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

TL;DR

This paper studies the existence, uniqueness, and explicit representation of solutions to the global characteristic Cauchy problem for linear wave equations on globally hyperbolic Lorentzian manifolds, with applications to quantum field theory.

Contribution

It establishes conditions for the existence and uniqueness of solutions, and derives an explicit formula involving null expansion, advancing understanding of wave equations on curved spacetimes.

Findings

Existence of continuous global solutions under support restrictions.

Uniqueness in Sobolev space $H^{1/2+ ext{epsilon}}$ with support conditions.

Explicit solution representation involving null expansion.

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 $H^{1/2+\varepsilon}_\mathrm{loc}$ 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.