Generalized embedding variables for geometrodynamics and spacetime diffeomorphisms: Ultralocal coordinate conditions

We investigate the embedding variable approach to geometrodynamics advocated in work by Isham, Kucha\v{r} and Unruh for a general class of coordinate conditions that mirror the Isham-Kucha\v{r} Gaussian condition but allow for arbitrary algebraic complexity. We find that the same essential structure present in the ultralocal Gaussian condition is repeated in the general case. The resultant embedding--extended phase space contains a full representation of the Lie algebra of the spacetime diffeomorphism group as well as a consistent pure gravity sector.