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.

给定一个洛伦兹流形，函数的光线变换就是其沿零坐标测地线的积分。本文关注的是光线在函数和张量上的注入性，直至问题的自然尺度。首先，我们研究整体双曲平稳洛伦兹流形上标量函数的光线变换的注入性，并证明如果流形上的柯西曲面上满足凸凸叶面条件或该流形为实解析和零大地测量学，则注入性成立没有切点。接下来，我们考虑在更严格的静态洛伦兹流形中的任意等级的张量场上进行光线变换，并表明如果在流形空间部分上定义的张量上的测地线变换在自然尺度上具有内射性，然后，张量上的光线转化也将达到其自然尺度。最后，我们将我们的结果应用于一些反问题，这些问题涉及从边界数据中恢复双曲型偏微分方程的系数。