Counterexamples to the Lorentzian Calder\'on problem

TL;DR

This paper demonstrates that different Lorentzian metrics can produce identical hyperbolic Dirichlet-to-Neumann maps, challenging uniqueness in inverse problems for Lorentzian geometry.

Contribution

It provides counterexamples showing non-uniqueness in the Lorentzian Calderón problem for smooth, globally hyperbolic metrics.

Findings

Two non-isometric Lorentzian metrics share the same Dirichlet-to-Neumann map.

Counterexamples exist on an infinite cylinder with timelike boundary.

The result challenges previous assumptions of uniqueness in inverse Lorentzian problems.

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.