Logarithmic local energy decay for scalar waves on a general class of\n asymptotically flat spacetimes

2015/09/28 by Georgios Moschidis, Moschidis, Georgios · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.1509.08495

openalex publication_date 2015/09/28 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

This paper establishes that on the domain of outer communications of a\ngeneral class of stationary and asymptotically flat Lorentzian manifolds of\ndimension d+1, d\≥3, the local energy of solutions to the scalar wave\nequation square_g\ψ=0 decays at least with an inverse logarithmic rate.\nThis class of Lorentzian manifolds includes (non-extremal) black hole\nspacetimes with no restriction on the nature of the trapped set. Spacetimes in\nthis class are moreover allowed to have a small ergoregion but are required to\nsatisfy an energy boundedness statement. Without making further assumptions,\nthis logarithmic decay rate is shown to be sharp. Our results can be viewed as\na generalisation of a result of Burq, dealing with the case of the wave\nequation on flat space outside compact obstacles, and results of\nRodnianski--Tao for asymptotically conic product Lorentzian manifolds. The\nproof will bridge ideas of Rodnianski--Tao with techniques developed in the\nblack hole setting by Dafermos--Rodnianski. As a soft corollary of our results,\nwe will infer an asymptotic completeness statement for the wave equation on the\nspacetimes considered, in the case where no ergoregion is present.\n

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• Energy bounds and ergoregion instability in Einstein-Maxwell-Scalar field models
• Logarithmic wave decay for short range wavespeed perturbations with radial regularity
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