  We investigate adiabatic solutions to general relativity for a spacetime with
spatial slices with boundary, by Manton approximation. This approximation tells
us for a theory with a Lagrangian in the natural form, a motion that is
described as a slow motion on the space of vacua-static solutions that minimize
the energy -- is a good approximate solution. To apply this to the case of
general relativity we first bring it to the natural form by splitting space and
time and choosing Gaussian normal coordinates, where a spacetime is described
by the metric on its spatial slices. Then following Manton we propose slow
solutions such that each slice is a slowly changing diffeomorphism of a
reference slice, and thus each solution is described by a vector field on the
spatial slice.
  These solutions will have the property that the action will become a
functional of the vector fields on the boundaries of the spatial slices.
Moreover using the Hodge-Morrey-Friedrichs decomposition we will show that the
constraints of general relativity will identify a unique solution for a given
boundary value. Then we comment on the structure of the space of vacua which we
show to be a (pseudo)-Riemannian homogeneous space. We illustrate our procedure
for a specific reference slice we choose: the 3d Euclidean round ball.
