  Given a Lorentzian manifold, the light ray transform of a function is its
integrals along null geodesics. This paper is concerned with the injectivity of
the light ray transform on functions and tensors, up to the natural gauge for
the problem. First, we study the injectivity of the light ray transform of a
scalar function on a globally hyperbolic stationary Lorentzian manifold and
prove injectivity holds if either a convex foliation condition is satisfied on
a Cauchy surface on the manifold or the manifold is real analytic and null
geodesics do not have cut points. Next, we consider the light ray transform on
tensor fields of arbitrary rank in the more restrictive class of static
Lorentzian manifolds and show that if the geodesic ray transform on tensors
defined on the spatial part of the manifold is injective up to the natural
gauge, then the light ray transform on tensors is also injective up to its
natural gauge. Finally, we provide applications of our results to some inverse
problems about recovery of coefficients for hyperbolic partial differential
equations from boundary data.
