We give a well posed initial value formulation of the Baumgarte-Shapiro-Shibata-Nakamura form of Einstein's equations with gauge conditions given by a Bona-Massó -like slicing condition for the lapse and a frozen shift. This is achieved by introducing extra variables and recasting the evolution equations into a first order symmetric hyperbolic system. We also consider the presence of artificial boundaries and derive a set of boundary conditions that guarantee that the resulting initial-boundary value problem is well posed, though not necessarily compatible with the constraints. In the case of dynamical gauge conditions for the lapse and shift we obtain a class of evolution equations which are strongly hyperbolic and so yield well posed initial value formulations.

We study torsional Alfvén oscillations of magnetars, that is neutron stars with a strong magnetic field. We consider the poloidal and toroidal components of the magnetic field and a wide range of equilibrium stellar models. We use a new coordinate system (X, Y), where and and a1 is the radial component of the magnetic field. In this coordinate system, the one+two‐dimensional evolution equation describing the quasi‐periodic oscillations (QPOs), see Sotani et al., is reduced to a one+one‐dimensional equation where the perturbations propagate only along the y‐axis. We solve the one+one‐dimensional equation for different boundary conditions and the open magnetic field lines, that is magnetic field lines that reach the surface and there match up with the exterior dipole magnetic field as well as closed magnetic lines, i.e. magnetic lines that never reach the stellar surface. For the open field lines, we find two families of QPO frequencies: a family of ‘lower’ QPO frequencies which is located near the x‐axis and a family of ‘upper’ frequencies located near the y‐axis. According to Levin, the fundamental frequencies of these two families can be interpreted as the turning point of the continuous spectrum. We find that the upper frequencies are multiples of the lower ones by a constant equalling 2n+ 1. For the closed lines, the corresponding factor is n+ 1. By using these relations, we can explain both the lower and the higher observed frequencies in SGR 1806−20 and SGR 1900+14.

Betrachtet wird die verallgemeinerte Schwingungsgleichung E: (D2 + aDq + b) x(t) = f(t); q ≠ (0, 2). Es wird gezeigt, daß es für q ε 1 und x, f ε L2C(R) beliebig viele Möglichkeiten der Definition von E gibt, entsprechend der Wahl der Äste von (iε)q zur Definition charakteristischer Funktionen p(ε) = (i)2 + a(iε)q + b. Dabei muß p lediglich meßbar sein. Neben allgemeinen Bedingungen für die Eindeutigkeit und die Kausalität von Lösungen werden auch für Anwendungen wichtige Ergebnisse erzielt: So folgt die Eindeutigkeit bereits, wenn p stetig ist und keine reellen Nullstellen hat. Wird ferner p auf den Hauptast beschränkt, so gibt es kausale Lösungen genau dann, wenn a,b > 0. Als Beispiel wird eine allgemeine analytische Lösung der kausalen Impulsantwort gegeben und diskutiert.

We present a mathematically rigorous proof that the r-mode spectrum of relativistic stars to the rotational lowest order has a continuous part. A rigorous definition of this spectrum is given in terms of the spectrum of a continuous linear operator. This study verifies earlier results by Kojima (1998) about the nature of the r-mode spectrum.Comment: 6 pages, no figure