Colloquium

Année 2017- 2018

Alan Weinstein - University of California, Berkeley

Hamiltonian Lie algebroids and general relativity

Jeudi 21 septembre 17:00-18:00

Jussieu, salle 15-25-502

In general relativity, the gravitational field is a lorentzian metric on a 4-dimensional space-time manifold. The Einstein field equations may be expressed, at least locally in time, as the trajectories of a hamiltonian system on a cotangent bundle T*R, where R is the (infinite dimensional) manifold of riemannian metrics on a 3-dimensional ``time slice", with initial values constrained to a certain submanifold C of T*R.

Geometric properties of C suggest that the constraints should be related to the symmetry group of the Einstein equations, consisting of the diffeomorphisms of space-time. But this group does not act on R, since it does not act on an individual time slice. Blohmann, Fernandes, and the speaker have shown that the algebraic structure of the constraints is in fact related to a groupoid of diffeomorphisms between pairs of time slices, but a direct connection between the constraints and this groupoid was not found.

I will report on ongoing work with Blohmann and Schiavina establishing this direct connection, centered around an extension to Lie algebroids (the infinitesimal version of Lie groupoids) of the theory of hamiltonian actions of Lie algebras. Aside from the application to relativity, the theory leads to interesting questions in symplectic topology. The talk will be aimed at a general audience, including graduate students, the only prerequisite being basic differential geometry.