Under smallness and decay conditions on nonlocal potentials, symmetric hyperbolic systems on curved spacetimes admit strong solutions to the Cauchy problem, with a sharp threshold beyond which solutions fail.
On the Geometry of Static Spacetimes
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abstract
We review geometrical properties of a static spacetime $(M,g)$, including geodesic completeness, causality, standard splittings, compact $M$, closed geodesics and geodesic connectedness. We pay special attention to the critical quadratic behavior at infinity of the coefficients $\beta$, $\beta^{-1}$ ($\beta = -g(K,K)$, being $K$ a timelike irrotational Killing vector field), which essentially control completeness, causality and geodesic connectedness. Recent references are specially discussed.
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The Cauchy Problem for Symmetric Hyperbolic Systems with Nonlocal Potentials
Under smallness and decay conditions on nonlocal potentials, symmetric hyperbolic systems on curved spacetimes admit strong solutions to the Cauchy problem, with a sharp threshold beyond which solutions fail.