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

Friedman-Robertson-Walker 时空上半经典爱因斯坦方程的解和稳定性 - 相空间方法

• 批准号: 279133405
• 负责人: Professor Dr. Hanno Gottschalk
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2015
• 资助国家: 德国
• 起止时间: 2014-12-31 至 2017-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/279133405?language=en
• 关键词: Solutions; stability; semi; classical; Einstein

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.

提议研究量化的相对论问题的耦合，以非自我互动量子场建模到在半经典爱因斯坦方程（SCE）方法后，机械处理的非量子的重力场。我们将为该方程式提供一个初始值公式，作为量子场模式的（潜在隐式）无限二维动力学系统，用于宇宙学弗里德曼·罗伯逊 - 沃克（FRW）的频率。我们打算从数学上严格描述相位空间，从共同耦合的情况开始，在这些情况下，对于特定的初始条件和一般耦合，已经为全球存在解决方案存在。特别是，我们将通过先验估计和Faedeo Galerkin方法研究局部和全球解决方案的存在及其稳定性。对上述解决方案的稳定性分析，并特别强调了无限尺寸动力学系统中稳定和不稳定的歧管，将应用于诸如Minkowski时空和渐近缩放行为之类的固定点，几乎是时空的奇异性（大爆炸）和宇宙的后期（宇宙的衰落）。我们还将研究对非最大对称空间时间的概括。同样，还将开发出一个与SCE的物理和数学性质链接的衍生方程系统的数值求解器。