Spectral invariants for stationary spacetimes

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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.