Abstract:
It is a quite common view among people, that are not aware of the developments in modern physics, that it is part of human nature to substitute religious faith in places where there is no knowledge. Therefore, an increase in knowledge would lead to a decrease in the necessity of faith. Further, it is argued that, ideally speaking, a full knowledge of the laws of nature would make obsolete any sort of religious faith and would ultimately allow a complete control of nature by man. Since referring to nature, such views must be founded in the natural sciences of which physics is the most fundamental. Therefore, the question whether such views are compatible with the current state of natural sciences is ultimately decided in physics. Indeed, it is likely that these simplistic views have their origin in the world view generated by the successes of Newtonian physics from the middle of the 17th century until the beginning of the 20th century which viewed the physical world as a type of mechanical clock in which the motion of the gears affect each other in a precise and predictable way. In particular, the paper points out that the above views are no longer supported by current physics and that abstracted world views cannot be considered as part of natural sciences, but only as belief systems.

Abstract:
This short paper should serve as 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 L^2-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 L^2-space and in this way generalizes a corresponding result of Kay (1985) in the case of the Schwarzschild black hole. It is believed that the employed methods are applicable to other separable wave equations.

Abstract:
The completeness of the quasinormal modes of the wave equation with Poeschl-Teller potential is investigated. A main result is that after a large enough time $t_0$, the solutions of this equation corresponding to $C^{\infty}$-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 $t_0$ 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.

Abstract:
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-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.

Abstract:
The paper provides a new framework for the description of linearized adiabatic lagrangian perturbations and stability of differentially rotating newtonian stars. In doing so it overcomes problems in a previous framework by Dyson and Schutz and provides the basis of a rigorous analysis of the stability of such stars. For this the governing equation of the oscillations is written as a first order system in time. From that system the generator of time evolution is read off and a Hilbert space is given where it generates a strongly continuous group. As a consequence the governing equation has a well-posed initial value problem. The spectrum of the generator relevant for stability considerations is shown to be equal to the spectrum of an operator polynomial whose coefficients can be read off from the governing equation. 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.

Abstract:
The reduced (in the angular coordinate $\phi$) wave equation and Klein-Gordon equation are considered on a Kerr background and in the framework of $C^{0}$-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 semigroup's 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 semigroup's generator and the corresponding Green's 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 {\it resonances}, i.e., poles of the analytic continuation of this resolvent. Finally, stability of the background with respect to reduced massive perturbations is proven for large enough masses.

Abstract:
This brief note wants to bring to attention that the formulation of physically reasonable initial-boundary value problems for wave equations in Lorentzian space-times is not unique, i.e., that there are inequivalent such formulations that lead to a different outcome of the stability discussion of the solutions. For demonstration, the paper uses the case of the wave equation on the right Rindler wedge in 2-dimensional Minkowski space. The used methods can be generalized to wave equations on stationary globally hyperbolic space-times with horizons in higher dimensions, such as Schwarzschild and Kerr space-times.

Abstract:
This course introduces the use of semigroup methods in the solution of linear and nonlinear (quasi-linear) hyperbolic partial differential equations, with particular application to wave equations and Hermitian hyperbolic systems. Throughout the course applications to problems from current relativistic (hyperbolic) physics are provided, which display the potential of semigroup methods in the solution of current research problems in physics.

Abstract:
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 $\mu$. Among others, the paper provides an improved bound for $\mu$ 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.

Abstract:
The paper considers the spectrum of axial perturbations of slowly uniformly rotating general relativistic stars in the framework of Y. Kojima. In a first step towards a full analysis only the evolution equations are treated but not the constraint. Then it is found that the system is unstable due to a continuum of non real eigenvalues. In addition the resolvent of the associated generator of time evolution is found to have a special structure which was discussed in a previous paper. From this structure it follows the occurrence of a continuous part in the spectrum of oscillations at least if the system is restricted to a finite space as is done in most numerical investigations. Finally, it can be seen that higher order corrections in the rotation frequency can qualitatively influence the spectrum of the oscillations. As a consequence different descriptions of the star which are equivalent to first order could lead to different results with respect to the stability of the star.