An algebra of proper observables at null infinity: Dirac brackets, Memory and Goldstone probes

TL;DR

This paper rigorously evaluates Dirac brackets for classical gravitational observables at null infinity, incorporating memory effects and identifying proper observables related to supertranslations and Goldstone modes.

Contribution

It develops a formalism for proper observables in gravitational phase space, clarifies the nature of Goldstone probes, and explains challenges in constructing memory-based Hilbert spaces.

Findings

Proper observables generate correct supertranslation transformations.

Goldstone probes can measure Goldstone modes, unlike traditional modes.

Distributional Dirac brackets include non-local corrections.

Abstract

We develop a rigorous evaluation of Dirac brackets for classical observables on the phase space of radiative gravitational modes at null infinity that naturally incorporates memory effects. Considering the Ashtekar-Streubel phase space, with boundary conditions in time given by vanishing {\it news} and purely electric {\it shear}, and taking into account the infinite dimensionality of the phase space, we identify the algebra of proper observables (understood as functions on phase space that can be associated with smooth symplectic flows). We show that the action of supertranslation charges generate the correct transformations on the shear. We also show that the conventional definition of the ``Goldstone mode'' adopted in the literature cannot be associated with a proper observable, but nevertheless there exists an infinite family of proper observables, which we call {\it Goldstone probes}, that are capable of measuring the Goldstone mode. We notice that there are no Goldstone probes constructed only out of the shear {\it or} the news, providing a possible explanation for why attempts to construct a (separable) Hilbert space with different memory states have failed so far. Finally, we derive formulas for distributional Dirac brackets between local shear and news, and show that they contain non-local corrections.