View source for Physics:Misner space

'''Misner space''' is an abstract mathematical [[Physics:Spacetime|spacetime]],<ref>{{cite book |first1=S. |last1=Hawking |first2=G. |last2=Ellis |title=The Large Scale Structure of Space-Time |publisher=Cambridge University Press |year=1973 |page=171 |isbn=0-521-20016-4 }}</ref> first described by [[Biography:Charles W. Misner|Charles W. Misner]].<ref>{{cite book |last=Misner |first=C. W. |year=1967 |chapter=Taub-NUT space as a counterexample to almost anything |title=Relativity Theory and Astrophysics I: Relativity and Cosmology |editor-first=J. |editor-last=Ehlers |series=Lectures in Applied Mathematics |volume=8 |publisher=American Mathematical Society |pages=160–169 |chapter-url=https://ntrs.nasa.gov/search.jsp?R=19660007407 }}</ref> It is also known as the Lorentzian [[Orbifold|orbifold]] <math>\mathbb{R}^{1,1}/\text{boost}</math>. It is a simplified, two-dimensional version of the [[Physics:Taub–NUT space|Taub–NUT space]]time. 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 (mathematics)|metric]]

:<math> ds^2= -dt^2 + dx^2,</math>

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

:<math> (t, x) \to (t \cosh (\pi) + x \sinh(\pi), x \cosh (\pi) + t \sinh(\pi)).</math>

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

:<math> ds^2= -2dt'd\varphi + t'd\varphi^2,</math>

The two coordinates are related by the map

:<math> t= 2 \sqrt{-t'} \cosh\left(\frac{\varphi}{2}\right)</math>

:<math> x= 2 \sqrt{-t'} \sinh\left(\frac{\varphi}{2}\right)</math>

and

:<math> t'= \frac{1}{4}(x^2 - t^2)</math>
:<math> \phi= 2 \tanh^{-1}\left(\frac{x}{t}\right)</math>

==Causality==

Misner space is a standard example for the study of causality since it contains both closed timelike curves and a compactly generated [[Physics:Cauchy horizon|Cauchy horizon]], while still being flat (since it is just Minkowski space). With the coordinates <math>(t', \varphi)</math>, the loop defined by <math>t = 0, \varphi = \lambda</math>, with tangent vector <math>X = (0,1)</math>, has the norm <math>g(X,X) = 0</math>, making it a closed null curve. This is the chronology horizon : there are no closed timelike curves in the region <math>t < 0</math>, while every point admits a closed timelike curve through it in the region <math>t > 0</math>.

This is due to the tipping of the light cones which, for <math>t < 0</math>, remains above lines of constant <math>t</math> but will open beyond that line for <math>t > 0</math>, causing any loop of constant <math>t</math> 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,<ref>{{cite journal | last=Hawking | first=S. W. | title=Chronology protection conjecture | journal=Physical Review D | publisher=American Physical Society (APS) | volume=46 | issue=2 | date=1992-07-15 | issn=0556-2821 | doi=10.1103/physrevd.46.603 | pages=603–611| pmid=10014972 | bibcode=1992PhRvD..46..603H }}</ref> by showing that in the semiclassical approximation, the expectation value of the stress-energy tensor for the vacuum <math>\langle T_{\mu\nu} \rangle_\Omega</math> is divergent.

== References ==
{{Reflist}}

==Further reading==
*{{cite journal |first1=M. |last1=Berkooz |first2=B. |last2=Pioline |first3=M. |last3=Rozali |url=http://iopscience.iop.org/1475-7516/2004/08/004/ |title=Closed Strings in Misner Space: Cosmological Production of Winding Strings |journal=Journal of Cosmology and Astroparticle Physics |year=2004 |volume=2004 |issue=8 |page=004 |doi=10.1088/1475-7516/2004/08/004 |arxiv=hep-th/0405126 |bibcode=2004JCAP...08..004B |s2cid=119408206 }}

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[[Category:General relativity]]

{{Sourceattribution|Misner space}}