  It is common knowledge that the Einstein-Hilbert action does not furnish a
well-posed variational principle. The usual solution to this problem is to add
an extra boundary term to the action, called a counter-term, so that the
variational principle becomes well-posed. When the boundary is spacelike or
timelike, the Gibbons-Hawking-York counter-term is the most widely used. For
null boundaries, we had proposed a counter-term in a previous paper. In this
paper, we extend the previous analysis and propose a counter-term that can be
used to eliminate variations of the "off-the-surface" derivatives of the metric
on any boundary, regardless of its spacelike, timelike or null nature.
