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The theory of small, adiabatic, dipole perturbations of a star away from hydrostatic equilibrium is developed within the framework of general relativity. The analysis is linearized in the perturbation amplitudes. The "odd-parity" perturbations describe differential rotation, while the "even-parity" perturbations describe pulsation. In the pulsational case there are two degrees of freedom associated with the fluid motion and no degrees of freedom in the gravitational field (no dipole gravitational waves). During the pulsation the star's external gravitational field is completely unperturbed, to first order in the amplitude, despite the fact that its surface performs a finite, back-and-forth motion. In both the pulsational and rotational cases, the Einstein field equations are put into simple forms suitable for numerical integration.
Campolattaro et al. (Sun,) studied this question.