Determination of the Spacetime from Local Time Measurements

TL;DR

This paper proves that local time measurements on a submanifold can determine the spacetime metric's jet, advancing inverse problems in Lorentzian geometry and their applications to general relativity.

Contribution

It establishes the unique determination of the spacetime metric's jet from local time data and explores global reconstruction under specific geometric and physical conditions.

Findings

Time measurements determine the metric's $C^$-jet in Fermi coordinates.

Results extend inverse problems from Riemannian to Lorentzian geometry.

Counterexamples show limitations when assumptions are violated.

Abstract

We consider an inverse problem for a Lorentzian spacetime $(M,g)$, and show that time measurements, that is, the knowledge of the Lorentzian time separation function on a submanifold $\Sigma\subset M$ determine the $C^\infty$-jet of the metric in the Fermi coordinates associated to $\Sigma$. We use this result to study the global determination of the spacetime $(M,g)$ when it has a real-analytic structure or is stationary and satisfies the Einstein-scalar field equations. In addition to this, we require that $(M,g)$ is geodesically complete modulo scalar curvature singularities. The results are Lorentzian counterparts of extensively studied inverse problems in Riemannian geometry - the determination of the jet of the metric and the boundary rigidity problem. We give also counterexamples in cases when the assumptions are not valid, and discuss inverse problems in general relativity.