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A bstract: 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 news and purely electric 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 generates 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 Goldstone probes , that are capable of measuring the Goldstone mode. We notice that there are no Goldstone probes constructed only out of the shear 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.

Support material for lectures at the May ‘25 Galileo Galilei Institute school on asymptotic symmetries and flat holography. Contains an introduction to Noether theorem for gauge theories and gravity, covariant phase space formalism, boundary and asymptotic symmetries, flux-balance laws on null hypersurfaces, future null infinity in Bondi–Sachs coordinates and with Penrose’s conformal compactification, BMS symmetries and their charges and fluxes. Includes an original and pedagogical derivation of the BMS group using only Minkowski, and an original derivation of an integral Hamiltonian generator for diffeomorphisms of a scalar field on a null hypersurface.

We propose a new set of BMS charges at null infinity, characterized by a super-translation flux that contains only the ‘hard’ term. This is achieved with a specific corner improvement of the symplectic 2-form, and we spell the conditions under which it is unique. The charges are associated to a Wald-Zoupas symplectic potential, and satisfy all standard criteria: they are covariant, provide a center-less realization of the symmetry algebra, have vanishing flux in non-radiative spacetimes, and vanish in Minkowski. We use them to define a notion of spatially localized energy density of gravitational waves, and explain how it can be measured doing experiments which are purely local in space and over an extended period of time.

Motivated by issues in the context of asymptotically flat spacetimes at null infinity, we discuss in the simplest example of a free massless scalar field in two dimensions several subtleties that arise when setting up the canonical formulation on a single or on two intersecting null hyperplanes with a special emphasis on the infinite-dimensional global and conformal symmetries and their canonical generators, the free data, a consistent treatment of zero modes, matching conditions, and implications for quantization of massless versus massive fields.

Black hole horizons in equilibrium and null infinity of asymptotically flat space-times are null 3-manifolds but have very different physical connotations. We first show that they share a large number of geometric properties, making them both weakly isolated horizons. We then use this new unified perspective to unravel the origin of the drastic differences in the physics they contain. Interestingly, the themes are woven together in a manner reminiscent of voices in a fugue.

Black hole horizons in equilibrium and null infinity of asymptotically flat space-times are null 3-manifolds but have very different physical connotations. We first show that they share a large number of geometric properties, making them both weakly isolated horizons. We then use this new unified perspective to unravel the origin of the drastic differences in the physics they contain. Interestingly, the themes are woven together in a manner reminiscent of voices in a fugue.