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 formula_1. 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 formula_4 with coordinates formula_5 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 formula_5, the loop defined by formula_12, with tangent vector formula_13, has the norm formula_14, making it a closed null curve. This is the chronology horizon : there are no closed timelike curves in the region formula_15, while every point admits a closed timelike curve through it in the region formula_16.
This is due to the tipping of the light cones which, for formula_15, remains above lines of constant formula_18 but will open beyond that line for formula_16, causing any loop of constant formula_18 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 formula_21 is divergent.