Solutions and stability of the semi-classical Einstein equation on Friedman-Robertson-Walker spacetimes - a phase space approach

采用半经典爱因斯坦方程的方法，研究了量子化的相对论物质作为非自相互作用的量子场与引力场的耦合。对于宇宙Friedman-Robertson-Walker(FRW)时空，我们将给出这个方程的初值公式，它是一个(潜在隐式的)无限维量子场模动力系统。我们打算从共形耦合的情况开始给出相空间的严格的数学描述，在这种情况下，对于特定的初始条件，已经建立了整体解的存在性，然后再到一般耦合。特别地，我们将通过先验估计和Faedeo Galerkin方法研究一般耦合问题局部解和整体解的存在性及其稳定性。上述解的稳定性分析，特别强调无限维动力系统中的稳定和不稳定流形，将应用于动力学的不动点，如Minkowski时空和接近时空奇点(大爆炸)和晚时间(宇宙的命运)的渐近标度行为。我们还将研究它在非最大对称时空中的推广。此外，还将开发一个数值解算器，以求解导出的方程组，该数值解算器与姐妹会议的物理和数学性质有内在联系。

It is proposed to study the coupling of quantized, relativistic matter modeled as a non- selfinteracting quantum field to the gravitational field treated non-quantum mechanically following the semi-classical Einstein equation (SCE) approach. We will provide an initial value formulation for this equation as a (potentially implicit) infinite-dimensional dynamical system of quantum field modes for the case of cosmological Friedman-Robertson-Walker (FRW) spacetimes. We intend to give a mathematically rigorous description of the phase space starting with the conformally coupled case where global existence of solutions has already been estab- lished for specific sets of initial conditions and proceeding to general couplings. In particular, we will investigate existence of local and global solutions and their stability for general coupling via a priori estimates and the Faedeo Galerkin method. A stability analysis of the afore mentioned solutions, with a special emphasis on stable and unstable manifolds in infinite dimensional dynamical systems, will be applied to fixed points of the dynamics like Minkowski spacetime and asymptotic scaling behaviour close to spacetime singularities (big bang) and late times (fate of the universe). We will also study the generalization to non maximally symmetric space-times. Also, a numerical solver for the derived system of equations will be developed that is intrinsically linked to the physical and mathematical nature of the SCE.