The study of oscillations of relativistic stars is motivated by the prospect of detecting such oscillations in electromagnetic or gravitational wave signals. In the same way that helioseismology is providing us with information about the interior of the Sun, the observational identification of oscillation frequencies of relativistic stars could constrain the high-density equation of state [13]. The oscillations could be excited after a core collapse or during the final stages of a neutron star binary merger. Rapidly rotating relativistic stars can become unstable to the emission of gravitational waves.

When the oscillations of an equilibrium star are of small magnitude compared to its radius, it will suffice to approximate them as linear perturbations. Such perturbations can be described in two equivalent ways. In the Lagrangian approach, one studies the changes in a given fluid element as it oscillates about its equilibrium position. In the Eulerian approach, one studies the change in fluid variables at a fixed point in space. Both approaches have their strengths and weaknesses.

In the Newtonian limit, the Lagrangian approach has been used to develop variational principles [215, 118], but the Eulerian approach proved to be more suitable for numerical computations of mode frequencies and eigenfunctions [162, 217, 158, 160, 159]. Clement [64] used the Lagrangian approach to obtain axisymmetric normal modes of rotating stars, while nonaxisymmetric solutions were obtained in the Lagrangian approach by Imamura et al. [156] and in the Eulerian approach by Managan [217] and Ipser and Lindblom [158]. While a lot has been learned from Newtonian studies, in the following we will focus on the relativistic treatment of oscillations of rotating stars.

3.1 Quasi-normal modes of oscillation

3.2 Effect of rotation on quasi-normal modes

3.3 Axisymmetric perturbations

3.3.1 Secular and dynamical axisymmetric instability

3.3.2 Axisymmetric pulsation modes

3.4 Nonaxisymmetric perturbations

3.4.1 Nonrotating limit

3.4.2 Slow rotation approximation

3.4.3 Post-Newtonian approximation

3.4.4 Cowling approximation

3.5 Nonaxisymmetric instabilities

3.5.1 Introduction

3.5.2 CFS instability of polar modes

3.5.3 CFS instability of axial modes

3.5.4 Effect of viscosity on the CFS instability

3.5.5 Gravitational radiation from CFS instability

3.5.6 Viscosity-driven instability

3.2 Effect of rotation on quasi-normal modes

3.3 Axisymmetric perturbations

3.3.1 Secular and dynamical axisymmetric instability

3.3.2 Axisymmetric pulsation modes

3.4 Nonaxisymmetric perturbations

3.4.1 Nonrotating limit

3.4.2 Slow rotation approximation

3.4.3 Post-Newtonian approximation

3.4.4 Cowling approximation

3.5 Nonaxisymmetric instabilities

3.5.1 Introduction

3.5.2 CFS instability of polar modes

3.5.3 CFS instability of axial modes

3.5.4 Effect of viscosity on the CFS instability

3.5.5 Gravitational radiation from CFS instability

3.5.6 Viscosity-driven instability

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