[Paper Review] Inverse problems in spacetime II: Reconstruction of a Lorentzian manifold from light observation sets

Yaroslav Kurylev, Matti Lassas+1 more|arXiv (Cornell University)|May 14, 2014

Geometric Analysis and Curvature Flows12 references8 citations

TL;DR

This paper addresses two inverse problems in Lorentzian geometry: reconstructing the conformal structure of a spacetime region from passive light observations along a time-like geodesic, and solving an active inverse problem for semilinear wave equations with quadratic non-linearities using a novel non-linearity-based method. In 4D, the source-to-solution operator uniquely determines the topological, differentiable, and conformal structures of the causal domain of dependence from the geodesic.

ABSTRACT

We study two inverse problems on a globally hyperbolic Lorentzian manifold $(M,g)$. The problems are: 1. Passive observations in spacetime: Consider observations in a neighborhood $V\subset M$ of a time-like geodesic $\mu$. Under natural causality conditions, we reconstruct the conformal type of the unknown open, relatively compact set $W\subset M$, when we are given $V$, the conformal class of $g|_V$, and the light observations sets $P_V(q)$ corresponding to all source points $q$ in $W$. The light observation set $P_V(q)$ is the intersection of $V$ and the light-cone emanating from the point $q$, i.e., the points in the set $V$ where light from a point source at $q$ is observed. 2. Active measurements in spacetime: We develop a new method for inverse problems for non-linear hyperbolic equations that utilizes the non-linearity as a tool. This enables us to solve inverse problems for non-linear equations for which the corresponding problems for linear equations are still unsolved. To illustrate this method, we solve an inverse problem for semilinear wave equations with quadratic non-linearities. We assume that we are given the neighborhood $V$ of the time-like geodesic $\mu$ and the source-to-solution operator that maps the source supported on $V$ to the restriction of the solution of the wave equation in $V$. When $M$ is 4-dimensional, we show that these data determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from $\mu$ and return back to $\mu$.

Motivation & Objective

• To reconstruct the conformal type of a spacetime region from passive observations of light cones emanating from sources within that region.
• To solve an inverse problem for non-linear hyperbolic wave equations where linear counterparts remain unsolved.
• To demonstrate that the source-to-solution operator for semilinear wave equations determines the full geometric structure of the spacetime in the causal domain of influence from a time-like geodesic.
• To develop a new method that leverages non-linearity as a tool in inverse problems, enabling solutions where linear methods fail.

Proposed method

• Uses light observation sets $ P_V(q) $, defined as the intersection of a neighborhood $ V $ with the future light cone from a source point $ q $, to infer geometric structure.
• Applies causality conditions and conformal invariance to reconstruct the conformal class of the unknown region $ W \subset M $ from data on $ V $.
• Introduces a novel method for non-linear inverse problems that exploits the non-linear terms in semilinear wave equations as a constructive tool.
• Utilizes the source-to-solution operator, mapping compactly supported sources on $ V $ to the restriction of the solution in $ V $, to recover spacetime structure.
• Employs microlocal analysis and propagation of singularities techniques to analyze the wave equation's solution behavior and infer geometric data.
• Establishes that in 4-dimensional spacetimes, the source-to-solution data determine the topological, differentiable, and conformal structures of the maximal domain where waves propagate from and return to the geodesic $ \mu $.

Experimental results

Research questions

• RQ1 Can the conformal structure of a spacetime region be reconstructed from passive observations of light signals emitted from points within that region?
• RQ2 Can non-linear terms in hyperbolic wave equations be used as a tool to solve inverse problems that remain open for the corresponding linear equations?
• RQ3 To what extent does the source-to-solution operator for semilinear wave equations determine the underlying spacetime geometry in 4D?
• RQ4 Is it possible to recover the full geometric structure—topological, differentiable, and conformal—of a spacetime region from wave data restricted to a neighborhood of a time-like geodesic?

Key findings

• The conformal type of an open, relatively compact subset $ W \subset M $ is uniquely reconstructed from the conformal class of the metric on $ V $, the neighborhood $ V $, and the light observation sets $ P_V(q) $ for all $ q \in W $, under causality assumptions.
• The method successfully solves an inverse problem for semilinear wave equations with quadratic non-linearities, a class for which the linear analog remains unsolved.
• In 4-dimensional spacetimes, the source-to-solution operator uniquely determines the topological, differentiable, and conformal structures of the maximal domain where waves can propagate from and return to the time-like geodesic $ \mu $.
• The non-linearity of the wave equation is not a hindrance but a key enabler in the inverse problem, allowing the recovery of geometric data that would be inaccessible via linear methods.