Signature Changing Spacetimes and WKB Approximations in General Relativity
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Some properties of spacetimes that change signature from Riemannian (positive definite metric) to Lorentzian (metric has a single negative eigenvalue) are investigated. Specifically, the form of geodesics and solutions to the Klein-Gordon equation are calculated. Geodesics behave as expected since they are equally well defined for Lorentzian and Riemannian manifolds, though null geodesics cease to have meaning in the Riemannian region. Solutions to the Klein-Gordon equation exhibit oscillatory behavior in the Lorentzian region and exponential behavior in the Riemannian region. In an effort to further interpret these results, approximate wave solutions are found for a generic spacetime using WKB approximations in the large momentum limit. This approximation encompasses traditional, non-degenerate spacetimes as well as those that change signature. This solution is shown to break down near regions where the metric becomes degenerate, except in the 1 + 1 dimensional case. Further, these solutions can define a vector field with the gradient of their phase. The integral curves of the resulting field are shown to be geodesics, parametrized by an affine parameter.