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Remarks on the determination of the Lorentzian metric by the lengths of geodesics or null-geodesics

Gregory Eskin

TL;DR

Problem: determine the Lorentzian metric on $\mathbb{R}\times\mathbb{R}^n$ from lengths of space-time geodesics. Approach: formulate the problem in a Hamiltonian framework with energy $H$ and derive that the geodesic length data $R(g,T,\hat y,\hat\eta)$ is tied to $H$ via $R=\sqrt{2H(y,\eta_0,\eta)}\,T$, then develop perturbation estimates for null-geodesics to compare nearby metrics. Main results: (i) a pointwise rigidity theorem showing equality of geodesic lengths fixes the metric at the base point (and boundary derivatives) when metrics are close, and (ii) a corresponding rigidity result for null-geodesics showing equality of Euclidean lengths implies equality of the underlying Hamiltonians; Significance: the work advances inverse boundary problems in Lorentzian geometry by proving local uniqueness from geodesic-length data.

Abstract

We consider a Lorentzian metric in $\mathbb{R}\times\mathbb{R}^n$. We show that if we know the lengths of the space-time geodesics starting at $(0,y,η)$ when $t=0$, then we can recover the metric at $y$. We prove the rigidity of Lorentzian metrics. We also prove a variant of the rigidity property for the case of null-geodesics: if two metrics are close and if corresponding null-geodesics have equal Euclidian lengths then the metrics are equal.

Remarks on the determination of the Lorentzian metric by the lengths of geodesics or null-geodesics

TL;DR

Problem: determine the Lorentzian metric on from lengths of space-time geodesics. Approach: formulate the problem in a Hamiltonian framework with energy and derive that the geodesic length data is tied to via , then develop perturbation estimates for null-geodesics to compare nearby metrics. Main results: (i) a pointwise rigidity theorem showing equality of geodesic lengths fixes the metric at the base point (and boundary derivatives) when metrics are close, and (ii) a corresponding rigidity result for null-geodesics showing equality of Euclidean lengths implies equality of the underlying Hamiltonians; Significance: the work advances inverse boundary problems in Lorentzian geometry by proving local uniqueness from geodesic-length data.

Abstract

We consider a Lorentzian metric in . We show that if we know the lengths of the space-time geodesics starting at when , then we can recover the metric at . We prove the rigidity of Lorentzian metrics. We also prove a variant of the rigidity property for the case of null-geodesics: if two metrics are close and if corresponding null-geodesics have equal Euclidian lengths then the metrics are equal.
Paper Structure (5 sections, 3 theorems, 86 equations)

This paper contains 5 sections, 3 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. The determination of the metric
  3. Estimates for the null-geodesics
  4. Rigidity of the Lorentzian metric
  5. Rigidity in the case of null-geodesics

Key Result

Theorem 2.1

Suppose $g_1$ and $g_2$ are two inverse metric tensors in $\mathbb{R}\times\mathbb{R}^n$. Suppose integrals $R(g_1,T,\hat{y},\hat{\eta})$ and $R(g_2,T,\hat{y},\hat{\eta})$ are equal for all $\hat{y},\hat{\eta}$. Then the metrics $g_1^{-1}$ and $g_2^{-1}$ are also equal.

Theorems & Definitions (3)

  • Theorem 2.1
  • Theorem 4.1
  • Theorem 5.1