Moritz Reintjes

A non-perturbative construction of the fermionic projector on globally hyperbolic manifolds I: Space-times of finite lifetime

The previous functional analytic construction of the fermionic projector on globally hyperbolic Lorentzian manifolds is extended to space-times of infinite lifetime. The construction is based on an analysis of families of solutions of the Dirac equation with a varying mass parameter. It makes use of the so-called mass oscillation property which implies that integrating over the mass parameter generates decay of the Dirac wave functions at infinity. We obtain a canonical decomposition of the solution space of the massive Dirac equation into two subspaces, independent of observers or the choice of coordinates. The constructions are illustrated in the examples of ultrastatic space-times and de Sitter space-time.

No regularity singularities exist at points of general relativistic shock wave interaction between shocks from different characteristic families

We give a constructive proof that coordinate transformations exist which raise the regularity of the gravitational metric tensor from C0,1 to C1,1 in a neighbourhood of points of shock wave collision in general relativity. The proof applies to collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. Our result here implies that spacetime is locally inertial and corrects an error in our earlier Proc. R. Soc. A publication, which led us to the false conclusion that such coordinate transformations, which smooth the metric to C1,1, cannot exist. Thus, our result implies that regularity singularities (a type of mild singularity introduced in our Proc. R. Soc. A paper) do not exist at points of interacting shock waves from different families in spherically symmetric spacetimes. Our result generalizes Israels celebrated 1966 paper to the case of such shock wave interactions but our proof strategy differs fundamentally from that used by Israel and is an extension of the strategy outlined in our original Proc. R. Soc. A publication. Whether regularity singularities exist in more complicated shock wave solutions of the Einstein–Euler equations remains open.

Points of general relativistic shock wave interaction are ‘regularity singularities’ where space–time is not locally flat

We show that the regularity of the gravitational metric tensor in spherically symmetric space–times cannot be lifted from C 0,1 to C 1,1 within the class of C 1,1 coordinate transformations in a neighbourhood of a point of shock wave interaction in General Relativity, without forcing the determinant of the metric tensor to vanish at the point of interaction. This is in contrast to Israel9s theorem, which states that such coordinate transformations always exist in a neighbourhood of a point on a smooth single shock surface. The results thus imply that points of shock wave interaction represent a new kind of regularity singularity for perfect fluids evolving in space–time, singularities that make perfectly good sense physically, that can form from the evolution of smooth initial data, but at which the space–time is not locally Minkowskian under any coordinate transformation. In particular, at regularity singularities, delta function sources in the second derivatives of the metric exist in all coordinate systems of the C 1,1 -atlas, but due to cancellation, the full Riemann curvature tensor remains supnorm bounded .