Equations governing the nonradial oscillations of a slowly rotating relativistic star

The nonradial perturbation equations of a slowly rotating star are derived in the framework of general relativity. We assume the stellar rotation is slow and include the rotational effect up to first order. The oscillation modes of a nonrotating star are completely separated by the spherical harmonic indices l, m and the parity. The eigenfrequencies are degenerate with respect to m at fixed l. In the presence of rotation, however, the degeneracy with respect to m is removed and different modes are mixed with each other. The odd-parity mode with l, m is coupled with the even-parity modes with l±1, m and vice versa. The basic equations derived here will give a new estimate for the instability of the relativistic rotating star due to gravitational radiation reaction.