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 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,[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
Hawking and Ellis, The large scale structure of space-time, section 5.8, p. 171
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
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
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Time travel
General terms and concepts
Chronology protection conjecture Closed timelike curve Novikov self-consistency principle Self-fulfilling prophecy Quantum mechanics of time travel
Time travel in fiction
Timelines in fiction
in science fiction in games
Temporal paradoxes
Grandfather paradox Causal loop
Parallel timelines
Alternative future Alternate history Many-worlds interpretation Multiverse Parallel universes in fiction
Philosophy of space and time
Butterfly effect Determinism Eternalism Fatalism Free will Predestination
Spacetimes in general relativity that
can contain closed timelike curves
Alcubierre metric BTZ black hole Gödel metric Kerr metric Krasnikov tube Misner space Tipler cylinder van Stockum dust Traversable wormholes
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