Lecture II: Hamiltonian formulation of general relativity (Courses in canonical gravity)

The Hamiltonian formulation of ordinary mechanics is given in terms of a set of canonical variables q and p at a given instant of time t. In eld theory, however, one is dealing with elds rather than a mechanical system then the canonical variables ϕ(x) are functions of position, and their canonical momenta are ϕ(x), both given at an instant of time. General relativity treats space and time on the same footing, that is not what is done in Hamiltonian formulations. Therefore, in order to discuss general relativity in a Hamiltonian fashion, one needs to break that equal footing. This requires a space-time splitting, since only time derivatives are transformed to momenta but not space derivatives. We assume a foliation of space-time in terms of space-like three dimen-sional surfaces S of space-time manifold M. Thus, we consider the case of a Lorentzian manifold M dieomorphic to R S, where the manifold S represents `space', and t 2 R represents `time'. It should be noted that, the particular slicing of space-time into `instants of time ' is an arbitrary choice, rather than something intrinsic to the world. In other words, there are lots of way to pick a dieomorphism ϕ:M ! R S: These give dierent ways to dene a time coordinate on the space-time manifold M, namely the pull-back by ϕ of the standard time coordinate t on R S: = ϕt: For simplicity, we assume a submanifold M is a slice of M if it equals f = constg for some time coordinate