Abstract:
The twistor approach to Lorentzian manifolds (spacetimes) is based
on the 5-dimensional 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 3-space),
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 3-dimensional real manifold,
compatible with aforementioned contact structure.
It permits to use surfaces (sky images) in the 3-manifold
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.
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