Misner space

Abstract mathematical spacetime

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

d s 2 = − d t 2 + d x 2 , {\displaystyle ds^{2}=-dt^{2}+dx^{2},}

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

( 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 )).}

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

d s 2 = − 2 d t ′ d φ + t ′ d φ 2 , {\displaystyle ds^{2}=-2dt'd\varphi +t'd\varphi ^{2},}

The two coordinates are related by the map

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)}

and

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)}

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 ( 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} .

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.

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 ⟨ T μ ν ⟩ Ω {\displaystyle \langle T_{\mu \nu }\rangle _{\Omega }} is divergent.

References

  1. ^ Hawking, S.; Ellis, G. (1973). The Large Scale Structure of Space-Time. Cambridge University Press. p. 171. ISBN 0-521-20016-4.
  2. ^ 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.
  3. ^ 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.

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.