  In this paper, we prove the existence theorem for longest paths in
sub-Lorentzian problems, which generalizes the classical theorem for globally
hyperbolic Lorentzian manifolds. We specifically address the case of invariant
structures on homogeneous spaces, as the conditions for the existence theorem
in this case can be significantly simplified. In particular, it turns out that
longest paths exist for any left-invariant sub-Lorentzian structures on Carnot
groups.
