Misner space

Misner space is an abstract mathematical spacetime, first described by Charles W. Misner. It is also known as the Lorentzian orbifold $$\mathbb{R}^{1,1}/\text{boost}$$. 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


 * $$ ds^2= -dt^2 + dx^2,$$

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


 * $$ (t, x) \to (t \cosh (\pi) + x \sinh(\pi), x \cosh (\pi) + t \sinh(\pi)).$$

It can also be defined directly on the cylinder manifold $$\mathbb{R} \times S$$ with coordinates $$(t', \varphi)$$ by the metric


 * $$ ds^2= -2dt'd\varphi + t'd\varphi^2,$$

The two coordinates are related by the map


 * $$ t= 2 \sqrt{-t'} \cosh\left(\frac{\varphi}{2}\right)$$


 * $$ x= 2 \sqrt{-t'} \sinh\left(\frac{\varphi}{2}\right)$$

and


 * $$ t'= \frac{1}{4}(x^2 - t^2)$$
 * $$ \phi= 2 \tanh^{-1}\left(\frac{x}{t}\right)$$

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 $$(t', \varphi)$$, the loop defined by $$t = 0, \varphi = \lambda$$, with tangent vector $$X = (0,1)$$, has the norm $$g(X,X) = 0$$, making it a closed null curve. This is the chronology horizon : there are no closed timelike curves in the region $$t < 0$$, while every point admits a closed timelike curve through it in the region $$t > 0$$.

This is due to the tipping of the light cones which, for $$t < 0$$, remains above lines of constant $$t$$ but will open beyond that line for $$t > 0$$, causing any loop of constant $$t$$ 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 $$\langle T_{\mu\nu} \rangle_\Omega$$ is divergent.