# Smoothness of the twistor space of a lorentzian manifold, or “convexity wrt null geodesics”

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Null lines in Minkowski space form a 5-dimensional manifold, represented as a (real) quadric $\mathbf{PN}\subset\mathbb{C}\mathbf{P}^3$. This is a well-known fact, on which R. Penrose’s twistor programme is based. It is also known that the space of null lines can be constructed locally (albeit without Cauchy–Riemann structure in the curved case) and that the resulting space $\mathfrak{N}$ of (non-parametrized) null geodesics possesses a natural contact structure, co-oriented one if original manifold is time-oriented.

As of global constructions, a Cauchy hypersurface $M$ permits to describe $\mathfrak{N} = ST^*M$; see e.g. arxiv:0810.5091. This requires global hyperbolicity, a strong condition on the original manifold, that isn’t anywhere near necessity for $\mathfrak{N}$ to be a manifold.

On the other hand, for a lorentzian manifold $X$ (or d+1 pseudo-Riemann for arbitrary dimension) let’s define $\mathfrak{S}_x$ (called the sky of $x$) as the projectivization of all null vectors in $T_x X$, diffeomorphic (and conformly equivalent) to the sphere $S^{d-1}$. Let $\mathfrak{S}X$ be the bundle of skies for all $x\in X$, with (d−1)-dimensional fibres. Its total space (2 d-dimensional) has a natural foliation with one-dimensional leaves, namely null geodesics. Now we build $\mathfrak{N}$ as the space of leaves (in other words, the quotient space of $\mathfrak{S}X$ by equivalence relation to lie on the same null geodesic) and for a strongly causal lorentzian manifold it should result in a Hausdorff space. But it isn’t necessary a manifold, as the following 1+1-dimensional example shows:

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Intuitively, the property looked for smoothness of $\mathfrak{N}$ is convexity of $X$ wrt null geodesics, but Ī̲’m unsure how to say it exactly. Convexity of subsets in aforementioned sense is well-known in lorentzian geometry, but Ī̲ failed to find in papers anything like “convex”, “light-convex” or “causally convex” referring to entire manifold. Can anybody suggest a strict formulation that is weaker than global hyperbolicity?

This post imported from StackExchange MathOverflow at 2015-12-22 18:43 (UTC), posted by SE-user Incnis Mrsi
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