Is General Relativity an 'Already Parametrized' Field Theory?

TL;DR

This paper argues that general relativity cannot be strictly considered a parametrized field theory due to fundamental obstructions, impacting canonical quantization approaches.

Contribution

It clarifies the conceptual distinction between general relativity and parametrized theories, challenging the common paradigm in quantum gravity research.

Findings

Obstruction identified in matching constraint surfaces

General relativity cannot be fully described as a parametrized theory

Implications for canonical quantization methods

Abstract

Beginning with the work of Dirac and Arnowitt, Deser, Misner in the late fifties and early sixties, and then after subsequent development by Kucha\v r, the canonical dynamical structure of general relativity has often been viewed as that of a parametrized field theory in which the many-fingered spacetime variables are hidden amongst the geometrodynamical field variables. This paradigm of general relativity as an ``already parametrized theory'' forms the basis for one of the most satisfactory resolutions of the problems of time and observables in classical and quantum gravity. However, despite decades of effort, no identification of many-fingered spacetime variables has ever been satisfactorily obtained for vacuum general relativity. We point out that there is an obstruction to identifying the constraint surface of general relativity (for the case of a closed universe) with that of any parametrized theory. Therefore, strictly speaking, general relativity cannot be viewed as a parametrized field theory. We discuss implications for the canonical quantization program.