# The DIANA seminar

This seminar is an informal forum where members of the DIANA group meet to discuss topics of interest. We meet on a weekly basis. The programme for these meetings will be advertised below, and by email.

The the seminar takes place (at least for now) every Friday at 9.45 am in Seminarraum 7 (2nd floor) or at 10.15 in Seminarraum 11, as announced by email.

Anyone interested is welcome to attend.

## Summer term 2018

Date Speaker Title
02. Mar. 2018 Scheduling
09. Mar. 2018 Clemens Sämann
Lorentzian length spaces
AbstractWe introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The role of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this way we recover many fundamental results in greater generality, while at the same time clarifying the minimal requirements for and the interdependence of the basic building blocks of the theory. A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT(k)-spaces, based on triangle comparison. Applications include Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity. This is joint work with Michael Kunzinger. Preprint: https://arxiv.org/abs/1711.08990
16. Mar. 2018 Benedict Schinnerl
Master Defense - Length Structures and Geodesics on Riemannian Manifolds of Low Regularity
AbstractIn the thesis I dealt with Riemannian Manifolds of low Regularity. In particular applying results from metric Geometry/Theory of Length spaces. Further I dealt with several results/counterexamples concerning the regularity of geodesics for metrics below $C^{1,1}$ differentiability, as well as results on the exponential map for the borderline case of $C^{1,1}$ metrics. In the talk I will focus more on the geodesic aspect of the thesis, in particular focusing on the connection of geodesics and shortest paths as well as the regularity of shortest paths for metrics of low regularity.
23. Mar. 2018 Melanie Graf
Rigidity of asymptotically $AdS_2 \times S^2$ spacetimes
AbstractWe will illustrate some of the properties of $AdS_2 \times S^2$ that one can recover for spacetimes that are asymptotically $AdS_2 \times S^2$ (in a suitable sense) and satisfy the null energy condition, namely that it possesses two transverse foliations by totally geodesic achronal null hypersurfaces, whose intersections are round 2-spheres. Without going into to much detail I shall try to present a road map on how to get this result. Next, I want to look at the additional condition $\nabla Ric =0$. It is known that this condition is very restrictive for the structure of any Riemannian manifold, forcing it be locally a product of Einstein manifolds. Under the added assumption that Ric is non-degenerate it is similarly restrictive in the Lorentzian case. This allows one to show that any asymptotically $AdS_2 \times S^2$ spacetime satisfying the null energy condition and $\nabla Ric=0$ is globally isometric to $AdS_2 \times S^2$. This is joint work in progress with Greg Galloway.
13. Apr. 2018 Stefan Palenta
Colliding plane gravitational waves and the circularly polarized impulsive case
AbstractThe talk will give a detailed introduction into spacetimes describing colliding plane gravitational waves. In this setup the Einstein equations reduce to the hyperbolic Ernst equation, whose solution in terms of the inverse scattering method will be sketched. As a byproduct of this solution scheme, a four parameter class of colliding wave solution is obtained, which includes the well-known Szekeres-class and the Nutku-Halil solution as limiting cases. Another limit is discussed which should be described as the collision of circularly polarized impulsive waves.
20. Apr. 2018 Roland Steinbauer
Inconclusive remarks on the memory effect for impulsive gravitational plane waves
