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Mathematical General Relativity

Mathematical General Relativity (GR) is concerned with the mathematical study of Albert Einsteins theory of space, time and gravitation, which was established almost exactly 100 years ago. My own research in this area is mostly concerned with questions of low regularity. In fact, the geometric foundation of GR is Lorentzian geometry which is usually formulated in the smooth category. On the other hand, the nature of GR as a physical theory and its analytic foundations (most notably hyperbolic PDE) demand the use of non-smooth functions and regularity issues become essential.

• One of the foundational tools in Lorentzian and semi-Riemannian geometry is the exponential map, which sends rays in the tangent space to geodesic on the manifold. If the metric is smooth then the exponential map is a local diffeomeorphism and a radial isometry but if the regularity of the metric is below $C^{1,1}$ (i.e., its first derivatives are Lipschitz continuous which guarantees unique local solvability of the geodesic equation) then classical counterexamples show the failure of the ususal convexity properties. These were complimented recently by Chrusciel and Grant who have studied metrics of Hölder class $C^{1,a}$ (a less than 1) where the lightcones `bubble up' (i.e. fail to be hypersurfaces).
  A positive result was achieved in [P34] together with Michael Kunzinger and his Ph.D student Milena Stojkovic, where we have shown that if the metric is $C^{1,1}$ then the exponential map retains the maximal regularity, i.e., it is a bi-Lipschitz homeomorphism. Based in these results and techniques (a careful regularisation approach combined with techniques from comparison geometry), and with James Vickers joining the team, we could show in [P35] that the bulk of the ususal smooth Lorentzian causality theory remains valid also in the regularity class $C^{1,1}$. This in turn enabled us to provide the first proof of a $C^{1,1}$-singularity theoreom, namely the Hawking one in [P38]. Furthermore we could also establish the Penrose singularity theorem in regularity $C^{1,1}$ in [P40]. Current research in this direction is concerned with the proof of the most general singularity theorem, namely the Hawking-Penrose one in this regularity class, and the use of techniques of comparison geometry to further lower the regularity assumptions in the singularity theorems.

• Also since a long time I have been involved in the study of impulisve gravitational waves. These are exact spacetimes which model short but intense bursts of gravitational radiation. From a mathematical viewpoint, they are key examples of spactimes of low regularity, which are either described by a distributional metric or a metric of Lipschitz regularity. I have primarily been concerned with the study of geodesics in these geometries and with questions of completeness. Starting from my Ph.D days I have in part together with Michael Kunzinger studied the simplest model, i.e., impulsive pp-waves using nonlinear distributional geometry, see [P1-P6]. Also long ago, joining with Jiri Podolsky from Charles University in Prague I have studied expanding impulsive gravitational waves in Minkowski background ([P14]). This cooperation has been stalled for a some time but was revived some 5 yeas ago when Jiri's Ph.D. student Robert Svarc took up this topic. Together we have studied gyratons, i.e., impulsive gravitational waves with spin in [P36], a study which nicely complements work I had done previoiusly together with Clemens Sämann on impulsive waves with a general wave surface ([P30,P39]). The oberservation that Filippov's solutions concept is very well-suited to describe geodesics in Lipschitz space times ([P33]) boosted our cooperation and made it possible to provide a mathematical study of completeness of non-expanding impulsive waves on all constant curvature backgrounds ([P32,37]). Current work is concerned with the study of completeness in expanding impulsive waves on all constant curvature backgrounds and with an investigation of the 'general Brinkmann metric' which combines the effects of a general wave surface with that of a spin.

• The study of geodesics in impulsive pp-waves using nonlinear distributional geometry based on algebras of generalized functions has triggered a research line in generalized functions which lead to the development of a theory suitable to be applied in a geometric setting. This has been applied to deal with the (distributional-)Schwarzschild geometry ([P10]), and more generally the topic of linear and nonlinear distributional general relativity has been reviewed in [P15] and more recently in [P21] (jointly with James Vickers). In particular, we have related the nonlinear framework to the `maximal reasonable' class of distributional metrics of Geroch and Traschen in [P26], see aslo [P23]. An up-to-date review on this topic has been provided by Eduard Nigsch and Clemens Sämann.

Algebras of generalized functions and applications

Algebras of generalized functions, also known as Colombeau Algebras, are sheaves of differential algebras which contain the vector space of Schwartz distributions as a subspace and the space of smooth functions as a subalgebra, hence provide a framework for multiplying distributions which benefits from maximal compatibility with classical analysis (read more...) In particular, I'm interested in all geometric aspects of this theory and its applications, most of all in general relativity and PDE.

• Michael Kunzinger and myself, partly in collaboration with James Vickers and Michael Oberguggenberger have developped the foundations of global analysis as well as (semi-)Riemannian geometry with generalized functions in Colombeau's so-called special setting ([P9, P11, P12, P17, P18, J24]), which we called non-linear distributional geometry. Some classical questions which we had encountered when dealing with flows of singular vector fields have been solved in collaboration with Hermann Schichl ([P20]).

• Together with Michael Grosser and again Michael Kunzinger and James Vickers I have developped a diffeomorphism invariant full Colombeau algebra on differentiable manifolds ([P8])---a task which needed restructuring some of the foundations of the theory in Euclidean space and which has been achieved in collaboration with Eva Farkas ([M1]). The idea of this research is explained in [P7] while an overview is presented in [P16]. Finally in [P27] I have reported on a significant shift of focus which is indispensable when dealing with the tensor case (see below). Significant earlier parts of this research have been collected in the monograph [M2] with Michael Grosser, Michael Kunzinger and Michael Oberguggenberger.
  One big challenge in the theory of diffeomorphism invariant full Colombeau algebras has been to construct spaces of generalized tensor fields. By the Schwartz impossibility result the obvious approach of taking the tensor product of the full algebra on a manifold with classical tensor fields is ruled out and one has to take a different route. By introducing a suitable class of transport operators (an idea put forward by James Vickers long time ago) and letting generalized fields depend on them Michael Grosser, Michael Kunzinger, James Vickers and myself could finally construct such spaces in [P28]. This line of research has been further developped by Eduard Nigsch, in part together with James Vickers and Michael Grosser.
  

• As an application in PDE I have, together with Eberhard Mayerhofer and James Grant studied local solvability of the wave equation on weakly singluar space-times ([P25]). This result has been extended to a global one together with Günther Hörmann and Michael Kunzinger in ([P29] and relations to first oder systems have been investigated in [P31] in collaboration with my Ph.D. student Clemens Hanel and Güther's Ph.D. student Christian Spreitzer.

Kinetic theory

I have also done some work in nonlinear PDE, more precisely in collissionless models in kinetic theory. Together with Gerhard Rein, Michael Kunzinger and Gerald Teschl I have studied the Vlasov-Klein-Gordon system: In [P13] we have proven the existence of local weak solutions, while in [P19] we have derived local classical solvability plus a continuation criterion. Again with Michael Kunzinger and Irina Kmit I have studied singular solutions of the Vlaosov-Poisson system ([P22]) with the aim of looking at the singular limits of the VP-system, i.e., the Euler equations and the n-body problem.

Gravitational waves

Quite early in my career I developped some interest in gravitational wave detection, which resulted in two fun-papers [Mi1] and [Mi3] where together with some fellow students as well as Karsten Danzmann we put forward the idea of a space-borne gravitational wave detector much larger than LISA. Of course such a project is not at all feasable.