On variational principle and canonical structure of gravitational theory in double-foliation formalism

In this paper, we analyze the variation of the gravitational action on a bounded region of space time whose boundary contains segments with various characters, including null. We develop a systematic approach to decompose the derivative of metric variations into orthogonal and tangential components with respect to the boundary and express them in terms of variations of geometric objects associated with the boundary hypersurface. We suggest that a double-foliation of spacetime provides a natural and useful set-up for treating the general problem and clarifies the assumptions and results in specialized ones. In this set-up, we are able to obtain the boundary action necessary for the variational principle to become well-posed as well as the canonical structure of the theory, while keeping the variations completely general. Especially, we show how one can remove the restrictions imposed on the mole variations in previous works due to the assumption that the boundary character is kept unaltered. As a result, we find that on null boundaries there exists a new canonical pair related to the change in character of the boundary. This set-up and the procedure of calculation are stated in a way that can be applied to other more generalized theories of gravity.