Vorträge in der Woche 03.11.2025 bis 09.11.2025

Donnerstag, 06.11.2025: The Conformal Method and Hamiltonian General Relativity

David Maxwell (Fairbanks, Alaska)

The conformal method is a workhorse tool for building solutions of the Einstein constraint equations, especially those with constant mean curvature. One frequent perspective on the technique is that it is simply a handy tool, a frequently effective mathematical artifice. In this talk we demonstrate instead that the conformal method is deeply rooted in the Hamiltonian approach to general relativity. We start with a careful analysis of the Gauss constraint $\mathrm{div} E = \rho$ of electromagnetism and describe an approach, grounded in Hamiltonian field theory, for generating its solutions. Then, after an unexpected detour into temporal wave gauge in general relativity, we exhibit a modern understanding of conformal method as an immediate generalization of the toy-model approach to the Gauss constraint. If time permits, we discuss how these ideas led to recent advances in formulating the conformal method in the non-vacuum setting, with applications to charged fluids. Finally, we speculate on how refinements to the Hamiltonian approach could lead to generalizations that apply in situations where the method is currently less effective.