Abstract:
We investigate properties of r-modes characterized by regular eigenvalue problem in slowly rotating relativistic polytropes. Our numerical results suggest that discrete r-mode solutions for the regular eigenvalue problem exist only for restricted polytropic models. In particular the r-mode associated with l=m=2, which is considered to be the most important for gravitational radiation driven instability, do not have a discrete mode as solutions of the regular eigenvalue problem for polytropes having the polytropic index N > 1.18 even in the post-Newtonian order. Furthermore for a N=1 polytrope, which is employed as a typical neutron star model, discrete r-mode solutions for regular eigenvalue problem do not exist for stars whose relativistic factor M/R is larger than about 0.1. Here M and R are stellar mass and stellar radius, respectively.

Abstract:
We calculate high-order quasinormal modes with large imaginary frequencies for electromagnetic and gravitational perturbations in nearly extremal Schwarzschild-de Sitter spacetimes. Our results show that for low-order quasinormal modes, the analytical approximation formula in the extremal limit derived by Cardoso and Lemos is a quite good approximation for the quasinormal frequencies as long as the model parameter $r_1\kappa_1$ is small enough, where $r_1$ and $\kappa_1$ are the black hole horizon radius and the surface gravity, respectively. For high-order quasinormal modes, to which corresponds quasinormal frequencies with large imaginary parts, on the other hand, this formula becomes inaccurate even for small values of $r_1\kappa_1$. We also find that the real parts of the quasinormal frequencies have oscillating behaviors in the limit of highly damped modes, which are similar to those observed in the case of a Reissner-Nordstr{\" o}m black hole. The amplitude of oscillating ${\rm Re(\omega)}$ as a function of ${\rm Im}(\omega)$ approaches a non-zero constant value for gravitational perturbations and zero for electromagnetic perturbations in the limit of highly damped modes, where $\omega$ denotes the quasinormal frequency. This means that for gravitational perturbations, the real part of quasinormal modes of the nearly extremal Schwarzschild-de Sitter spacetime appears not to approach any constant value in the limit of highly damped modes. On the other hand, for electromagnetic perturbations, the real part of frequency seems to go to zero in the limit.

Abstract:
We discuss the modal properties of the $r$-modes of relativistic superfluid neutron stars, taking account of the entrainment effects between superfluids. In this paper, the neutron stars are assumed to be filled with neutron and proton superfluids and the strength of the entrainment effects between the superfluids are represented by a single parameter $\eta$. We find that the basic properties of the $r$-modes in a relativistic superfluid star are very similar to those found for a Newtonian superfluid star. The $r$-modes of a relativistic superfluid star are split into two families, ordinary fluid-like $r$-modes ($r^o$-mode) and superfluid-like $r$-modes ($r^s$-mode). The two superfluids counter-move for the $r^s$-modes, while they co-move for the $r^o$-modes. For the $r^o$-modes, the quantity $\kappa\equiv\sigma/\Omega+m$ is almost independent of the entrainment parameter $\eta$, where $m$ and $\sigma$ are the azimuthal wave number and the oscillation frequency observed by an inertial observer at spatial infinity, respectively. For the $r^s$-modes, on the other hand, $\kappa$ almost linearly increases with increasing $\eta$. It is also found that the radiation driven instability due to the $r^s$-modes is much weaker than that of the $r^o$-modes because the matter current associated with the axial parity perturbations almost completely vanishes.

Abstract:
We investigate the modal properties of inertial modes of rotating neutron stars with the core filled with neutron and proton superfluids, taking account of entrainment effects between the superfluids. In this paper, the entrainment effects are modeled by introducing a parameter $\eta$ so that no entrainment state is realized at $\eta=0$. We find that inertial modes of rotating neutron stars with the superfluid core are split into two families, which we call ordinary fluid inertial modes ($i^o$-mode) and superfluid inertial modes ($i^s$-mode). The two superfluids in the core counter-move for the $i^s$-modes. For the $i^o$-modes, $\kappa_0=\lim_{\Omega\to 0}\omega/\Omega$ is only weakly dependent on the entrainment parameter $\eta$, where $\Omega$ and $\omega$ are the angular frequency of rotation and the oscillation frequency observed in the corotating frame of the star, respectively. For the $i^s$-modes, on the other hand, $|\kappa_0|$ almost linearly increases as $\eta$ increases. Avoided crossings as functions of $\eta$ are therefore quite common between $i^o$- and $i^s$-modes. We find that some of the $i^s$-modes that are unstable against the gravitational radiation reaction at $\eta=0$ become stable when $\eta$ is larger than $\eta_{crit}$, the value of which depends on the mode. Since the radiation driven instability associated with the current multipole radiation is quite weak for the inertial modes and the mutual friction damping in the superfluid core is strong, the instability caused by the inertial modes will be easily suppressed unless the entrainment parameter $\eta$ is extremely small and the mutual friction damping is sufficiently weak.

Abstract:
We investigate quasinormal modes of a massless charged scalar field on a small Reissner-Nordstr\"om-anti-de Sitter (RN-AdS) black hole both with analytical and numerical approaches. In the analytical approach, by using the small black hole approximation (r_+ << L), we obtain the quasinormal mode frequencies in the limit of r_+/L -> 0, where r_+ and L stand for the black hole event horizon radius and the AdS scale, respectively. We then show that the small RN-AdS black hole is unstable if its quasinormal modes satisfy the superradiance condition and that the instability condition of the RN-AdS black hole in the limit of r_+/L -> 0 is given by Q>(3/eL)Q_c, where Q, Q_c, and e are the charge of the black hole, the critical (maximum) charge of the black hole, and the charge of the scalar field, respectively. In the numerical approach, we calculate the quasinormal modes for the small RN-AdS black holes with r_+ << L and confirm that the RN-AdS black hole is unstable if its quasinormal modes satisfy the superradiance condition. Our numerical results show that the RN-AdS black holes with r_+ =0.2L, 0.1L, and 0.01L become unstable against scalar perturbations with eL=4 when the charge of the black hole satisfies Q > 0.8Q_c, 0.78Q_c, and 0.76Q_c, respectively.

