  The Weiss variation of the Einstein-Hilbert action with an appropriate
boundary term has been studied for general boundary surfaces; the boundary
surfaces can be spacelike, timelike, or null. To achieve this we introduce an
auxiliary reference connection and find that the resulting Weiss variation
yields the Einstein equations as expected, with additional boundary
contributions. Among these boundary contributions, we obtain the dynamical
variable and the associated conjugate momentum, irrespective of the spacelike,
timelike or, null nature of the boundary surface. We also arrive at the
generally non-vanishing covariant generalization of the Einstein
energy-momentum pseudotensor. We study this tensor in the Schwarzschild
geometry and find that the pseudotensorial ambiguities translate into
ambiguities in the choice of coordinates on the reference geometry. Moreover,
we show that from the Weiss variation, one can formally derive a gravitational
Schr{\"o}dinger equation, which may, despite ambiguities in the definition of
the Hamiltonian, be useful as a tool for studying the problem of time in
quantum general relativity. Implications have been discussed.
