# What Are Observables in Hamiltonian Einstein-Maxwell Theory?

## TL;DR

This paper investigates the nature of observables in Hamiltonian Einstein-Maxwell theory, proposing a bifurcated definition that distinguishes invariance under internal gauge symmetries from covariance under external ones, aligning with geometric intuition.

## Contribution

It introduces a bifurcated definition of observables in Hamiltonian gauge theories, reconciling internal invariance with external covariance, and applies it to Einstein-Maxwell theory.

## Key findings

- Observables are invariant under internal gauge transformations.
- Observables are covariant under external spacetime transformations.
- The bifurcated definition aligns with geometric and physical expectations.

## Abstract

Is change missing in Hamiltonian Einstein-Maxwell theory? Given the most common definition of observables (having weakly vanishing Poisson bracket with each first-class constraint), observables are constants of the motion and nonlocal. Unfortunately this definition also implies that the observables for massive electromagnetism with gauge freedom (Stueckelberg) are inequivalent to those of massive electromagnetism without gauge freedom (Proca). The alternative Pons-Salisbury-Sundermeyer definition of observables, aiming for Hamiltonian-Lagrangian equivalence, uses the gauge generator G, a tuned sum of first-class constraints, rather than each first-class constraint separately, and implies equivalent observables for equivalent massive electromagnetisms. For General Relativity, G generates 4-dimensional Lie derivatives for solutions. The Lie derivative compares different space-time points with the same coordinate value in different coordinate systems, like 1 a.m. summer time vs. 1 a.m. standard time, so a vanishing Lie derivative implies constancy rather than covariance. Requiring equivalent observables for equivalent formulations of massive gravity confirms that $G$ must generate the $4$-dimensional Lie derivative (not $0$) for observables. These separate results indicate that observables are invariant under internal gauge symmetries but covariant under external gauge symmetries, but can this bifurcated definition work for mixed theories such as Einstein-Maxwell theory? Pons, Salisbury and Shepley have studied $G$ for Einstein-Yang-Mills. For Einstein-Maxwell, both $F_{\mu\nu}$ and $g_{\mu\nu}$ are invariant under electromagnetic gauge transformations and covariant (changing by a Lie derivative) under $4$-dimensional coordinate transformations. Using the bifurcated definition, these quantities count as observables, as one would expect on non-Hamiltonian grounds.