Restoration of four-dimensional diffeomorphism covariance in canonical general relativity: An intrinsic Hamilton-Jacobi approach

TL;DR

This paper develops a Hamilton-Jacobi formalism for general relativity that preserves four-dimensional diffeomorphism invariance using intrinsic coordinates, challenging traditional approaches and offering new insights into quantum gravity.

Contribution

It introduces an intrinsic Hamilton-Jacobi approach that maintains full diffeomorphism covariance without privileging any geometric structure, providing a new framework for quantum gravity.

Findings

Variables are invariant under full diffeomorphisms.

Intrinsic coordinates enable new phase space variables and constraints.

Reinterpretation of multi-fingered time as full diffeomorphism invariants.

Abstract

Classical background independence is reflected in Lagrangian general relativity through covariance under the full diffeomorphism group. We show how this independence can be maintained in a Hamilton-Jacobi approach that does not accord special privilege to any geometric structure. Intrinsic spacetime curvature based coordinates grant equal status to all geometric backgrounds. They play an essential role as a starting point for inequivalent semi-classical quantizations. The scheme calls into question Wheeler's geometrodynamical approach and the associated Wheeler-DeWitt equation in which three-metrics are featured geometrical objects. The formalism deals with variables that are manifestly invariant under the full diffeomorphism group. Yet, perhaps paradoxically, the liberty in selecting intrinsic coordinates is precisely as broad as is the original diffeomorphism freedom. We show how various ideas from the past five decades concerning the true degrees of freedom of general relativity can be interpreted in light of this new constrained Hamiltonian description. In particular, we show how the Kucha\v{r} multi-fingered time approach can be understood as a means of introducing full four-dimensional diffeomorphism invariants. Every choice of new phase space variables yields new Einstein-Hamilton-Jacobi constraining relations, and corresponding intrinsic Schr\"odinger equations. We show how to implement this freedom by canonical transformation of the intrinsic Hamiltonian. We also reinterpret and rectify significant work by B. Dittrich on the construction of `Dirac observables'.