TU Berlin Fakultät II
Institut für Mathematik
     

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Lorentzian Geometry

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P. Janik (secretary), Tel.: (++49 30) 314-222 51, Fax: 314-792 82

Technische Universität Berlin
School for Mathematics and Natural Sciences
Department of Mathematics, MA 8-3
Straße des 17. Juni 136
10623 Berlin
Germany
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News

Now available: Advances in Lorentzian Geometry: Proceedings of the Lorentzian Geometry Conference in Berlin, AMS/IP Studies in Advanced Mathematics, S.-T. Yau (series ed.), M. Plaue, A. Rendall and M. Scherfner (eds.), 2011


Topics

Lorentzian geometry is a vivid field of mathematical research that can be seen as part of differential geometry as well as mathematical physics. It represents the mathematical foundation of the general theory of relativity - which is probably one of the most successful and beautiful theories of physics.
In this project, we fuel our research with ideas from theoretical physics. For example, properties of Lorentzian manifolds like causality are linked with properties of a so-called observer field that represents the velocity field of the matter content in the respective spacetime. Geometrically, this observer field represents a time orientation, and the irreducible parts of its covariant derivative can be physically interpreted as rotation, shear and expansion of the spacetime model.
These quantities are called kinematical invariants and play a prominent role in the construction of cosmological models, thermodynamics in curved spacetime and so on. We believe that the kinematical invariants also play an important role in the study of Lorentzian geometry in its own right.

diagram

Our research covers - amongst other things - the following topics:
  • Closed timelike curves
  • Kinematical invariants and causality
  • Visualization of Lorentzian manifolds (gallery)
  • Lorentzian two-manifolds with applications to optimization
  • Exact solutions of the Einstein equations
  • Differentiable structures and four-manifolds
  • Spacetime symmetries
  • Geometry of spacelike hypersurfaces

Guests (Recent)

  • Graham Hall, Univ. of Aberdeen
  • Francois Vigneron, Univ. Paris Est
  • Ryszard Deszcz, Univ. Wroclaw
  • Frédéric Robert, Laboratoire J. A. Dieudonné
  • Henri Anciaux, Univ. Tours

Books

Cover Cover Cover
Advances in Lorentzian Geometry,
M. Plaue and M. Scherfner (eds.),
Shaker (2008)
Gödel-type Spacetimes:
History and New Developments
,
Collegium Logicum. Annals of
the Kurt Gödel Society, M. Plaue
and M. Scherfner (eds.), 2010
Advances in Lorentzian Geometry:
Proceedings of the Lorentzian
Geometry Conference in Berlin
,
AMS/IP Studies in Advanced Mathematics,
S.-T. Yau (series ed.), M. Plaue, A. Rendall
and M. Scherfner (eds.), 2011

Conferences


Talks

Conferences

Seminars

  • A. Dirmeier: "Causality and Topology of Lorentzian Manifolds Admitting a Cartan Flow I and II" (TU Berlin, 2011-06-15 and 2011-06-22, 14:15, MA 851)
  • T. Wowereit: "Algebraische Krümmungsstrukturen vierdimensionaler Lorentzmannigfaltigkeiten I und II" (TU Berlin, 2011-02-09, 14:15, MA 851 and 2011-02-16, 16:15, MA 851)
  • A. Dirmeier: "New Results on Global Hyperbolicity and Metric Completeness I and II" (TU Berlin, 2010-05-18 and 2010-05-25, 16:30, MA 649)
  • S. Ullrich: "Über die Fermiableitung" (TU Berlin, 2009-12-16, 16:15, MA 749)
  • M. Plaue: "Towards Understanding the Conformal Geometry and Causality of Kinematical Spacetimes" (TU Berlin, 2009-09-23, 16:15, MA 550)
  • S. Barthel: "Anwendungen algebraisch- und differentialtopologischer Methoden in der Theorie der Gravitationslinsen" (TU Berlin, 2009-07-03)
  • G. Hall: "Projective Structure on 4-dimensional Lorentz Manifolds" (TU Berlin, 2009-03-30)
  • S. Barthel: "Über den Abbildungsgrad von 'Linsenabbildungen' " (TU Berlin, 2009-02-13)
  • C. Reiher: "Zur Berechnung von Killingfeldern in speziellen Lorentz-Geometrien" (TU Berlin, 2008-10-29)
  • A. Dirmeier: "Über konforme Vektorfelder" (TU Berlin, 2007-11-13)
  • A. Croci: "Frei fallende skalare Ladungen in statischen und homogenen Gravitationsfeldern" (TU Berlin, 2007-11-20)
  • A. Dirmeier: "Konforme Vektorfelder im Newman-Penrose-Formalismus" (TU Berlin, 2008-04-16)

Teaching

The following topics are available for a bachelor/master/diploma thesis:
  • Existence of conformal vector fields in semi-Riemannian geometry and special implications thereof
  • Killing vector fields on special Lorentzian manifolds,
  • Visualization of closed timelike curves,
  • Spacelike hypersurfaces in de Sitter space,
  • any other interesting topic that you come up with.
You can also talk about these topics in the differential geometry seminar for students to obtain a course certificate.

