

Complex analysis and mathematical physics
February 16, 2016 16:00, Moscow, Steklov Mathematical Institute, Room 430 (8 Gubkina)






Conformal reference frames for Lorentzian manifolds
I. V. Maresin^{} ^{} Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

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Abstract:
The twistor approach to Lorentzian manifolds (spacetimes) is based
on the 5dimensional space $\mathfrak{N}$ of all null geodesics.
For the Minkowski space case it is described as “null projective twistors”
(a real hypersufrace in the complex projective 3space),
but there is no canonical complex structure in general, curved case.
One can rely on skies $\mathfrak{S}_x$ instead for each $x\in X$.
The very definition of $\mathfrak{N}$ is problematical,
unless the spacetime $X$ meets some convexity conditions.
We can resort to the foliation of null geodesics in the bundle of skies,
whereas $\mathfrak{N}$ (which could be defined as the space of leaves of the former)
is endowed with a contact structure.
This structure can be presented in terms of complex linear bundles
and the Lorentz vector representation $({^1/_2},{^1/_2})$.
We shall consider a conformal reference frame, that is,
a projection of the bundle of skies (as well as of $\mathfrak{N}$)
onto a 3dimensional real manifold,
compatible with aforementioned contact structure.
It permits to use surfaces (sky images) in the 3manifold
to describe the spacetime, as well as the differential geometry on it.
A kind of “partial” almost complex structure on the bundle of skies appears,
that compensates us for the loss of the global twistors’ complex structure.

