Solutions and stability of the semi-classical Einstein equation on Friedman-Robertson-Walker spacetimes - a phase space approach
Final Report Abstract
The semi-classical Einstein equation couples quantum fields to general relativity via the expected value of the energy impulse tensor of the quantum field. However, this coupling is subject to singularities which can be corrected by renormalization. It should be noted, however, that this should not contradict Einstein's postulate of general covariance. This excludes renormalization procedures like Wick ordering. Therefore, it is common practice to renormalize on general Lorentz manifolds using the covariant Hadamard parametrix, which covers only the singular part of the quantum state. However, this leads - except in special cases - to higher than second order differential equations in the metric of the underlying spacetime. In this project, the resulting differential equations were investigated for cosmological Friedmann-Lemaître-Robertson-Walker spacetimes with vanishing spatial curvature. The dynamic variables of the system consist of the space-time expansion factor, its first three derivatives and the (infinitely many) dynamic degrees of freedom of the quantum field. However, this system cannot immediately be understood as a dynamic system, since the Hadamard parametrix contains derivatives of the scale factor of arbitrary order. However, in this project we were able to show that a transformation of the system using a new set of dynamic variables, introduced here for the first time, leads to a consistent dynamic system of fourth order in the expansion factor. For this new system, we were able to prove the existence and uniqueness for two classes of states. One class has properties similar to those of the massive Minkowski vacuum and the other like a massless Minkowski-thermal state. The solution theory for this infinitely dimensional dynamic system requires the development of novel methods, similar to Ovsyannikov‘s method. The system can also be solved for any coupling of the scalar curvature to the field equations. Earlier results were limited to one special case - conformal coupling. Further results are the existence of classes of so-called Hadamard states which satisfy the semi-classical Einstein equation and the existence of maximum solutions for vacuum-like states. This establishes, for the first time since the semi-classical Einstein equation was introduced 60 years ago, a complete mathematical theory of this equation for cosmological spacetimes.
Publications
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The Cosmological Semiclassical Einstein Equation as an Infinite-Dimensional Dynamical System, 2018 (33p)
H. Gottschalk, D. Siemssen
