Inverse scattering problems for non-linear wave equations on Lorentzian manifolds

TL;DR

This work addresses inverse scattering for semilinear wave equations on globally hyperbolic Lorentzian manifolds with at least one asymptotically Minkowskian end. It introduces scattering functionals as robust, finite-data surrogates for the scattering operator and develops Penrose-based, near-field reductions together with microlocal analysis of κ-th order wave interactions. The main result proves that the scattering functionals determine the manifold’s topology, differentiable structure, and conformal class, with the metric and nonlinear coefficient recovered up to a conformal gauge: $g^{(1)}=e^{2\gamma}\Psi^*g^{(2)}$ and $a^{(1)}=e^{(\kappa-3)\gamma}\Psi^*a^{(2)}$. The framework extends to FLRW-type cosmologies and other asymptotically Minkowskian geometries, highlighting a pathway for geometric reconstruction from restricted nonlinear scattering data in curved spacetimes.

Abstract

We show that an inverse scattering problem for a semilinear wave equation can be solved on a manifold having an asymptotically Minkowskian infinity, that is, scattering functionals determine the topology, differentiable structure, and the conformal type of the manifold. Moreover, the metric and the coefficient of the non-linearity are determined up to a multiplicative transformation. The manifold on which the inverse problem is considered is allowed to be an open, globally hyperbolic manifold which may have non-trivial topology or several infinities (i.e., ends) of which at least one has to be of the asymptotically Minkowskian type. To formulate the inverse problems we define a new type of data, non-linear scattering functionals, which are defined also in the cases where the classically defined scattering operators are not well-defined. This makes it possible to solve inverse problems also in cases where some of the incoming waves lead to a blow-up of the scattered solution. We use non-linear interaction of waves as a beneficial tool that helps to solve the inverse problem. The corresponding inverse problem for the linear wave equation still remains unsolved.

Figures (10)

• Figure 1: Left: The Penrose map is a conformal map $\Phi:\mathbb{R}\times \mathbb{R}^3\to \mathbb{R}\times\mathbb{S}^3$ and its image ${{{\widehat{N}}}} = \Phi(\mathbb{R}\times\mathbb{R}^3)\subset \mathbb{R}\times\mathbb{S}^3$ is the Penrose compactification of the Minkowski space, see \ref{['eq:Penrose coordinates']}. In the figure $\mathbb{R}\times\mathbb{S}^3$ is visualized as a cylindrical surface $\mathbb{R}\times \mathbb{S}^1$, and ${{{\widehat{N}}}}$ is visualized as the area shaded by the red lines, that is, ${{{\widehat{N}}}}$ is visualized as a subset that is cut from the cylinder by two "circles", one of which passes through the points $i_0$ and $i_-$ and the other passes through $i_0$ and $i_+$. The lower part of the boundary of ${{{\widehat{N}}}}$ is the past conformal infinity and the upper part of the boundary of ${{{\widehat{N}}}}$ is the future conformal infinity. Right: The Penrose compactification ${\widehat{N}}$ is the extended to a manifold ${N_\mathrm{ext}}$ by gluing to ${\widehat{N}}$ non-physical extensions on the other sides of the future and the past (light-like) infinities. In the figure, the boundary of extended space-time ${N_\mathrm{ext}}$ is marked by blue (color online) curves. The shaded region is the Penrose compactification ${\widehat{N}}$ of the Minkowski space. The space ${\widehat{N}}$ is extended to the Lorentzian manifold ${N_\mathrm{ext}}={\widehat{N}}\cup {{{N}}}^+\cup {{{N}}}^-$ where ${{{N}}}^+$ and ${{{N}}}^-$ are the non-physical parts of extended manifold that are the mirror images of the space ${\widehat{N}}$ on the other side of the future and the past light-like infinities. The boundary of ${N_\mathrm{ext}}$ is marked in the figure by blue curves.
• Figure 2: Visualization of the definition of the asymptotically Minkowskian infinity $E\subset M$. The figures show the Penrose diagrams that are 2-dimensional analogs of the cylinders shown in Figure \ref{['fig:penrose2']}. The map $F$ takes $V$ conformally to $\widehat{V}\cap {{{\widehat{N}}}}$ where ${{{\widehat{N}}}} =\Phi(\mathbb{R}^{1+3})\subset \mathbb{R}\times\mathbb{S}^3$ is the Penrose compactification of the Minkowski space and $\widehat{V}\subset \mathbb{R}\times\mathbb{S}^3$ is a neighborhood of $\mathcal{I}^+\cup \mathcal{I}^-\cup i_0$. The Lorentzian metric on $\widehat{V}$ coincides with the standard metric of $\mathbb{R}\times\mathbb{S}^3$ outside ${{{\widehat{N}}}}$.
• Figure 3: Left: The set $P(R)$ is shown as the horizontal bold gray line. The grayed region depicts the set $S(R)$, while its restriction $S_-(R)$ to the past null infinity $\mathcal{I}^-$ is shown as the diagonal gray line. Right: Sets and the support of the cut-off function $\rho$ used in the proof of Theorem \ref{['thm:nonlinear_goursat copy']}.
• Figure 4: Left: Visualization of setting where scattering functionals are defined. The in-going radiation field $h_-$ is supported on a relatively compact subset of $\mathcal{I}^-$. When $\|h_-\|<\varepsilon(t_1,{q_1})$, the solution $u$ of the scattering problem is defined in the past of the point $q_1$. The scattering functional $S_{t_1,{q}}(h_-)$ takes the value of the out-going radiation field $h_+$ evaluated at the point $q<q_1$. Middle: In our spacetime sources will be produced in the nonphysical past (the lower triangular region below past null infinity). The nonlinear interaction of waves produces new waves in the physical region (shaded red), and the interactions are observed in the nonphysical future. The sources and receivers are separated by the point $i_0$. Right: Schematic picture on the reconstruction. Sources located in $\Omega_\mathrm{in}$ produce waves that interact inside the red shaded region $D$ and cause signals that can be observed in the causally separated domain $\Omega_\mathrm{out}$. The closures of the domains $\Omega_\mathrm{in}$, $\Omega_\mathrm{out}$ and $D$ are disjoint. This causes difficulties that are encountered also in the figure on the figure on the left, and we overcome this issue by introducing a reconstruction algorithm which works in the situation when the light-like geodesics connecting $\Omega_\mathrm{in}$ to $D$ do not have conjugate or cut points.
• Figure 5: Left: Morris-Thorne wormhole manifold, see MorrisThorneMorrisThorne1, is a static universe (that is non-physical due to negative mass) of the form $M=\mathbb{R}\times N_0$, where $N_0$ is illustrated in the figure. Right: Penrose diagramm of a (non-physical) traversable wormhole, see GarattiniBambivisser. Note that this resembles the Penrose diagram of a Schwarzschild blackhole, though in that case the consideration of the point $i_0$ is more complicated due to singularities in the compactification, see blackhole-compactification.

Theorems & Definitions (35)

• Theorem 1
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Theorem 2
• Remark 2
• Lemma 1
• Lemma 2