Abstract:
We investigate the properties of $r$-mode oscillations of a slowly rotating neutron star with a solid crust, by taking account of the effects of the Coriolis force. For the modal analysis we employ three-component neutron star models that are composed of a fluid core, a solid crust and a surface fluid ocean. For the three-component models, we find that there exist two kinds of $r$-modes, that is, those confined in the surface fluid ocean and those confined in the fluid core, which are most important for the $r$-mode instability. The $r$-modes do not have any appreciable amplitudes in the solid crust if rotation rate of the star is sufficiently small. We find that the core $r$-modes are strongly affected by mode coupling with the crustal torsional (toroidal) modes and lose their simple properties of the eigenfunction and eigenfrequency as functions of the angular rotation velocity $\Omega$. This indicates that the extrapolation formula, which is obtained in the limit of $\Omega\to 0$, cannot be used to examine the $r$-mode instability of rapidly rotating neutron stars with a solid crust unless the effects of mode coupling with the crustal torsional modes are correctly taken into account.

Abstract:
We investigate the effects of the purely toroidal magnetic field on the equilibrium structures of the relativistic stars. The master equations for obtaining equilibrium solutions of relativistic rotating stars containing purely toroidal magnetic fields are derived for the first time. To solve these master equations numerically, we extend the Cook-Shapiro-Teukolsky scheme for calculating relativistic rotating stars containing no magnetic field to incorporate the effects of the purely toroidal magnetic fields. By using the numerical scheme, we then calculate a large number of the equilibrium configurations for a particular distribution of the magnetic field in order to explore the equilibrium properties. We also construct the equilibrium sequences of the constant baryon mass and/or the constant magnetic flux, which model the evolution of an isolated neutron star as it loses angular momentum via the gravitational waves. Important properties of the equilibrium configurations of the magnetized stars obtained in this study are summarized as follows ; (1) For the non-rotating stars, the matter distribution of the stars is prolately distorted due to the toroidal magnetic fields. (2) For the rapidly rotating stars, the shape of the stellar surface becomes oblate because of the centrifugal force. But, the matter distribution deep inside the star is sufficiently prolate for the mean matter distribution of the star to be prolate. (3) The stronger toroidal magnetic fields lead to the mass-shedding of the stars at the lower angular velocity. (4) For some equilibrium sequences of the constant baryon mass and magnetic flux, the stars can spin up as they lose angular momentum.

Abstract:
We investigate the modal properties of the $r$-modes of rotating neutron stars with the core filled with neutron and proton superfluids, taking account of entrainment effects between the superfluids. The stability of the $r$-modes against gravitational radiation reaction is also examined considering viscous dissipation due to shear and a damping mechanism called mutual friction between the superfluids in the core. We find the $r$-modes in the superfluid core are split into ordinary $r$-modes and superfluid $r$-modes, which we call, respectively, $r^o$- and $r^s$-modes. The two superfluids in the core flow together for the $r^o$-modes, while they counter-move for the $r^s$-modes. For the $r^o$-modes, the coefficient $\kappa_0\equiv\lim_{\Omega\to 0}\omega/\Omega$ is equal to $2m/[l^\prime(l^\prime+1)]$, almost independent of the parameter $\eta$ that parameterizes the entrainment effects between the superfluids, where $\Omega$ is the angular frequency of rotation, $\omega$ the oscillation frequency observed in the corotating frame of the star, and $l^\prime$ and $m$ are the indices of the spherical harmonic function representing the angular dependence of the $r$-modes. For the $r^s$-modes, on the other hand, $\kappa_0$ is equal to $2m/[l^\prime(l^\prime+1)]$ at $\eta=0$ (no entrainment), and it almost linearly increases as $\eta$ is increased from $\eta=0$. The mutual friction in the superfluid core is found ineffective to stabilize the $r$-mode instability caused by the $r^o$-mode except in a few narrow regions of $\eta$. The $r$-mode instability caused by the $r^s$-modes, on the other hand, is extremely weak and easily damped by dissipative processes in the star.

Abstract:
We investigate the properties of r-mode and inertial mode of slowly rotating, non-isentropic, Newtonian stars, by taking account of the effects of the Coriolis force and the centrifugal force. For the non-isentropic models we consider only two cases, that is, the models with the stable fluid stratification in the whole interior and the models that are fully convective. For simplicity we call these two kinds of models "radiative" and "convective" models in this paper. For both cases, we assume the deviation of the models from isentropic structure is small. Examining the dissipation timescales due to the gravitational radiation and several viscous processes for the polytropic neutron star models, we find that the gravitational radiation driven instability of the r-modes remains strong even in the non-isentropic models. Calculating the rotational modes of the radiative models as functions of the angular rotation frequency $\Omega$, we find that the inertial modes are strongly modified by the buoyant force at small $\Omega$, where the buoyant force as a dominant restoring force becomes comparable with or stronger than the Coriolis force. Because of this property we obtain the mode sequences in which the inertial modes at large $\Omega$ are identified as g-modes or the r-modes with l=|m| at small $\Omega$. We also note that as $\Omega$ increases from $\Omega=0$ the retrograde g-modes become retrograde inertial modes, which are unstable against the gravitational radiation reaction.