Boundary dynamics in gravitational theories

We present a foliation-focused critical review of the boundary conditions and dynamics of 4D gravitational theories. A general coordinate transformation introduces a new foliation and changes the hypersurface on which a natural boundary condition is imposed; in this sense gauge transformations must be viewed as changing the boundary conditions. The issue of a gauge invariant boundary condition is nontrivial and has been extensively studied in the literature. We turn around the difficulty in obtaining such a boundary condition (and subtleties observed in the main body) and take it as one of the indications of an enlarged Hilbert space so as to include the states satisfying different boundary conditions. Through the systematical reduction procedure we obtain, up to some peculiarities, the explicit form of the reduced Lagrangian that describes the dynamics of the physical states. We examine the new insights offered by the 3D Lagrangian on BMS-type symmetry and black hole information. In particular we confirm that the boundary dynamics is an indispensable part of the system information.