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arxiv: 2604.20320 · v1 · submitted 2026-04-22 · 🧮 math.AP · math.DG

Counterexamples to the Lorentzian Calder\'on problem

Lauri Oksanen , Miika Sarkkinen This is my paper

Pith reviewed 2026-05-10 00:08 UTC · model grok-4.3

classification 🧮 math.AP math.DG
keywords Lorentzian metricsDirichlet-to-Neumann mapCalderón problemglobal hyperbolicitywave equationinverse problemstimelike boundary
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The pith

Two non-isometric smooth Lorentzian metrics on an infinite cylinder can produce identical hyperbolic Dirichlet-to-Neumann maps.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs two distinct smooth metrics on an infinite cylinder that are globally hyperbolic and Lorentzian yet produce exactly the same boundary response data for the wave equation. This establishes a concrete case of non-uniqueness for the inverse problem of recovering the interior geometry from timelike boundary observations. A sympathetic reader cares because the result isolates a setting where the Lorentzian Calderón problem fails to determine the metric uniquely, even under smoothness and global hyperbolicity. The infinite extent of the cylinder and the timelike character of its boundary are essential to the construction that keeps the metrics different inside while matching all measured wave data on the boundary.

Core claim

Two non-isometric, smooth, globally hyperbolic Lorentzian metrics on an infinite cylinder with timelike boundary can induce identical hyperbolic Dirichlet-to-Neumann maps.

What carries the argument

The hyperbolic Dirichlet-to-Neumann map that records the normal derivative of solutions to the wave equation at the timelike boundary for given Dirichlet data.

If this is right

  • The Lorentzian Calderón problem lacks uniqueness on this class of non-compact manifolds with timelike boundaries.
  • Additional conditions such as compactness or a compact time interval are needed to restore uniqueness for metric recovery.
  • Wave propagation data on the boundary cannot distinguish these two interior geometries despite their difference.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The counterexamples suggest that finite-time observations or different boundary topologies might eliminate the non-uniqueness.
  • Similar constructions could be tested on other non-compact Lorentzian manifolds to map the boundary between unique and non-unique regimes.

Load-bearing premise

The construction is limited to an infinite cylinder geometry with timelike boundary together with the global hyperbolicity and smoothness of the two metrics.

What would settle it

An explicit calculation or numerical check that the two constructed metrics produce measurably different boundary responses on the cylinder would disprove the claimed equality of the maps.

Figures

Figures reproduced from arXiv: 2604.20320 by Lauri Oksanen, Miika Sarkkinen.

Figure 1
Figure 1. Figure 1: A cross-section of the Minkowski domain with a timelike hy￾perboloidal boundary. The shaded region is the domain U in (2.2). The hyperboloid asymptotes to the boundaries of the causal future and past of U but never touches them. where (t, x) are the natural coordinates on R 1+n . The Minkowski space is the paradigm case of a globally hyperbolic manifold where the level surfaces of t give a smooth folia￾tio… view at source ↗
Figure 2
Figure 2. Figure 2: The Kruskal diagram of the maximally extended Schwarzschild spacetime. The black hole is the region {r < rS, T > 0} and the white hole is the region {r < rS, T < 0}. The wavy lines r = 0 are the black and white hole singularities that are not part of the spacetime. with l ∈ [0, ∞], γ(0) ∈ ∂M, and ˙γ(0) pointing inside. To study light rays in M, it is convenient to introduce the conformal time coordinate (2… view at source ↗
Figure 3
Figure 3. Figure 3: The spacetime cylinder with a timelike boundary in the Big Bounce spacetime. The region U is the upward-opening truncated cone, and U ′ is the downward-opening one. 3. Proofs We now prove Proposition 1.1, starting with three lemmas. Then we turn to the proof of Proposition 1.2. Lemma 3.1. Let (M, g) be a time-oriented Lorentzian manifold and let U ⊂ M be open. Then J ± g (U) = I ± g (U). Proof. Let us prov… view at source ↗
read the original abstract

We show that two non-isometric, smooth, globally hyperbolic Lorentzian metrics can have the same hyperbolic Dirichlet-to-Neumann map on an infinite cylinder with timelike boundary.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper constructs two non-isometric, smooth, globally hyperbolic Lorentzian metrics on an infinite cylinder with timelike boundary that induce identical hyperbolic Dirichlet-to-Neumann maps, thereby furnishing counterexamples to uniqueness in the Lorentzian Calderón problem under these geometric hypotheses.

Significance. If the construction holds, the result is significant for inverse problems in Lorentzian geometry: it shows that the hyperbolic DN map fails to determine the metric uniquely even for smooth metrics when the underlying manifold is non-compact (an infinite cylinder) and the boundary is timelike. The explicit deformation-based construction, which preserves both global hyperbolicity and the DN map while altering the metric, supplies concrete, verifiable examples rather than an abstract non-uniqueness argument; this strengthens the contribution and highlights the necessity of additional assumptions (compactness, specific curvature conditions, or spacelike boundaries) for uniqueness theorems.

minor comments (2)
  1. The abstract and introduction could briefly indicate the dimension of the cylinder and the precise form of the deformation used to equate the DN maps, for immediate readability.
  2. A short remark comparing the timelike-boundary setting here to existing uniqueness results that require spacelike boundaries or compact Cauchy surfaces would help situate the counterexample within the broader literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central result is an explicit existence construction of two non-isometric smooth globally hyperbolic Lorentzian metrics on the infinite cylinder with timelike boundary that share the same hyperbolic Dirichlet-to-Neumann map. This is achieved by direct geometric deformation that preserves the required hypotheses while equating the maps via invariance; no fitted parameters are renamed as predictions, no self-definitional loops appear in the equations, and no load-bearing uniqueness theorems are imported from the authors' prior work. The derivation is self-contained against the stated geometric assumptions and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard background results in Lorentzian geometry and hyperbolic PDE theory; no new free parameters or invented entities are introduced in the abstract statement.

axioms (1)
  • domain assumption Existence and uniqueness of solutions to the wave equation on globally hyperbolic Lorentzian manifolds with timelike boundary.
    Implicit in the definition of the hyperbolic Dirichlet-to-Neumann map.

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Reference graph

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