AbstractIn a recent paper [ZDH:18, see the attachment] the authors have investigated particle motion in plane impulsive gravitational waves from the perspective of the so-called "memory effect". Thereby they have rediscovered some of our earlier results and some "new" results which are claimed resp. seem to differ from what we had found out. The approach of [ZHD:18] is heavily built upon the use of symmetries and---of course---formal calculations involving products of distributions. These symmetries and the respective particle motion have been investigated in the extended (i.e., non-impulsive) case in the interconnected series of papers [ZDGH:17a, ZDGH:17b,DGHZ:17]. I will try to take you on a walk through these works and hopefully lay the foundations for an informed discussion of these matters. [ZDH:18] Zhang, P.-M.; Duval, C.; Horvathy, P. A. Memory effect for impulsive gravitational waves. Classical Quantum Gravity 35 (2018), 065011, 20 pp. [ZDGH:17a] Zhang, P.-M.; Duval, C.; Gibbons, G. W.; Horvathy, P. A. The memory effect for plane gravitational waves. Phys. Lett. B 772 (2017), 743–746. [ZDGH:17b] Zhang, P.-M.; Duval, C.; Gibbons, G. W.; Horvathy, P. A. Soft gravitons and the memory effect for plane gravitational waves. Phys. Rev. D 96, (2017), 064013, 20pp. [DGHZ:17] Duval, C.; Gibbons, G. W.; Horvathy, P. A.; Zhang, P.-M. Carroll, symmetry of plane gravitational waves. Classical Quantum Gravity 34 (2017), no. 17, 175003, 11 pp.
27. Apr. 2018 No seminar
04. May. 2018 Ondřej Hruška
B-metrics with a cosmological constant and their physical interpretation
AbstractWe study exact solutions of Einstein’s field equations, namely the B-metrics by the classification of Ehlers and Kundt (1962). They can be obtained from general Plebański–Demiański line element, which represents solution of algebraic type D with a cosmological constant and non-null electromagnetic field. We examine the behaviour of Minkowski, de Sitter and anti-de Sitter geometries in Plebański–Demiański coordinates as backgrounds of the B-metrics. Afterwards, we summarize the physical interpretation of BI-metric with vanishing cosmological constant made by J. R. Gott (1974), which corresponds to a part of spacetime containing tachyon singularity. We also present our generalization of Gott’s work for non-zero cosmological constant.
11. May. 2018 No seminar
18. May. 2018 Simon Reif
Completeness of the trajectories of particles coupled to a general force field
AbstractI will present a paper by Candela, Sanchez and Romero. They studied extendability of the solutions of a certain second order differential equation on a Riemannian manifold. I will give an overview of the paper and after that I will present you the main results in detail. First I prove the autonomous case. The non-autonomous case will be reduced to the autonomous one by working on the 1-dim higher manifold M \times \mathbb{R}. I will also show you examples including an application to realistic pp-waves.
25. May. 2018 Jiří Veselý
t.b.a.
01. Jun. 2018 No seminar
08. Jun. 2018 Volker Branding
The supersymmetric nonlinear sigma model as a geometric variational problem
AbstractWe will discuss the functional of the supersymmetric nonlinear sigma model as a geometric variational problem. In the case of a Riemannian domain its critical points couple the harmonic map equation to spinor fields, these became known as Dirac-harmonic maps and variants thereof in the mathematics literature. We will give an overview of the current known results on the geometric and analytic properties of the critical points. Finally, we will present recent work on the existence problem for the hyperbolic version of the problem, which arises if the domain is a globally hyperbolic manifold.
15. Jun. 2018 No seminar
22. Jun. 2018 Eduard Nigsch
The curvature of conical metrics
AbstractThe curvature of the conical metric $ds^2 = dr^2 + A^2 r^2 d\phi^2$, $\lvert A \rvert < 1$, cannot be calculated classically because this metric is not regular enough at the tip of the cone. Interpreted as a distributional metric tensor, this metric falls outside the class of gt-regular metrics for which the curvature tensor is still well-defined as a distribution. One possibility to give a meaning to this curvature is to resort to regularization procedures and remove the regularizationin the end. Although this was done in the setting of full Colombeau algebras already in 1996 by Clarke, Vickers and Wilson, who obtained that the scalar curvature is proportional to a delta distribution, their result was not automatically diffeomorphism invariant (although this was shown later). In my talk I will sketch a theory of nonlinear generalized tensor fields which allows to recover the same result, but in much more generality (i.e., with a rather large class of regularizations) and with built-in diffeomorphism invariance.
29. Jun. 2018 Roman Popovych
t.b.a.