  We establish a uniform estimate for the injectivity radius of the past null
cone of a point in a general Lorentzian manifold foliated by spacelike
hypersurfaces and satisfying an upper curvature bound. Precisely, our main
assumptions are, on one hand, upper bounds on the null curvature of the
spacetime and the lapse function of the foliation, and sup-norm bounds on the
deformation tensors of the foliation. Our proof is inspired by techniques from
Riemannian geometry, and it should be noted that we impose no restriction on
the size of the curvature or deformation tensors, and allow for metrics that
are "far" from the Minkowski one. The relevance of our estimate is illustrated
with a class of plane-symmetric spacetimes which satisfy our assumptions but
admit no uniform lower bound on the curvature not even in the L2 norm. The
conditions we put forward, therefore, lead to a uniform control of the
spacetime geometry and should be useful in the context of general relativity.
