# Note on the symplectic structure of asymptotically flat gravity and BMS symmetries

## TL;DR

This paper demonstrates that by splitting gravitational fields into bulk and boundary components and applying covariant phase space methods, one can derive correct BMS-related Poisson brackets without ad-hoc boundary terms, clarifying their role in asymptotically flat gravity.

## Contribution

It introduces a systematic approach to derive boundary Poisson brackets in asymptotically flat gravity, eliminating the need for arbitrary boundary modifications.

## Key findings

- Derived boundary Poisson brackets without ad-hoc terms
- Showed BMS charges generate BMS transformations canonically
- Clarified the symplectic structure at null infinity

## Abstract

The Poisson brackets of the gravitational field at null infinity play a pivotal role in establishing the equivalence between the Ward identities involving BMS charges and the soft graviton theorem. In recent literature it was noticed that, in order to reproduce the action of BMS transformations via such Poisson brackets, one needs to add "ad-hoc" boundary terms in the symplectic form. In this note we show that, introducing a suitable splitting of the gravitational field in bulk and boundary degrees of freedom and using techniques of covariant phase space formalism, it is possible to obtain the correct Poisson brackets between the boundary fields without any additional assumption. The same Poisson brackets are used to show that BMS charges canonically generate BMS transformations on the gravitational phase space.