Misner space

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Misner space[1] is an abstract mathematical spacetime, discovered by Charles Misner of the University of Maryland.[2] It is also known as the Lorentzian orbifold . It is a simplified, two-dimensional version of the Taub-NUT spacetime. It contains a non-curvature singularity and is an important counterexample to various hypotheses in general relativity.

Metric[edit]

The simplest description of Misner space is to consider two-dimensional Minkowski space with the metric

with the identification of every pair of spacetime points by a constant boost

It can also be defined directly on the cylinder manifold with coordinates by the metric

The two coordinates are related by the map

and

Causality[edit]

Misner space is a standard example for the study of causality since it contains both closed timelike curves and a compactly generated Cauchy horizon, while still being flat (since it is just Minkowski space). With the coordinates , the loop defined by , with tangent vector , has the norm , making it a closed null curve. This is the chronology horizon : there are no closed timelike curves in the region , while every point admits a closed timelike curve through it in the region .

This is due to the tipping of the light cones which, for , remains above lines of constant but will open beyond that line for , causing any loop of constant to be a closed timelike curve.

Chronology protection[edit]

Misner space was the first spacetime where the notion of chronology protection was used for quantum fields,[3] by showing that in the semiclassical approximation, the expectation value of the stress-energy tensor for the vacuum is divergent.

References[edit]

  1. ^ Hawking and Ellis, The large scale structure of space-time, section 5.8, p. 171
  2. ^ Misner, "Taub-NUT space as a counterexample to almost anything," in Relativity theory and astrophysics I: relativity and cosmology, ed. J. Ehlers, 1967, p. 160; publicly available at https://ntrs.nasa.gov/search.jsp?R=19660007407
  3. ^ S. Hawking, Chronology protection conjecture, Phys. Rev. D, vol. 46, 2, 1992
  • S. Hawking, G. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, 1973
  • M. Berkooz, B. Pioline, M. Rozali. Closed Strings in Misner Space: Cosmological Production of Winding Strings, Journal of Cosmology and Astroparticle Physics.
  • Misner, C.W. (1967), 'Taub-NUT space as a counterexample to almost anything', Relativity Theory and Astrophysics I: Relativity and Cosmology, ed. J.Ehlers, Lectures in Applied Mathematics, Volume 8 (American Mathematical Society), 160-9