Thèse de doctorat
Résumé : The present thesis aims at providing a unified description of radiative phase spaces in General Relativity for any value of the cosmological constant using covariant phase space methods. We start by considering generic asymptotically locally (A)dS spacetimes with leaky boundary conditions in the Starobinsky/Fefferman-Graham gauge and in arbitrary dimensions. The boundary structure is allowed to fluctuate and plays the role of source yielding some flux of gravitational radiation at the boundary. The holographic renormalization procedure is employed to remove divergences from the presymplectic structure, which leads to finite surface charges for the whole class of boundary diffeomorphisms and Weyl rescalings. The charge algebra represents this asymptotic symmetry algebra under the Barnich-Troessaert bracket, up to a field-dependent 2-cocycle in odd spacetime dimensions. We then propose a boundary gauge fixing isolating the radiative components among the boundary degrees of freedom without constraining the Cauchy problem in asymptotically de Sitter spacetimes. This additional gauge fixing reduces the set of allowed boundary diffeomorphisms to the infinite-dimensional $\Lambda$-BMS algebroid, which is the counterpart to the Generalized BMS algebra of smooth supertranslations and super-Lorentz transformations in the flat limit. In a second round, the analysis is repeated in the Bondi gauge, which is better suited to discuss radiative phenomena as well as construct a flat limit process at the level of the solution space. Thanks to a diffeomorphism we translate the results previously obtained in the Starobinsky/Fefferman-Graham coordinates and identify the analogues of the Bondi news, mass and angular momentum aspects in the presence of a non-vanishing cosmological constant. We give a prescription to perform the flat limit at the level of the phase space and demonstrate how to use this connection to renormalize the corresponding phase space of asymptotically locally flat spacetimes at null infinity. The latter is made necessary as soon as the boundary structure of the gravitational field is allowed to vary under arbitrary super-Lorentz transformations. The last part of the manuscript is devoted to discussing the various implications of these super-Lorentz transformations as genuine asymptotic symmetries of asymptotically flat Einstein's gravity. In particular, we derive a closed-form expression of the orbit of gravitational vacua under the Generalized BMS symmetries. Transitions among these vacua are related to the refraction/velocity kick memory effect and the displacement memory effect. Finally, we give a physical prescription to define finite Hamiltonian generators canonically conjugated to Generalized BMS transformations on the subclass of physical solutions that are stationary at early and late times and comment on the enhancement of the infrared structure of gravity in the presence of super-Lorentz transformations.