If we assume a small variation in the fluid or/and in the spacetime we must deal with the perturbed Einstein equations
Following the procedure of the previous section one can decompose the perturbation equations into spherical harmonics. This decomposition leads to two classes of oscillations according to the parity of the harmonics (exactly as for the black hole case). The first ones called even (or spheroidal, or polar) produce spheroidal deformations on the fluid, while the second are the odd (or toroidal, or axial) which produce toroidal deformations.
For the polar case one can use certain combinations of the metric perturbations as unknowns, and the linearized field equations inside the star will be equivalent to the following system of three wave equations for unknowns :[118, 6]. The functions and correspond to the perturbations of the spacetime while the function is proportional to the density perturbation and is only defined on the background star. With we define the speed of sound and with a prime we denote differentiation with respect to :  it was shown that (for background stars whose boundary density is positive) the above system – together with the geometrical transition conditions at the boundary of the star and regularity conditions at the center – admits a well posed Cauchy problem. The constraint is preserved under the evolution. We see that two variables propagate along light characteristics and the density propagates with the sound velocity of the background star.
It is possible to eliminate the constraint – first done by Moncrief  – if one solves the constraint (54) for and puts the corresponding expression into . (The characteristics for change then to sound characteristics inside the star and light characteristics outside.) This way one has just to solve two coupled wave equations for and with unconstrained data, and to calculate using the constraint from the solution of the two wave equations. Again the explicit form of the equation can be found in .
Turning next to quasi-normal modes in the spirit of Section 2, we can Laplace transform the two wave equations and obtain a system of ordinary differential equations which is of fourth order. The Green function can be constructed from solutions of the homogeneous equations (having the appropriate behavior at the center and infinity) and its analytic continuation may have poles defining the quasi-normal mode frequencies.
From the form of the above equations one can easily see two limiting cases. Let us first assume that the gravitational field is very weak. Then Equation (51) and (52) can be omitted (actually in the weak field limit [200, 6]) and we find that one equation is enough to describe (with acceptable accuracy) the oscillations of the fluid. This approach is known as the Cowling approximation . Inversely, we can assume that the coupling between the two Equations (51) and (52) describing the spacetime perturbations with the Equation (53) is weak and consequently derive all the features of the spacetime perturbations from only the two of them. This is what is called the “inverse Cowling approximation” (ICA) .
For the axial case the perturbations reduce to a single wave equation for the spacetime perturbations which describes toroidal deformationsaxial oscillations are a trivial solution of zero frequency to the perturbation equations and the variations of pressure and density are zero. Nevertheless, the variation of the velocity field is not zero and produces non-oscillatory eddy motions. This means that there are no oscillatory velocity fields. In the relativistic case the picture is identical  nevertheless; in this case there are still QNMs, the ones that we will describe later as “spacetime or -modes” .
When the star is set in slow rotation then the axial modes are no longer degenerate, but instead a new family of modes emerges, the so-called -modes. An interesting property of these modes that has been pointed out by Andersson [14, 94] is that these modes are generically unstable due the Chandrasekhar–Friedman–Schutz instability [53, 95] and furthermore it has been shown [23, 142] that these modes can potentially restrict the rotation period of newly formed neutron stars and also that they can radiate away detectable amounts of gravitational radiation . The equations describing the perturbations of slowly rotating relativistic stars have been derived by Kojima [120, 121], and Chandrasekhar and Ferrari .
© Max Planck Society and the author(s)