  We study homologically maximizing timelike geodesics in conformally flat
tori. A causal geodesic $\gamma$ in such a torus is said to be homologically
maximizing if one (hence every) lift of $\gamma$ to the universal cover is
arclength maximizing. First we prove a compactness result for homologically
maximizing timelike geodesics. This yields the Lipschitz continuity of the time
separation of the universal cover on strict sub-cones of the cone of future
pointing vectors. Then we introduce the stable time separation $\mathfrak{l}$.
As an application we prove relations between the concavity properties of
$\mathfrak{l}$ and the qualitative behavior of homologically maximizing
geodesics.
