Spectral invariants for stationary spacetimes

谱理论、谱几何和谱逆问题是纯数学和应用数学中经典的研究领域。所研究的是几何与拉普拉斯算子的特征值之间的关系。用广义相对论重新解释，这对应于超静时空中波动方程解空间上时间平移的产生器。换句话说，特征值本质上是积时空中驻波的频率。从物理学的观点来看，这样的积时空在爱因斯坦方程的解中是极其罕见的。例如，我们在地球引力场下接触到的度规就不是这种类型的。该项目是将光谱几何中已知的基本波和热迹不变量推广到具有边界的静止时空的情况。这个项目是在交叉研究领域的数学分析和几何和拓扑。它将为系统研究黑洞或其他旋转重物体附近的波传播提供一个基础。

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.