On the global “two-sided” characteristic Cauchy problem for linear wave equations on manifolds

2017/08/31 by Umberto Lupo · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Black Holes and Theoretical Physics #Cauchy distribution #Geometric Analysis and Curvature Flows #Hypersurface #Initial value problem #Mathematical analysis #Mathematics #Pure mathematics #Sobolev space #Uniqueness #gr-qc #math-ph #math.AP #math.MP #msc:35L15 #msc:53C50 #msc:58J45 #msc:58J47 #msc:58Z05 #msc:81T20

paper · pdf · doi:10.1007/s11005-018-1088-6

published in Letters in Mathematical Physics 108(10), 2315-2362 (Springer Science+Business Media) · 41 pages, 1 figure. Some typos fixed, close to published version

openalex created_date 2017/08/31 · arxiv created 2018/04/30 · openalex publication_date 2018/04/30 · arxiv updated 2018/05/01 · openalex updated_date 2026/08/05

Abstract

The global characteristic initial value problem for linear wave equations on globally hyperbolic Lorentzian manifolds is examined, for a class of smooth initial value hypersurfaces satisfying favourable global properties. First it is shown that, if geometrically well-motivated restrictions are placed on the supports of the (smooth) initial datum and of the (smooth) inhomogeneous term, then there exists a continuous global solution which is smooth "on each side" of the initial value hypersurface. A uniqueness result in Sobolev regularity H^1/2+ε_loc is proved among solutions supported in the union of the causal past and future of the initial value hypersurface, and whose product with the indicator function of the causal future (resp. past) of the hypersurface is past compact (resp. future compact). An explicit representation formula for solutions is obtained, which prominently features an invariantly defined, densitised version of the null expansion of the hypersurface. Finally, applications to quantum field theory on curved spacetimes are briefly discussed.

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