Related Publications

  • J. Dietz, A. Dirmeier and M. Scherfner, Geometric Analysis of Particular Compactly Constructed Time Machine Spacetimes, In Press, J. Geom. Phys. (2012), (doi:10.1016/j.geomphys.2011.11.013)
  • A. Dirmeier, M. Plaue and M. Scherfner, Growth Conditions, Riemannian Completeness and Lorentzian Causality, J. Geom. Phys. 62 (3), (2012), (doi:10.1016/j.geomphys.2011.04.017)
  • J. Dietz, A. Dirmeier and M. Scherfner, Geometric Analysis of Ori-type Spacetimes, Submitted, (2012), (arXiv:1201.1929 [gr-qc])
  • M. Guerses, M. Plaue, M. Scherfner, On a particular type of product manifolds and shear-free cosmological models, Class. Quant. Grav. 28 (17) (2011)
  • A. Dirmeier: Causal Classification of Conformally Flat Lorentzian Cylinders, arXiv:1103.5960 [math.DG], (2011)
  • R. Deszcz, M. Głogowska, M. Plaue, K. Sawicz and M. Scherfner, On hypersurfaces in space forms satisfying particular curvature conditions of Tachibana type, Krag. J. Math. 35 (2) (2011)
  • M. Guerses, M. Plaue, M. Scherfner, T. Schönfeld and L. A. M. de Sousa jr., On spacetimes with given kinematical invariants: Construction and examples, in Gödel-type Spacetimes: History and New Developments (2010), (arXiv:0801.3364v2 [gr-qc])
  • A. Dirmeier, M. Plaue, M. Scherfner, On conformal vector fields parallel to the observer field, Adv. Lorentz. Geom. 1, (2008), (arXiv:0802.3642v2 [math-ph])
  • R. Deszcz, M. Scherfner, On a particular class of warped products with fibres locally isometric to generalized Cartan hypersurfaces, Colloq. Math. 109 (2007), 13-29
  • Z. Hu, M. Scherfner, S. Zhai, On spacelike hypersurfaces with constant scalar curvature in the de Sitter space, J. Diff. Geom. Appl. 25 (2007), 594-611
  • T. Chrobok, Y. Obukhov, M. Scherfner, Rotation in string cosmology, Class. Quantum Grav. 20 (2003)
  • T. Chrobok, Y. Obukhov, M. Scherfner, Shearfree rotating inflation, Phys. Rev. D 66 (2002)
  • T. Chrobok, Y. Obukhov, M. Scherfner, On closed rotating worlds, Phys. Rev. D 63 (2001)
  • T. Chrobok, Y. Obukhov, M. Scherfner, On the construction of shearfree cosmological models, Mod. Phys. Lett. A 16, No. 20 (2001)
  • M. Scherfner, On the construction of spacetime manifolds using kinematical invariants, Proc. MGM IX, Ed. Ruffini, Singapore, World Scientific, (2001)
  • M. Scherfner, Kinematical invariants in Gödel-type models, Proc. Coll. on Cosmic Rotation, Wissenschaft und Technik Verlag Berlin (2000)

Preprints

  • R. Deszcz, M. Plaue, M. Scherfner, On Roter type warped products with 1-dimensional fibres, (preprint 1012)
  • A. Dirmeier, Lorentzian Bochner Technique and the Raychaudhuri Equation, (preprint 2012)
  • S. Born, M. Kossowski, M. Scherfner, Characteristic classes for pseudo-Riemannian manifolds with volume-resolvable metric singularities, (preprint 2012)
  • A. Dirmeier, M. Plaue, M. Scherfner, Global Hyperbolicity of Spacetimes with an Isotropic Hubble Law, (preprint 2012)
  • M. Guerses, M. Plaue, M. Scherfner, Goedel-type metrics in Einstein-aether theory II: nonflat background, (preprint 2012)

A. Dirmeier . 24.02.2012.