  Static observers remain on Killing-vector world lines and measure the
rest-mass+kinetic energies of particles moving past them, and the flux of that
mechanical energy through space and time. The total mechanical energy is the
total flux through a spacelike cut at one time. The difference between the
total mass-energy and the total mechanical energy is the total gravitational
energy, which we prove to be negative for certain classes of systems. For
spherical systems, Misner, Thorne and Wheeler define the total gravitational
energy this way.
  To obtain the gravitational energy density analogous to that of
electromagnetism we first use Einstein's equations with integrations by parts
to remove second order derivatives. Next we apply a conformal transformation to
reexpress the scalar 3-curvature of the 3-space. The resulting density is
non-local.
  We repeat the argument for mechanical energies as measured by stationary
observers moving orthogonally to constant time slices like the "zero angular
momentum" observers of Bardeen who exist even within ergospheres.
