We study shadows of regular rotating black holes described by the axially symmetric solutions asymptotically Kerr for a distant observer, obtained from regular spherical solutions of the Kerr–Schild class specified by T t t = T r r ( p r = − ε ) . All regular solutions obtained with the Newman–Janis algorithm belong to this class. Their basic generic feature is the de Sitter vacuum interior. Information about the interior content of a regular rotating de Sitter-Kerr black hole can be in principle extracted from observation of its shadow. We present the general formulae for description of shadows for this class of regular black holes, and numerical analysis for two particular regular black hole solutions. We show that the shadow of a de Sitter-Kerr black hole is typically smaller than that for the Kerr black hole, and the difference depends essentially on the interior density and on the pace of its decreasing.

Despite the fact that different particle species can diffuse with respect to each other in neutron star (NS) cores, the effect of particle diffusion on various phenomena associated with NS oscillations is usually ignored. Here we demonstrate that the diffusion can be extremely powerful dissipative mechanism in superconducting NSs. In particular, it can be much more efficient than the shear and bulk viscosities. This result has important implications for the damping times of NS oscillations, development and saturation of dynamical instabilities in NSs, and for the excitation and coupling of oscillation modes during the late inspiral of binary NSs.

We study the effects of diffusion on damping of oscillations in the neutron star cores. This dissipation mechanism is usually ignored in the literature. As we show, the effect of diffusion is always smaller than viscous dissipation if the normal (nonsuperfluid and nonsuperconducting) matter of neutron stars is considered. However, we argue that for superconducting stars the role of diffusion may increase.

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 rmode 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 rmodes 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 non-analytic functions of the stellar angular velocity, Ω. We also derive the explicit expressions for the r-mode eigenfunctions and eigenfrequencies for very small values of Ω. These expressions explain the asymptotic behavior of the numerically calculated eigenfrequencies and eigenfunctions in the limit Ω → 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 σ = −4/3 Ω, are found.