  Motivated by the application to spacetimes of general relativity we
investigate the geometry and regularity of Lorentzian manifolds under certain
curvature and volume bounds. We establish several injectivity radius estimates
at a point or on the past null cone of a point. Our estimates are entirely
local and geometric, and are formulated via a reference Riemannian metric that
we canonically associate with a given observer $(p,T)$ --where $p$ is a point
of the manifold and $T$ is a future-oriented time-like unit vector prescribed
at $p$. The proofs are based on a generalization of arguments from Riemannian
geometry. We first establish estimates on the reference Riemannian metric, and
then express them in term of the Lorentzian metric. In the context of general
relativity, our estimates should be useful to investigate the regularity of
spacetimes satisfying Einstein field equations.
