Spectral theory of non-symmetric operators in time-dependent external fields and in globally hyperbolic spacetimes
Final Report Abstract
The fermionic projector can be applied in quantum field theory to introduce the fermionic vacuum state, the construction of which, non-perturbative and independent of the choice of a time function, was previously unknown in (timedependent) globally hyperbolic spacetimes. Within our DFG-project, we succeeded to give a non-perturbative construction of the fermionic projector on a globally hyperbolic Lorentzian manifold, which has previously been constructed in a formal power series expansion only. First, we propose a functional analytic construction of the fermionic projector on globally hyperbolic Lorentzian manifolds of finite lifetime, and we show that the proposed operator gives the correct splitting of the solution space of the Dirac equation in closed FRW spacetimes. Second, we extend the construction of the fermionic projector to globally hyperbolic spacetimes of infinite lifetime, by analyzing families of solutions of the Dirac equation with a varying mass parameter and by introducing and using the “mass oscillation property” which generates suitable decay of wave functions at infinity. After my work on the non-perturbative construction of the fermionic projector, I started to work on the question whether the gravitational metric tensor g of shock wave solutions of the Einstein equations is smooth enough for space time to be locally inertial. We succeeded to 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 neighborhood 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. The result implies that spacetime is locally inertial and corrects an earlier error, 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), 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. Whether regularity singularities exist in more complicated shock wave solutions of the Einstein Euler equations remains open. We therefore set out in an ongoing project to study possible physical implications of these singularities, assuming their existence.
Publications
- “No Regularity Singularities Exist at Points of General Relativistic Shock Wave Interaction between Shocks from Different Characteristic Families”, Proc. R. Soc. A 471:20140834
M. Reintjes and B. Temple
(See online at https://doi.org/10.1098/rspa.2014.0834) - "Spacetime is Locally Inertial at Points of General Relativistic Shock Wave Interaction between Shocks from Different Characteristic Families”
M. Reintjes
- “A Non-Perturbative Construction of the Fermionic Projector on Globally Hyperbolic Manifolds I - Space-Times of Finite Lifetime”, Adv. Theor. Math. Phys. 19.3 (2015)
M. Reintjes and F. Finster
- “A Non-Perturbative Construction of the Fermionic Projector on Globally Hyperbolic Manifolds II - Space-Times of Infinite Lifetime”
M. Reintjes and F. Finster
