Spectral invariants for stationary spacetimes

谱理论、谱几何和逆谱问题是纯数学和应用数学中经典的和研究得很好的研究领域。研究了拉普拉斯算子的几何与本征值之间的关系。用广义相对论重新解释，这对应于超静态时空上波动方程解空间上的时间平移的生成器。换句话说，本征值本质上是乘积时空中驻波的频率。从物理学的角度来看，这样的乘积时空在爱因斯坦方程的解中是极其罕见的。例如，我们接触到的地球引力场的度规不是这种类型。该项目是将谱几何中已知的基本波和热迹不变量推广到有边界的静止时空的情况。该项目位于数学分析和几何与拓扑学的研究领域的交叉点。它将为系统研究黑洞或其他旋转的重物附近的波传播提供基础之一。

Spectral theory, spectral geometry, and the inverse spectral problem are classical and well studied research areas in pure and applied mathematics. What is studied is the relation between geometry and the eigenvalues of the Laplace operator. Re-interpreted in terms of general relativity this corresponds to the generator of time-translations on the solution space of the wave-equation on an ultrastatic spacetime. In other words eigenvalues are essentially frequencies of standing waves in product spacetimes. From the point of view of physics such product spacetimes are extremely rare amongst the solutions of Einstein's equations. For example the metric we are exposed to by the gravitational field of the Earth is not of this type.The project is to generalise the basic wave and heat-trace invariants known in spectral geometry to the case of stationary spacetimes with boundary. This project is at the intersection of the research areas Mathematical Analysis and Geometry and Topology. It will provide one of the building blocks of a systematic study of wave-propagation near black holes or other rotating heavy objects.