Signature Change at Material Layers and Step Potentials

Abstract

For a contravariant 4-metric which changes signature from Lorentzian to Riemannian across a spatial hypersurface, the mixed Einstein tensor is manifestly non-singular. In Gaussian normal coordinates, the metric contains a step function and the Einstein tensor contains the Dirac delta function with support at the junction. The coefficient of the Dirac function is a linear combination of the second fundamental form (extrinsic curvature) of the junction. Thus, unless the junction has vanishing extrinsic curvature, the physical interpretation of the metric is that it describes a layer of matter (with stresses but no energy or momentum) at the junction. In particular, such metrics do not satisfy the vacuum Einstein equations, nor the Einstein-Klein-Gordon equations and so on. Similarly, the d'Alembertian of a Klein-Gordon field contains the Dirac function with coefficient given by the momentum of the field. Thus, if the momentum of the field does not vanish at the junction, the physical interpretation is that there is a source (with step potential) at the junction. In particular, such fields do not satisfy the massless Klein-Gordon equation. These facts contradict claims in the literature.