Abstract
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.
Highlights
Let (N, g) be a smooth Lorentzian manifold of dimension 1+n, n ≥ 2, with signature (−, +, . . . , +)
Let us remark that given a spacelike Cauchy surface N and a timelike Killing vector field E, it is possible to fix the parametrization for the null geodesics in a natural way as follows
Let Sm = Sm(M) denote the bundle of symmetric tensors of rank m on M and define the geodesic ray transform of ω ∈ Cc∞(M; Sm) along γ in M as follows: τ+ (x,v)
Summary
Let (N , g) be a smooth Lorentzian manifold of dimension 1+n, n ≥ 2, with signature (−, +, . . . , +). We are interested in studying the injectivity of the so-called light ray transform on functions and tensors over such Lorentzian manifolds. We note that such parametrizations can depend on the null geodesic itself. Given any choice of such parametrization along β, we define the light ray transform of α ∈ Cc∞(N ; Sm) along β as follows: Lβ α = α(β(s), β(s)) ds. By the strong causality condition the null geodesic β(s) will lie outside of any compact set K ⊂ N , for |s| large enough Together with the linearity of the map L, this implies that the choice of the parametrization in (1) is of no significance provided that we are concerned with injectivity of the light ray transform on N
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