Counterexamples to the Lorentzian Calderón problem

1. Problem Statement

The Lorentzian Calderón problem is an inverse boundary value problem for hyperbolic partial differential equations on Lorentzian manifolds. The goal is to determine the underlying geometry (the Lorentzian metric $g$) of a spacetime $(M, g)$ based on measurements taken at the boundary. Specifically, one seeks to recover the metric up to boundary-fixing diffeomorphisms given the hyperbolic Dirichlet-to-Neumann (DN) map $\Lambda^{\text{Hyp}}_g$.

• The Setup: Let $M$ be a non-compact, connected smooth manifold of dimension $1+n$ ($n \ge 1$) equipped with a globally hyperbolic Lorentzian metric $g$. Let $\mathcal{M} \subset M$ be a "cylinder" (a submanifold of the same dimension with a smooth timelike boundary $\partial \mathcal{M}$).
• The Data: The DN map is defined by solving the wave equation $\square_g u = 0$ with Dirichlet boundary data $u|_{\partial \mathcal{M}} = \phi$ and zero initial conditions ($u|_{\tau \ll 0} = 0$). The map $\Lambda^{\text{Hyp}}_g$ sends the boundary data $\phi$ to the normal derivative $\partial_\nu u|_{\partial \mathcal{M}}$.
• The Question: If two globally hyperbolic metrics $g$ and $g'$ on $\mathcal{M}$ yield the same DN map ($\Lambda^{\text{Hyp}}_g = \Lambda^{\text{Hyp}}_{g'}$), are they necessarily isometric via a diffeomorphism that fixes the boundary?

While recent positive results exist under stringent geometric conditions, this paper investigates whether uniqueness holds generally, specifically for data on an infinite cylinder with a timelike boundary.

2. Methodology

The authors employ a strategy based on the finite speed of propagation inherent to hyperbolic equations and the causal structure of Lorentzian manifolds.

A. Causal Isolation

The core mechanism relies on the existence of a non-empty open set $U \subset \mathcal{M}$ that is causally disconnected from the boundary $\partial \mathcal{M}$. Specifically, the authors look for regions where:
$$J^-_g(U) \cap \partial \mathcal{M} = \emptyset \quad \text{or} \quad J^+_g(U) \cap \partial \mathcal{M} = \emptyset$$
where $J^\pm$ denote the causal future/past. In such regions, signals from the boundary cannot reach $U$, and signals from $U$ cannot reach the boundary.

B. Metric Perturbation

The authors construct a counterexample by perturbing the metric $g$ to a new metric $g'$ only within the causally isolated region $U$.

1. Construction: They define $g'$ such that $g' = g$ outside a compact subset of $U$, but $g'$ differs significantly (e.g., having different curvature) inside $U$.
2. Global Hyperbolicity: They ensure $g'$ remains globally hyperbolic and that the original Cauchy temporal function $\tau$ remains valid for $g'$.
3. Source-to-Solution Map: They prove that if the source-to-solution map (mapping internal sources to solutions) is identical for $g$ and $g'$, then the boundary DN maps must also be identical.

C. Key Propositions

• Proposition 1.1: If there exists a causally isolated open set $U$, one can construct a metric $g'$ such that the source-to-solution maps coincide ($L_g = L_{g'}$), yet $g$ and $g'$ are not isometric.
• Proposition 1.2: If the source-to-solution maps coincide ($L_g = L_{g'}$), then the hyperbolic DN maps coincide ($\Lambda^{\text{Hyp}}_g = \Lambda^{\text{Hyp}}_{g'}$).

3. Key Contributions and Results

The paper provides explicit counterexamples demonstrating that the Lorentzian Calderón problem is not uniquely solvable in general, even for smooth, globally hyperbolic spacetimes.

A. Three Specific Counterexamples

The authors construct specific spacetimes containing cylinders with timelike boundaries where a causally isolated region exists:

1. Minkowski Hyperboloid:

   • Geometry: A domain in Minkowski space defined by $|x|^2 - a|x| - t^2 \le 0$.
   • Mechanism: The boundary is a hyperboloid. There exists a diamond-shaped region $U$ in the interior such that no null geodesic starting from $U$ can reach the boundary (and vice versa).
   • Result: Uniqueness fails; the metric can be altered inside $U$ without affecting boundary data.

2. Black and White Hole Spacetimes (Kruskal Extension):

   • Geometry: The maximally extended Schwarzschild spacetime.
   • Mechanism: The authors consider a cylinder $\mathcal{M}$ defined by $r < r_0$ (where $r_0 > r_S$).
     • The Black Hole region ($r < r_S, T > 0$) is causally disconnected from the exterior boundary in the future ($J^+(B) \cap \partial \mathcal{M} = \emptyset$).
     • The White Hole region ($r < r_S, T < 0$) is causally disconnected from the exterior boundary in the past ($J^-(W) \cap \partial \mathcal{M} = \emptyset$).
   • Result: The geometry of the black or white hole interior cannot be uniquely determined from boundary measurements on the cylinder.

3. Big Bounce Cosmology (FLRW):

   • Geometry: A Friedmann–Lemaître–Robertson–Walker metric with scale factor $a(t) = \cosh(Ht)$.
   • Mechanism: In a cylinder of radius $R > \pi/H$, the conformal time $\eta$ is bounded ($0 < \eta < \pi/H$). Light rays can only travel a maximum conformal distance of $\pi/H$.
   • Result: There exist central regions ($U$ and $U'$) near the "top" and "bottom" of the time evolution that are causally disconnected from the spatial boundary $\partial \mathcal{M}$. Perturbations in these regions are invisible to the boundary.

B. Technical Proofs

• Non-Isometry: The authors use the Morse–Sard theorem to ensure the perturbed metric $g'$ has a scalar curvature critical value set different from $g$, proving they are not isometric.
• Equality of Maps: They utilize the finite propagation speed to show that solutions to the wave equation with sources supported in the "visible" region are identical for both metrics, and by density arguments (Lemma 3.4), this extends to the boundary DN map.

4. Significance

• Limitation of Inverse Problems: The paper establishes that unlike the Riemannian Calderón problem (where uniqueness often holds under mild conditions), the Lorentzian version is fundamentally limited by the causal structure. If a region is causally isolated from the boundary, its geometry is invisible to boundary measurements.
• Physical Interpretation: This result has profound implications for spacetime cloaking and inverse scattering. It suggests that in relativistic settings, one can hide geometric changes (or even entire regions like black hole interiors) from external observers if those regions are causally disconnected from the observation surface.
• Clarification of Conditions: The work explains why recent positive results in the literature require such stringent geometric conditions (e.g., specific curvature bounds or global visibility). Without these conditions, uniqueness is impossible.
• Methodological Advancement: The paper bridges inverse problems with Lorentzian causality theory, providing a rigorous framework (Propositions 1.1 and 1.2) for constructing counterexamples in globally hyperbolic spacetimes.

In summary, Oksanen and Särkkänen demonstrate that the Lorentzian Calderón problem is ill-posed in the sense of uniqueness for general globally hyperbolic spacetimes with timelike boundaries, due to the existence of causally isolated regions where the metric can be arbitrarily modified without affecting boundary data.