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Misner space
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<h2 id="metric">Metric</h2> <p>The simplest description of Misner space is to consider two-dimensional Minkowski space with the <a href="/facts/Metric_(mathematics)/RkUptSo8">metric</a> </p> d s 2 = − d t 2 + d x 2 , {\displaystyle ds^{2}=-dt^{2}+dx^{2},} <p>with the identification of every pair of spacetime points by a constant boost </p> ( t , x ) → ( t cosh ( π ) + x sinh ( π ) , x cosh ( π ) + t sinh ( π ) ) . {\displaystyle (t,x)\to (t\cosh(\pi )+x\sinh(\pi ),x\cosh(\pi )+t\sinh(\pi )).} <p>It can also be defined directly on the cylinder manifold R × S {\displaystyle \mathbb {R} \times S} with coordinates ( t ′ , φ ) {\displaystyle (t',\varphi )} by the metric </p> d s 2 = − 2 d t ′ d φ + t ′ d φ 2 , {\displaystyle ds^{2}=-2dt'd\varphi +t'd\varphi ^{2},} <p>The two coordinates are related by the map </p> t = 2 − t ′ cosh ( φ 2 ) {\displaystyle t=2{\sqrt {-t'}}\cosh \left({\frac {\varphi }{2}}\right)} x = 2 − t ′ sinh ( φ 2 ) {\displaystyle x=2{\sqrt {-t'}}\sinh \left({\frac {\varphi }{2}}\right)} <p>and </p> t ′ = 1 4 ( x 2 − t 2 ) {\displaystyle t'={\frac {1}{4}}(x^{2}-t^{2})} ϕ = 2 tanh − 1 ( x t ) {\displaystyle \phi =2\tanh ^{-1}\left({\frac {x}{t}}\right)} <h2 id="causality">Causality</h2> <p>Misner space is a standard example for the study of causality since it contains both closed timelike curves and a compactly generated <a href="/facts/Cauchy_horizon/6gn8k4hc">Cauchy horizon</a>, while still being flat (since it is just Minkowski space). With the coordinates ( t ′ , φ ) {\displaystyle (t',\varphi )} , the loop defined by t = 0 , φ = λ {\displaystyle t=0,\varphi =\lambda } , with tangent vector X = ( 0 , 1 ) {\displaystyle X=(0,1)} , has the norm g ( X , X ) = 0 {\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 t < 0 {\displaystyle t<0} , while every point admits a closed timelike curve through it in the region t > 0 {\displaystyle t>0} . </p><p>This is due to the tipping of the light cones which, for t < 0 {\displaystyle t<0} , remains above lines of constant t {\displaystyle t} but will open beyond that line for t > 0 {\displaystyle t>0} , causing any loop of constant t {\displaystyle t} to be a closed timelike curve. </p> <h2 id="chronology-protection">Chronology protection</h2> <p>Misner space was the first spacetime where the notion of chronology protection was used for quantum fields,<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> by showing that in the semiclassical approximation, the expectation value of the stress-energy tensor for the vacuum ⟨ T μ ν ⟩ Ω {\displaystyle \langle T_{\mu \nu }\rangle _{\Omega }} is divergent. </p> <h2 id="further-reading">Further reading</h2> <ul><li>Berkooz, M.; Pioline, B.; Rozali, M. (2004). <a href="http://iopscience.iop.org/1475-7516/2004/08/004/">"Closed Strings in Misner Space: Cosmological Production of Winding Strings"</a>. <i>Journal of Cosmology and Astroparticle Physics</i>. 2004 (8): 004. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/hep-th/0405126">hep-th/0405126</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2004JCAP...08..004B">2004JCAP...08..004B</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F1475-7516%2F2004%2F08%2F004">10.1088/1475-7516/2004/08/004</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119408206">119408206</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Hawking, S.; Ellis, G. (1973). The Large Scale Structure of Space-Time. Cambridge University Press. p. 171. ISBN 0-521-20016-4. <a href="0-521-20016-4" target="_blank">0-521-20016-4</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Misner, C. W. (1967). "Taub-NUT space as a counterexample to almost anything". In Ehlers, J. (ed.). Relativity Theory and Astrophysics I: Relativity and Cosmology. Lectures in Applied Mathematics. Vol. 8. American Mathematical Society. pp. 160–169. <a href="https://ntrs.nasa.gov/search.jsp?R=19660007407" target="_blank">https://ntrs.nasa.gov/search.jsp?R=19660007407</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Kaku, Michio (28 December 2004). Parallel Worlds: The Science Of Alternative Universes And Our Future In The Cosmos. Penguin. pp. 136–138. <a href="/wiki/Penguin_Books" target="_blank">/wiki/Penguin_Books</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Hawking, S. W. (1992-07-15). "Chronology protection conjecture". Physical Review D. 46 (2). American Physical Society (APS): 603–611. Bibcode:1992PhRvD..46..603H. doi:10.1103/physrevd.46.603. ISSN 0556-2821. PMID 10014972. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>