Non-analytic behavior of the relativistic r-modes in slowly rotating neutron stars

An inconsistency between the theoretical analysis and numerical calculations of the relativistic $r$-modes puzzles the neutron star community since the Kojima's finding of the continuous part in the $r$-mode oscillation spectrum in 1997. In this paper, after a brief review of the Newtonian $r$-mode theory and of the literature devoted to the continuous spectrum of $r$-modes, we apply our original approach to the study of relativistic oscillation equations. Working within the Cowling approximation, we derive the general equations, governing the dynamics of discrete relativistic $r$-modes for both barotropic (isentropic) and nonbarotropic stars. A detailed analysis of the obtained equations in the limit of extremely slow stellar rotation rate reveals that, because of the effect of inertial reference frame-dragging, the relativistic $r$-mode eigenfunctions and eigenfrequencies become {\it non-analytic} functions of the stellar angular velocity, $\Omega$. We also derive the explicit expressions for the $r$-mode eigenfunctions and eigenfrequencies for very small values of $\Omega$. These expressions explain the asymptotic behavior of the numerically calculated eigenfrequencies and eigenfunctions in the limit $\Omega\to 0$. All the obtained $r$-mode eigenfrequencies take discrete values in the frequency range, usually associated with the continuous part of the spectrum. No indications of the continuous spectrum, at least in the vicinity of the Newtonian $l=m=2$ $r$-mode frequency $\sigma=-4/3 \ \Omega$, are found.