Signature change at material layers and step potentials

For a contravariant 4-metric which changes signature from Lorentzian to Riemannian across a spatial hypersurface, the mixed Einstein tensor is manifestly nonsingular. 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. PACS: 04.20.Cv, 02.40.Ky Recently there has been growing interest in metrics which change signature, particularly from Riemannian to Lorentzian across a spatial hypersurface. Such metrics can be thought of as classical idealisations of tunnelling solutions in quantum cosmology, where the idea originally arose [1]. In order to satisfy the Einstein equations, the junction of signature change must be totally geodesic, i.e. have vanishing second fundamental form (extrinsic curvature) [1]. More generally, the momenta of the physical fields must vanish at the junction. Although this is well known in some quarters, much of the recent literature on signature change contains claims to the contrary. In particular, Ellis et al. and subsequent authors [2–11] have claimed that there are signature-changing solutions to the Einstein equations, without surface layers, with nonvanishing extrinsic curvature at the junction. The incorrectness of such claims will be demonstrated here simply by calculating the Einstein tensor of the proposed solutions, in the coordinates that Ellis et al. used. Also, Dray et al. and subsequent authors [12–18] have claimed that there are solutions to the massless Klein-Gordon equation across a change of signature with non-vanishing momentum at the junction. The incorrectness of such claims will be demonstrated here simply by calculating the d’Alembertian of the proposed solutions, in the same coordinates as above. These points have mostly been made before [19–30] but it is worth repeating them since many authors continue to claim the contrary. Moreover, rather than just showing that the proposed solutions do not satisfy the equations, this article will go a stage further