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

TL;DR

This paper extends the embedding variable approach in geometrodynamics to a broader class of coordinate conditions, maintaining the core algebraic structure and representing spacetime diffeomorphisms within an extended phase space.

Contribution

It generalizes the ultralocal Gaussian coordinate conditions to more complex algebraic forms, preserving the Lie algebra structure and a consistent gravity sector.

Findings

The algebraic structure of ultralocal Gaussian conditions is preserved in the general case.

The extended phase space fully represents the Lie algebra of spacetime diffeomorphisms.

A consistent pure gravity sector is maintained in the generalized framework.

Abstract

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.