Misner space

Misner space[1] is an abstract mathematical spacetime, discovered by Charles Misner of the University of Maryland.[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

1. Hawking and Ellis, The large scale structure of space-time, section 5.8, p. 171
2. Misner, "Taub-NUT space as a counterexample to almost anything," in Relativity theory and astrophysics I: relativity and cosmology, ed. J. Ehlers, 1967, p. 160; publicly available at https://ntrs.nasa.gov/search.jsp?R=19660007407
3. Hawking, S. W. (1992-07-15). "Chronology protection conjecture". Physical Review D. American Physical Society (APS). 46 (2): 603–611. doi:10.1103/physrevd.46.603. ISSN 0556-2821. PMID 10014972.

• 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