Representation of spacetime diffeomorphisms in canonical geometrodynamics under harmonic coordinate conditions

Abstract

Using a method developed by Isham and Kuchar (1985), the Lie algebra of the spacetime diffeomorphism group is mapped homomorphically into the Poisson algebra of dynamical variables on the phase space of canonical geometrodynamics. To accomplish this the phase space must first be extended by including embedding variables and their conjugate momenta. Also, the relationship between the embeddings and the spacetime metric must be limited by coordinate conditions. This is done by adding a source term to the Hilbert action of general relativity in such a way that the standard vacuum Einstein equations result when harmonic conditions are imposed as a supplementary set of constraints. In the process the usual super-Hamiltonian and supermomentum constraints of geometrodynamics become temporarily altered, but they are restored to their familiar form at the end by imposing the harmonic constraints, constraints that are preserved in the dynamical evolution generated by the total Hamiltonian. It is shown that the same method can also be applied to other generally covariant systems, namely the free relativistic particle and the bosonic string.