Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

Abstract

We consider inverse problems in space-time (M, g), a 4-dimensional Lorentzian manifold. For semilinear wave equations g u + H(x, u) = f , where g denotes the usual Laplace-Beltrami operator, we prove that the source-to-solution map L : f → u|V , where V is a neighborhood of a time-like geodesic µ, determines the topological, differentiable structure and the conformal class of the metric of the space-time in the maximal set where waves can propagate from µ and return back. Moreover, on a given space-time (M, g), the source-to-solution map determines some coefficients of the Taylor expansion of H in u .

Determination of Space‐Time Structures from Gravitational Perturbations

“…In this article we consider a general stress-energy tensor, but we consider the response to both gravitational and electromagnetic perturbations. Similar inverse problems have been studied for semilinear wave equations with quadratic nonlinearity in Kurylev-Lassas-Uhlmann [28], with the general nonlinear term in Lassas-Uhlmann-Wang [30] and with quadratic derivative terms in Wang-Zhou [39]. In particular, in [30] the authors improved the previous results in [28] so that the isometry class of the metric and some information about the nonlinear terms can be determined in many cases.…”

“…Similar inverse problems have been studied for semilinear wave equations with quadratic nonlinearity in Kurylev-Lassas-Uhlmann [28], with the general nonlinear term in Lassas-Uhlmann-Wang [30] and with quadratic derivative terms in Wang-Zhou [39]. In particular, in [30] the authors improved the previous results in [28] so that the isometry class of the metric and some information about the nonlinear terms can be determined in many cases. In de Hoop-Uhlmann-Wang [11], the nonlinear responses of two scalar waves at an interface of different media was considered and the related inverse problem was addressed.…”

“…In particular, K is the set in M carrying the singularities produced by three wave interactions. The proof of the proposition below is the the same as that of proposition 4.1 of [30] for the scalar case, so we just state the result without proving it. j h1 x .j / ; .j / .R g / h q 0 but the tangent vectors at q 0 are linearly dependent, then U .4/ is smooth away from J g g .q 0 /.…”

“…In Section 6, we follow the methods in [26,27] to prove the determination of the conformal class of the metric. To determine the conformal factor, we analyze the conformal transformation of the principal symbols of the newly generated waves as in [30] and prove Theorem 2.1 when the measurements are in wave gauge. For other gauges, e.g., the Fermi coordinates associated with freely falling observers, we need an additional step to show that such singularities are observable after the gauge transformation.…”

“…So in the above proposition (as well as the rest of the analysis), we basically only look at the new singularities produced by the interaction of four distorted plane waves. In section 3.5 of [30], the principal symbols of the terms (5.7) are found explicitly, and we recall them here. Consider the symbols at .q; / P y g q 0 n , which is joined with .q 0 ; / P q 0 by bicharacteristics.…”