# Timelike Loop Spaces as Projective Null Twistor Spaces

Let $\mathcal{M}$ be a spacetime, and let $\Omega\mathcal{M}$ denote the loop space of the spacetime. My idea is that the set of all closed timelike curves of $\mathcal{M}$ forms the projective null twistor space. The following is an excerpt from my own essay:

Consider the projective null twistor space of $\mathcal{M}$. This space has light rays equivalent to points, and light cones as Riemann spheres. Let us define $\Omega_t\mathcal{M}$ as the timelike loop space of $\mathcal{M}$, i.e., the set of all timelike curves in $\mathcal{M}$. Consider a closed timelike curve in $\mathcal{M}$. Restrict the curve so that one can deform it into a line. Define a subclass of spacetime $\mathcal{M}$ as the class $\mathbb{A}\subset\Omega_t\mathcal{M}$ of closed timelike curves in $\mathcal{M}$ such that all curves $\varsigma\in\mathbb{A}\subset\Omega_t\mathcal{M}$ are homeomorphic to a portion of the real line $\mathbb{P}$ which lies on the null cone of any point. This class is trivial, as homeomorphisms are equivalence relations. Thus, the class of CTCs forms a light ray in spacetime. This corresponds to a point in $\Omega_t\mathcal{M}$. Similarly, consider a region on the spacelike hypersurface such that the region is bounded by the sections of light rays that fall on the hypersurface. This corresponds to a Riemann sphere in $\Omega_t\mathcal{M}$. Thus, $\Omega_t\mathcal{M}$ is isomorphic to the projective null twistor space of $\mathcal{M}$.

Evolve $\Omega_t\mathcal{M}=\Sigma_T$ (This relation holds since Riemann spheres on $\Omega_t\mathcal{M}\mbox{ and }\Sigma_T$ correspond to sections of light cones on $\mathcal{M}$ and light rays $\mathcal{M}$ to points on $\Omega_t\mathcal{M}\mbox{ and }\Sigma_T$. Therefore, by the above theorem and the transitivity of homeomorphisms, $\Omega_t\mathcal{M}=\Sigma_T$.) with respect to time, where $\Sigma_0\times\mathbb{R}^1=\mathcal{M}$, with respect to time to get the the positive'' part of $\mathcal{M}$, i.e., the part $\mathcal{M}_+$ of $\mathcal{M}$ with all points $x(t)$ in $\mathcal{M}_+$ having $t>T$.

Does this method of thinking make sense? I have done some research on the topic and it seems to be consistent with earlier established results.

• The question is unclear. Does $\mathcal{M}$ denote Minkowski space (that is globally hyperbolic)? Or, maybe, its conformal extension? Some curved space and/or its conformal extension? – Incnis Mrsi Aug 24 '14 at 8:52