Horst R. Beyer

On the well posedness of the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein's field equations

We give a well posed initial value formulation of the Baumgarte-Shapiro-Shibata-Nakamura form of Einsteins equations with gauge conditions given by a Bona-Masso-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.

On the stability of the massive scalar field in Kerr space-time

The current early stage in the investigation of the stability of the Kerr metric is characterized by the study of appropriate model problems. Particularly interesting is the problem of the stability of the solutions of the Klein-Gordon equation, describing the propagation of a scalar field in the background of a rotating (Kerr-) black hole. Results suggest that the stability of the field depends crucially on its mass μ. Among others, the paper provides an improved bound for μ above which the solutions of the reduced, by separation in the azimuth angle in Boyer-Lindquist coordinates, Klein-Gordon equation are stable. Finally, it gives new formulations of the reduced equation, in particular, in form of a time-dependent wave equation that is governed by a family of unitarily equivalent positive self-adjoint operators. The latter formulation might turn out useful for further investigation. On the other hand, it is proved that from the abstract properties of this family alone it cannot be concluded that the corresponding solutions are stable.

On the stability of the Kerr metric

Abstract: The reduced (in the angular coordinate ϕ) wave equation and Klein–Gordon equation are considered on a Kerr background and in the framework of C0-semigroup theory. Each equation is shown to have a well-posed initial value problem, i.e., to have a unique solution depending continuously on the data. Further, it is shown that the spectrum of the semigroups generator coincides with the spectrum of an operator polynomial whose coefficients can be read off from the equation. In this way the problem of deciding stability is reduced to a spectral problem and a mathematical basis is provided for mode considerations. For the wave equation it is shown that the resolvent of the semigroups generator and the corresponding Greens functions can be computed using spheroidal functions. It is to be expected that, analogous to the case of a Schwarzschild background, the quasinormal frequencies of the Kerr black hole appear as resonances, i.e., poles of the analytic continuation of this resolvent. Finally, stability of the solutions of the reduced Klein–Gordon equation is proven for large enough masses.

On the completeness of the quasinormal modes of the Poeschl-Teller potential.

Abstract:The completeness of the quasinormal modes of the wave equation with Pöschl–Teller potential is investigated. A main result is that after a large enough time t0, the solutions of this equation corresponding to &C∞-data with compact support can be expanded uniformly in time with respect to the quasinormal modes, thereby leading to absolutely convergent series. Explicit estimates for t0 depending on both the support of the data and the point of observation are given. For the particular case of an “early” time and zero distance between the support of the data and observational point, it is shown that the corresponding series is not absolutely convergent, and hence that there is no associated sum which is independent of the order of summation.

The oscillation and stability of differentially rotating spherical shells: the normal-mode problem

An understanding of the dynamics of differentially rotating systems is key to many areas of astrophysics. We investigate the oscillations of a simple system exhibiting differential rotation, and discuss issues concerning the role of corotation points and the emergence of dynamical instabilities. This problem is of particular relevance to the emission of gravitational waves from oscillating neutron stars, which are expected to possess significant differential rotation immediately after birth or binary merger.

The spectrum of radial adiabatic stellar oscillations

The stability analysis with respect to ‘‘small’’ radial adiabatic perturbations of spherically symmetric stellar equilibrium models which are polytropic with a constant adiabatic index only near the center and the boundary of the star leads to the consideration of a class of singular minimal Sturm–Liouville operators. It is shown that the physical boundary conditions choose in a unique way the corresponding Friedrichs extensions. Moreover, all linear self‐adjoint extensions of the members of the class are determined and are shown to have a purely discrete spectrum.

The spectrum of adiabatic stellar oscillations

The stability analysis for spherically symmetric stellar equilibrium models with respect to ‘‘small’’ adiabatic Lagrangian perturbations leads to the consideration of a class of densely defined, linear symmetric operators in Hilbert space, which are induced by certain singular vector–integro–partial differential operator. The extension properties of these operators as well as the spectral properties of the linear self‐adjoint extensions which are chosen by physical boundary conditions are investigated. For this, the equilibrium models are assumed to be polytropic, with a constant adiabatic index only near the center and near the boundary of the star. Among others it is shown that the operators of the class having a polytropic index near the boundary which is ≥1 are in particular essentially self‐adjoint and have a closure with a pure point spectrum.

Torsional oscillations of slowly rotating relativistic stars

We study low-amplitude crustal oscillations of slowly rotating relativistic stars consisting of a central fluid core and an outer thin solid crust. We estimate the effect of rotation on the torsional toroidal modes and on the interfacial and shear spheroidal modes. The results compared against the Newtonian ones for wide range of neutron star models and equations of state.

A note on the Klein–Gordon equation in the background of a rotating black hole

This short paper should serve as a basis for further analysis of a previously found new symmetry of the solutions of the wave equation in the gravitational field of a Kerr black hole. Its main new result is the proof of essential self-adjointness of the spatial part of a reduced normalized wave operator of the Kerr metric in a weighted L2-space. As a consequence, it leads to a purely operator theoretic proof of the well posedness of the initial value problem of the reduced Klein–Gordon equation in that field in that L2-space and in this way generalizes a corresponding result of Kay [“The double-wedge algebra for quantum fields on Schwarzschild and Minkowski spacetimes,” Commun. Math. Phys. 100, 57 (1985)] in the case of the Schwarzschild black hole. It is believed that the employed methods are applicable to other separable wave equations.

A framework for perturbations and stability of differentially rotating stars

The paper provides a new framework for the description of linearized adiabatic Lagrangian perturbations and stability of differentially rotating Newtonian stars. It overcomes problems in a previous framework by Dyson & Schutz and provides the basis of a rigorous analysis of the stability of such stars. For this, the governing equations of the oscillations are written as a first–order system in time. From that system, the formal generator of time evolution is read off and a Hilbert space is given, in which generates a strongly continuous group. As a consequence, the governing linearized equations have a well–posed initial–value problem. The spectrum of the (in general non–normal) generator relevant for stability considerations is shown to coincide with the spectrum of an operator polynomial whose coefficients can be read off from the governing equations. Finally, we give for the first time sufficient criteria for stability in the form of inequalities for the coefficients of the polynomial. These show that a negative canonical energy of the star does not necessarily indicate instability. It is still unclear whether these criteria are strong enough to prove stability for realistic stars. However, their usefulness has already been demonstrated in another paper, where they lead to a new result in the discussion of the stability of rotating (Kerr) black holes. That stability is a classical open problem in general relativity.