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.
The simplest description of Misner space is to consider two-dimensional Minkowski space with the metric
with the identification of every pair of spacetime points by a constant boost
It can also be defined directly on the cylinder manifold R × × --> S {\displaystyle \mathbb {R} \times S} with coordinates ( t ′ , φ φ --> ) {\displaystyle (t',\varphi )} by the metric
The two coordinates are related by the map
and
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 φ --> ) {\displaystyle (t',\varphi )} , the loop defined by t = 0 , φ φ --> = λ 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.
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 μ --> ν ν --> ⟩ ⟩ --> Ω Ω --> {\displaystyle \langle T_{\mu \nu }\rangle _{\Omega }} is divergent.
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