We give a functional analytic construction of the fermionic projector on a globally hyperbolic Lorentzian manifold of finite lifetime. The integral kernel of the fermionic projector is represented by a two-point distribution on the manifold. By introducing an ultraviolet regularization, we get to the framework of causal fermion systems. The connection to the "negative-energy solutions" of the Dirac equation and to the WKB approximation is explained and quantified by a detailed analysis of closed Friedmann-Robertson-Walker universes.

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 Israel's 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.

We present the authors’ new theory of the RT-equations (‘regularity transformation’ or ‘Reintjes–Temple’ equations), nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections Γ to optimal regularity, one derivative smoother than the Riemann curvature tensor Riem( Γ ). As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at general relativistic shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations by application of elliptic regularity theory in L p spaces. The theory and results announced in this paper apply to arbitrary L ∞ connections on the tangent bundle T M of arbitrary manifolds M , including Lorentzian manifolds of general relativity.

We prove that spacetime is locally inertial at points of shock wave collision in General Relativity. The result applies for collisions between shock waves coming from different characteristic families in spherically symmetric spacetimes. We give a constructive proof that there exists coordinate transformations which raise the regularity of the gravitational metric tensor from C 0,1 to C 1,1 in a neighborhood of such points of shock wave interaction and a C 1,1 metric regularity suffices for locally inertial frames to exist. This result was first announced in [16] and the proofs are presented here. This result corrects an error in our earlier publication [15], which led us to the wrong conclusion that such coordinate transformations, which smooth the metric to C 1,1 , cannot exist. Our result here proves that regularity singularities, (a type of mild singularity introduced in [15]), do not exist at points of two interacting shock waves from different families in spherically symmetric spacetimes, and this generalizes Israel's famous 1966 result to the case of such shock wave interactions. The strategy of proof here is an extension of the strategy outlined in [15], but differs fundamentally from the method used by Israel. The question whether regularity singularities exist in more complicated shock wave solutions of the Einstein Euler equations still remains open. 42 1 2 M. REINTJES 8.2. Bootstrapping to C 1 -Regularity 61 8.3. Proof of Proposition 8.1 64 9. The Shock-Collision-Case: The Matching Conditions 65 10. The Shock-Collision-Case: The Proof of Theorem 1.1 72 11. Conclusion 74 Appendix A. The Integrability Condition 75 Acknowledgments 77 Funding 77 References 78which in matrix notation is given bywhich in matrix notation is given by ∂ ∂t J t 0 J r 0 + J r 1 J t 1 ∂ ∂r J t 0 J r 0 = 1 J t 1 J t 1,t J t 1,r J r 1,t J r 1,r J t 0 J r 0 . (8.2) J 0 t,r = J t 1 |J| 2 J r 1 J r 0,r − J r 0 J r 1,r − J r 1 |J| 2 J r 1 J t 0,r − J r 0 J t 1,r .

We give a constructive proof that coordinate transformations exist which raise the regularity of the gravitational metric tensor from C 0,1 to C 1,1 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 C 1,1 , 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 Israel's 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.