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On the splitting problem for Lorentzian manifolds with an $\mathbb{R}$-action with causal orbits

We study the interplay between the global causal and geometric structures of a spacetime $(M,g)$ and the features of a given smooth $\mathbb{R}$-action $ρ$ on $M$ whose orbits are all causal curves, building on classic results about Lie group actions on manifolds described by Palais. In the first part of this paper, we prove that $ρ$ is free and proper (so that $M$ splits topologically) provided that $(M,g)$ is strongly causal and $ρ$ does not have what we call weakly ancestral pairs, a notion which admits a natural interpretation in terms of "cosmic censorship". Accordingly, such condition holds automatically if $(M,g)$ is globally hyperbolic. We also prove that $M$ splits topologically if $(M,g)$ is strongly causal and $ρ$ is the flow of a complete conformal Killing causal vector field. In the second part, we investigate the class of Brinkmann spacetimes, which can be regarded as null analogues of stationary spacetimes in which $ρ$ is the flow of a complete parallel null vector field. Inspired by the geometric characterization of stationary spacetimes in terms of standard stationary ones [24], we obtain an analogous geometric characterization of when a Brinkmann spacetime is isometric to a standard Brinkmann spacetime. This result naturally leads us to discuss a conjectural null analogue for Ricci-flat $4$-dimensional Brinkmann spacetimes of a celebrated rigidity theorem by Anderson [1], and highlight its relation with a long-standing 1962 conjecture by Ehlers and Kundt [13]. If true, our conjecture provides strong mathematical support to the idea that gravitational plane waves are the most natural "boundary conditions" at infinity for vacuum solutions of the Einstein equation modeling regions outside gravitationally radiating sources.