Misner space is an abstract mathematical spacetime,[1] first described by Charles W. Misner.[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.
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
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.
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.