Unique continuation from infinity for linear waves

Abstract

We prove various uniqueness results from null infinity, for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. We find that the parts of infinity where we must impose a vanishing condition depend strongly on the background geometry. In particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than the ones in the Minkowski space-time. The results are nearly optimal in many respects. They can be considered analogues of uniqueness from infinity results for second order elliptic operators. This work is partly motivated by questions in general relativity.

arXiv comment47 pages; updated references; proofs in sections 4 and 5 slightly simplified

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• Wave equations with initial data on compact Cauchy horizons
• Unique continuation from infinity in asympotically Anti-de Sitter spacetimes II: Non-static boundaries
• The bulk-boundary correspondence for the Einstein equations in asymptotically Anti-de Sitter spacetimes
• On Schwarzschild causality in higher dimensions
• A gauge-invariant unique continuation criterion for waves in asymptotically Anti-de Sitter spacetimes
• Stationarity of asymptotically flat non-radiating electrovacuum spacetimes
• Carleman Estimate for Ultrahyperbolic Operators and Improved Interior Control for Wave Equations
• A uniqueness theorem for 3D semilinear wave equations satisfying the null condition
• On the rigidity from infinity for nonlinear Alfvén waves