Carrollian Structure of the Null Boundary Solution Space

We study pure $D$ dimensional Einstein gravity in spacetimes with a generic null boundary. We focus on the symplectic form of the solution phase space which comprises a $2D$ dimensional boundary part and a $2(D(D-3)/2+1)$ dimensional bulk part. The symplectic form is the sum of the bulk and boundary parts, obtained through integration over a codimension 1 surface (null boundary) and a codimension 2 spatial section of it, respectively. Notably, while the total symplectic form is a closed 2-form over the solution phase space, neither the boundary nor the bulk symplectic forms are closed due to the symplectic flux of the bulk modes passing through the boundary. Furthermore, we demonstrate that the $D(D-3)/2+1$ dimensional Lagrangian submanifold of the bulk part of the solution phase space has a Carrollian structure, with the metric on the $D(D-3)/2$ dimensional part being the Wheeler-DeWitt metric, and the Carrollian kernel vector corresponding to the outgoing Robinson-Trautman gravitational wave solution.