In this paper, we obtain general conditions under which the wave equation is well-posed in spacetimes with metrics of Lipschitz regularity. In particular, the results can be applied to spacetimes where there is a loss of regularity on a hypersurface such as shell-crossing singularities, thin shells of matter and surface layers. This provides a framework for regarding gravitational singularities, not as obstructions to the world lines of point-particles, but rather as an obstruction to the dynamics of test fields.
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