Coupling of radial and nonradial oscillations of relativistic stars: Gauge-invariant formalism

Linear perturbation theory is appropriate to describe small oscillations of stars, while a mild nonlinearity is still tractable perturbatively but requires one to consider mode coupling, i.e., to take into account second order effects. It is natural to start to look at this problem by considering the coupling between linear radial and nonradial modes. A radial pulsation may be thought of as an important component of an overall mildly nonlinear oscillation, e.g., of a protoneutron star. Radial pulsations of spherical compact objects do not per se emit gravitational waves but, if the coupling between the existing first order radial and nonradial modes is efficient in driving and possibly amplifying the nonradial oscillations, one may expect the appearance of nonlinear harmonics, and gravitational radiation could then be produced to a significant level. More in general, mode coupling typically leads to an interesting phenomenology, thus it is worth investigating in the context of star perturbations. In this paper we develop the relativistic formalism to study the coupling of radial and nonradial first order perturbations of a compact spherical star. From a mathematical point of view, it is convenient to treat the two sets of perturbations as separately parametrized, using a 2-parameter perturbative expansion of the metric, the energy-momentum tensor and Einstein equations in which $\lambda$ is associated with the radial modes, $ϵ$ with the nonradial perturbations, and the $\lambda ϵ$ terms describe the coupling. This approach provides a well-defined framework to consider the gauge dependence of perturbations, allowing us to use $ϵ$ order gauge-invariant nonradial variables on the static background and to define new second order $\lambda ϵ$ gauge-invariant variables representing the result of the nonlinear coupling. We present the evolution and constraint equations for our variables outlining the setup for numerical computations, and briefly discuss the surface boundary conditions in terms of the second order $\lambda ϵ$ Lagrangian pressure perturbation.