Expanding on Classical GR

This chapter covers the reformulation of classical general relativity in both the Lagrangian and Hamiltonian frameworks. In the first case this involves the Einstein-Hilbert action, which is shown to be a function of the metric and its derivatives. This also introduces the idea of a sum-over-geometries, describing the dynamics of spacetime as a kind of path integral, analogous to the description of particle and field dynamics in quantum field theory. The Arnowitt-Deser-Misner (ADM) splitting of spacetime into a foliation of hypersurfaces is introduced, and the Hamiltonian, diffeomorphism, and Gauss constraints derived. The technical difficulty of writing the constraints in operator form is used to motivate an alternative formulation more similar to a quantum field theory. Tetrads, the spin connection, and an appropriate choice of gauge group are discussed, and the Palatini formulation is described.