Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON THE GEOMETRY OF LORENTZ SPACES AS A LIMIT SPACE

  • Yun, Jong-Gug (Department of Mathematics Education Korea National University of Education)
  • Received : 2013.06.25
  • Published : 2014.07.31
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Abstract

In this paper, we prove that there is no branch point in the Lorentz space (M, d) which is the limit space of a sequence {($M_{\alpha},d_{\alpha}$)} of compact globally hyperbolic interpolating spacetimes with $C^{\pm}_{\alpha}$-properties and curvature bounded below. Using this, we also obtain that every maximal timelike geodesic in the limit space (M, d) can be expressed as the limit curve of a sequence of maximal timelike geodesics in {($M_{\alpha},d_{\alpha}$)}. Finally, we show that the limit space (M, d) satisfies a timelike triangle comparison property which is analogous to the case of Alexandrov curvature bounds in length spaces.

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References

  1. S. B. Alexander and R. L. Bishop, Lorentz and semi-Riemannian spaces with Alexandrov curvature bounds, Comm. Anal. Geom. 16 (2008), no. 2, 251-282. https://doi.org/10.4310/CAG.2008.v16.n2.a1
  2. L. Bombelli and J. Noldus, The moduli space of isometry classes of globally hyperbolic spacetimes, Classical Quantum Gravity 21 (2004), no. 18, 4429-4453. https://doi.org/10.1088/0264-9381/21/18/010
  3. J. Noldus, A Lorentzian Gromov-Hausdorff notion of distance, Classical Quantum Gravity 21 (2004), no. 4, 839-850. https://doi.org/10.1088/0264-9381/21/4/007
  4. J. Noldus, Lorentzian Gromov Hausdorff theory as a tool for quantum gravity kinematics, Ph.D. thesis, Gent University, 2004.
  5. J. Noldus, The limit space of a Cauchy sequence of globally hyperbolic spacetimes, Classical Quantum Gravity 21 (2004), no. 4, 851-874. https://doi.org/10.1088/0264-9381/21/4/008
  6. C. Plaut, Metric curvature, convergence, and topological finiteness, Duke Math. J. 66 (1992), no. 1, 43-57. https://doi.org/10.1215/S0012-7094-92-06602-6
  7. K. Shiohama, An Introduction to the Geometry of Alexandrov Spaces, GARC, Seoul National University, 1993.
  8. R. Sorkin and E. Woolgar, A causal order for spacetimes with Lorentzian metrics: Proof of compactness of the space of causal curves, Classical Quantum Gravity 13 (1996), no. 7, 1971-1993. https://doi.org/10.1088/0264-9381/13/7/023
  9. J.-G. Yun, Lorentzian Gromov-Hausdorff convergence and limit curve theorem, Commun. Korean Math. Soc. 28 (2013), no. 3, 589-596. https://doi.org/10.4134/CKMS.2013.28.3.589