Misner space is an abstract mathematical spacetime,^[1] first described by Charles W. Misner.^[2] It is also known as the Lorentzian orbifold ${\displaystyle \mathbb {R} ^{1,1}/{\text{boost}}}$. 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

${\displaystyle ds^{2}=-dt^{2}+dx^{2},}$

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

${\displaystyle (t,x)\to (t\cosh(\pi )+x\sinh(\pi ),x\cosh(\pi )+t\sinh(\pi )).}$

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

${\displaystyle ds^{2}=-2dt'd\varphi +t'd\varphi ^{2},}$

The two coordinates are related by the map

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

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

and

${\displaystyle t'={\frac {1}{4}}(x^{2}-t^{2})}$

${\displaystyle \phi =2\tanh ^{-1}\left({\frac {x}{t}}\right)}$

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

This is due to the tipping of the light cones which, for ${\displaystyle t<0}$, remains above lines of constant ${\displaystyle t}$ but will open beyond that line for ${\displaystyle t>0}$, causing any loop of constant ${\displaystyle t}$ 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,^[3] by showing that in the semiclassical approximation, the expectation value of the stress-energy tensor for the vacuum ${\displaystyle \langle T_{\mu \nu }\rangle _{\Omega }}$ is divergent.

References

Further reading

• Berkooz, M.; Pioline, B.; Rozali, M. (2004). "Closed Strings in Misner Space: Cosmological Production of Winding Strings". Journal of Cosmology and Astroparticle Physics. 2004 (8): 004. arXiv:hep-th/0405126. Bibcode:2004JCAP...08..004B. doi:10.1088/1475-7516/2004/08/004. S2CID 119408206.