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An Invitation to Lorentzian Geometry

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Abstract

The intention of this article is to give a flavour of some global problems in General Relativity. We cover a variety of topics, some of them related to the fundamental concept of Cauchy hypersurfaces:

  1. (1)

    structure of globally hyperbolic spacetimes,

  2. (2)

    the relativistic initial value problem,

  3. (3)

    constant mean curvature surfaces,

  4. (4)

    singularity theorems,

  5. (5)

    cosmic censorship and Penrose inequality,

  6. (6)

    spinors and holonomy.

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Fig. 1
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Notes

  1. Typically, an a priori stronger hypothesis that causality is used to define global hyperbolicity, namely, strong causality. In [24] it has been shown that both notions agree. For simplicity, we renounce giving details here and simply use the more recent definition of global hyperbolicity instead.

  2. Time machines are in contradiction to the unspoken fundamental assumption of the free will of the experimentalist taken by the vast majority of physicists, in the sense that any observer, in contrast to physical nature around him, is assumed to be able to take decisions like preparing a spin-up or a spin-down state in a manner which is in principle unpredictable for others, compare the discussion of Bell’s inequality and the EPR paradox. Note that without that assumption, time machines do not contradict any other principles of physics—with the possible exception of predictability of nature, cf. the article of Krasnikov [70] as well as its critical reception in [77].

  3. He considered a different, but equivalent, notion of global hyperbolicity, based on the compactness of the space of causal curves connecting each two points, but this is not specially relevant at this point.

  4. Because of the gradient condition |∇u|<f(u).

  5. Typically, this data must satisfy: (i) (Σ,h,σ) is asymptotically flat, (ii) T satisfies the dominant property, and the equations for T constitute a quasilinear, diagonal, second order hyperbolic system, (iii) the fall-off of the initial value of T on Σ is fast enough for the h-distance, and h is assumed to be complete.

  6. More precisely, the latter means that the spacetime is strongly asymptotically predictable, see [114]. Recall that WCCC cannot be regarded as a particular case of SCCC.

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Acknowledgements

The second-named author is partially supported by the Grants MTM2010–18099 (MICINN) and P09-FQM-4496 (J. Andalucía) with FEDER funds.

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  1. Fakultät für Mathematik, Universität Regensburg, Universitätsstraße 31, 93053, Regensburg, Germany

    Olaf Müller

  2. Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, Campus de Fuentenueva s/n., 18071, Granada, Spain

    Miguel Sánchez

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Müller, O., Sánchez, M. An Invitation to Lorentzian Geometry. Jahresber. Dtsch. Math. Ver. 115, 153–183 (2014). https://doi.org/10.1365/s13291-013-0076-0

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