General relativity: Dynamics without symmetry

The concept of conditional symmetry is introduced for a parametrized relativistic particle model and generalized to geometrodynamics. Its role in maintaining a one-system interpretation of the quantized theory is emphasized. It is shown that geometrodynamics does not have any conditional symmetry: Such a symmetry should be generated by a dynamical variable Kgab, pab which is linear and homogeneous in the gravitational momentum pab and which has a weakly vanishing Poisson bracket with the super-Hamiltonian and supermomentum. The generators K fall into equivalence classes modulo the supermomentum constraint. It is shown that each equivalence class can be represented by a member which is a spatial invariant. The remaining weak equations are turned into strong equations by the method of Lagrange multipliers. The local structure of the super-Hamiltonian and supermomentum imposes locality restrictions on the multipliers. These restrictions imply that the generator must be weakly equivalent to a local generator. A recursive argument then shows that the local generator must actually be weakly ultralocal. This uniquely determines the generator as the conformal Killing (super)vector of the local supermetric. However, the curvature scalar in the super-Hamiltonian breaks the conditional symmetry of the supermetric term and turns geometrodynamics into a theory without any symmetry. This result is generalized to inhomogeneous generators.