Change in Hamiltonian General Relativity with Spinors

TL;DR

This paper explores how to identify and define local change in Hamiltonian General Relativity with spinors by using a specific gauge formalism and constraint analysis, addressing longstanding issues of change and time dependence.

Contribution

It introduces a $3+1$ formalism that captures local change in Hamiltonian GR with spinors, eliminating extraneous gauge freedoms and clarifying the role of time dependence.

Findings

Change is characterized by the absence of a stronger-than-Killing timelike symmetry.

A gauge generator $G$ is constructed that implements time coordinate changes.

The formalism clarifies the nature of change in Hamiltonian GR with spinors.

Abstract

In Hamiltonian GR, change has seemed to be missing, defined only asymptotically, or otherwise obscured at best. By construing change as essential time dependence, can one find change locally in Hamiltonian GR with spinors? This paper is motivated by tendencies in space-time philosophy to slight fermionic/spinorial matter, in Hamiltonian GR to misplace changes of time coordinate, and in treatments of the Einstein-Dirac equation to include a gratuitous local Lorentz gauge symmetry. Spatial dependence is dropped in most of the paper. To include all and only the coordinate freedom, the Einstein-Dirac equation is investigated using the Schwinger time gauge and Kibble-Deser symmetric triad condition as a $3+1$ version of the DeWitt-Ogievetsky-Polubarinov nonlinear group realization formalism that dispenses with a tetrad and local Lorentz gauge freedom. Change is the lack of a time-like stronger-than-Killing field for which the Lie derivative of the metric-spinor complex vanishes. An appropriate $3+1$-friendly form of the Rosenfeld-Anderson-Bergmann-Castellani gauge generator $G$, a tuned sum of first class-constraints, changes the canonical Lagrangian by a total derivative and implements changes of time coordinate for solutions.