Misner space

Misner space is an abstract mathematical spacetime, discovered by Charles Misner of the University of Maryland.  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

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

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

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

References 

  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

Category:General relativity