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

The gravitational memory problem is typically phrased in terms of the failure of constructing a (separable) Hilbert space for radiative gravitational modes at null infinity containing distinct memory states. There is, in a sense, a classical version of this problem: so far, there is no completely satisfactory analysis of the algebra of proper observables (i.e., functions associated with regular symplectic flows) on a phase space with boundary conditions allowing for arbitrary memory states. In this talk, I present results from a recent work where we developed a rigorous study of Dirac brackets on the Ashtekar-Streubel phase space, including shears with purely electric asymptotic limits, and classified the algebra of proper observables. Among the results, we found that the supertranslation charge acts in the correct manner (i.e., with the right "factor of 2"). Also, we see that any subalgebra constructed only out of shears or only out of news commutes with all memory observables, shedding some light on the quantum version of the problem. We show that the Goldstone mode is not a proper observable, but there is a large class of proper observables, the "Goldstone probes", which do not commute with the memory and are thus capable of indirectly measuring the Goldstone mode. Finally, we derive formulas for the distributional brackets (e.g., {\sigma(x),N(x')}), which contain non-local corrections compared to the standard formulas in the literature.