Domenico Giulini

Idea and structures of geometrodynamics

Abstract:

The field equations of General Relativity (GR) can be cast into the form of (six underdetermined hyperbolic) evolution equations with (four underdetermined elliptic) constraints. The degrees of freedom for the gravitational field are then fully encoded in terms of a three-dimensional Riemannian geometry. The evolution equations for this 3-geometry are such that its time dependence may be interpreted as coming about through an appropriate motion (one-parameter family of embeddings) of a spacelike hypersurface through a Lorentzian spacetime. Converseley, the latter kinematical requirement may be used to motivate Einstein's equations of GR. All this will be diuscussed as well as certain interesting structural elements concerning the geometry and topology of the space of Riemannian 3-geometries which relates to the general theory of 3-manifolds.