invariant, and it is also covariant under
two-dimensional coordinate transformations that leave the angular coordinates
unchanged. The formalism is applied to the typical problem of calculating the
gravitational waves produced by material sources moving in the Schwarzschild
spacetime. We examine the radiation escaping to future null infinity as well as
the radiation crossing the event horizon. The waveforms, the energy radiated,
and the angular-momentum radiated can all be expressed in terms of two
gauge-invariant scalar functions that satisfy one-dimensional wave equations.
The first is the Zerilli-Moncrief function, which satisfies the Zerilli
equation, and which represents the even-parity sector of the perturbation. The
second is the Cunningham-Price-Moncrief function, which satisfies the
Regge-Wheeler equation, and which represents the odd-parity sector of the
perturbation. The covariant forms of these wave equations are presented here,
complete with covariant source terms that are derived from the stress-energy
tensor of the matter responsible for the perturbation. Our presentation of the
formalism is concluded with a separate examination of the monopole and dipole
components of the metric perturbation.
