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.

建议按照半经典爱因斯坦方程（SCE）方法研究模拟为非自相互作用量子场的量化相对论物质与非量子机械处理的引力场的耦合。我们将为这个方程提供一个初始值公式，作为宇宙弗里德曼-罗伯逊-沃克（FRW）时空情况下的量子场模式的（可能隐式的）无限维动力系统。我们打算对相空间进行严格的数学描述，从共形耦合的情况开始，其中已经针对特定的初始条件集建立了解的全局存在性，然后继续到一般耦合。特别是，我们将通过先验估计和 Faedeo Galerkin 方法研究局部和全局解的存在性及其一般耦合的稳定性。上述解决方案的稳定性分析，特别强调无限维动力系统中的稳定和不稳定流形，将应用于动力学的固定点，如明可夫斯基时空和接近时空奇点（大爆炸）和晚期的渐近标度行为次（宇宙的命运）。我们还将研究非最大对称时空的推广。此外，还将开发导出方程组的数值求解器，该求解器与 SCE 的物理和数学性质有内在联系。