Bernd G. Schmidt

Quasi-Normal Modes of Stars and Black Holes

AbstractPerturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades. They are of particular importance today, because of their relevance to gravitational wave astronomy. In this review we present the theory of quasi-normal modes of compact objects from both the mathematical and astrophysical points of view. The discussion includes perturbations of black holes (Schwarzschild, Reissner-Nordström, Kerr and Kerr-Newman) and relativistic stars (non-rotating and slowly-rotating). The properties of the various families of quasi-normal modes are described, and numerical techniques for calculating quasi-normal modes reviewed. The successes, as well as the limits, of perturbation theory are presented, and its role in the emerging era of numerical relativity and supercomputers is discussed.

Singular space-times

A classification scheme for boundary points of incomplete space-times is described. For all classes explicit examples are presented to illustrate the different behaviour of the geometry near those boundary points.

Asymptotic structure of symmetry reduced general relativity

Gravitational waves with a space-translation Killing field are considered. Because of the symmetry, the four-dimensional Einstein vacuum equations are equivalent to the three-dimensional Einstein equations with certain matter sources. This interplay between four- and three-dimensional general relativity can be exploited effectively to analyze issues pertaining to four dimensions in terms of the three-dimensional structures. An example is provided by the asymptotic structure at null infinity: While these space-times fail to be asymptotically flat in four dimensions, they can admit a regular completion at null infinity in three dimensions. This completion is used to analyze the asymptotic symmetries, introduce the analogue of the four-dimensional Bondi energy momentum, and write down a flux formula. The analysis is also of interest from a purely three-dimensional perspective because it pertains to a diffeomorphism-invariant three-dimensional field theory with local degrees of freedom, i.e., to a midisuperspace. Furthermore, because of certain peculiarities of three dimensions, the description of null infinity has a number of features that are quite surprising because they do not arise in the Bondi-Penrose description in four dimensions.

Existence and properties of spherically symmetric static fluid bodies with a given equation of state

It is shown that for a given equation of state and a given value of the central pressure there exists a unique global solution of the Einstein equations representing a spherically symmetric static fluid body. For the proof a new theorem on singular ordinary differential equations is established which is of interest in its own right. For a given equation of state and central pressure, the fluid will either fill the entire space or be finite in extent with a vacuum exterior. Criteria are given which allow one to decide for certain equations of state which of these two cases occurs. This generalizes well known results in Newtonian theory and is proved by showing that the relativistic model inherits the property of having a finite radius from a Newtonian model. Parameter-dependent families of relativistic solutions are constructed which have a Newtonian limit in a rigorous sense. The relationship between relativistic and Newtonian equations of state is examined by looking at the example of a degenerate Fermi gas.

Einstein’s Field Equations and Their Physical Implications: Selected Essays in Honour of Jürgen Ehlers

Selected Solutions of Einsteins Field Equations: Their Role in General Relativity and Astrophysics.- The Cauchy Problem for the Einstein Equations.- Post-Newtonian Gravitational Radiation.- Duality and Hidden Symmetries in Gravitational Theories.- Time-Independent Gravitational Fields.- Gravitational Lensing from a Geometric Viewpoint.

Isometries compatible with gravitational radiation

Isometries compatible with asymptotic flatness and admitting radiation are studied by using Bondi’s formalism. In axially symmetric space‐times, the only second allowable symmetry that does not exclude radiation is boost symmetry. The boost‐rotation symmetric solutions describe ‘‘uniformly accelerated particles’’ of various kinds. The news function is restricted by a differential equation; however, it need not vanish, as has been claimed in the literature. If two Killing fields corresponding to null rotations at null infinity are present, then it is shown that the vacuum field equations imply a further isometry. The resulting space‐time is a plane wave.

Static self-gravitating elastic bodies in Einstein gravity

We prove that given a stress-free elastic body there exists, for sufficiently small values of the gravitational constant, a unique static solution of the Einstein equations coupled to the equations of relativistic elasticity. The solution constructed is a small deformation of the relaxed configuration. This result yields the first proof of existence of static solutions of the Einstein equations without symmetries. c � 2008 Wiley Periodicals, Inc.

Behavior of Einstein-Rosen waves at null infinity

The asymptotic behavior of Einstein-Rosen waves at null infinity in 4 dimensions is investigated in {\it all} directions by exploiting the relation between the 4-dimensional space-time and the 3-dimensional symmetry reduction thereof. Somewhat surprisingly, the behavior in a generic direction is {\it better} than that in directions orthogonal to the symmetry axis. The geometric origin of this difference can be understood most clearly from the 3-dimensional perspective.

Singularities: The state of the art

A brief, but precise and unified account is given of the results that have been rigorously established at the time of writing concerning the existence and nature of singularities in classical general relativity.

Critical behavior in vacuum gravitational collapse in 4+1 dimensions

We show that the (4 + 1)-dimensional vacuum Einstein equations admit gravitational waves with radial symmetry. The dynamical degrees of freedom correspond to deformations of the three-sphere orthogonal to the (t,r) plane. Gravitational collapse of such waves is studied numerically and shown to exhibit discretely self-similar type II critical behavior at the threshold of black hole formation.