  The ADM energy for asymptotically flat spacetimes or its generalizations to
asymptotically non-flat spacetimes measure the energy content of a stationary
spacetime, such as a single black hole. Such a stationary energy is given as a
geometric invariant of the spatial hypersurface of the spacetime and is
expressed as an integral on the boundary of the hypersurface. For
non-stationary spacetimes, there is a refinement of the ADM energy, the
so-called Dain's invariant that measures the non-stationary part, the
gravitational radiation component, of the total energy. Dain's invariant uses
the metric and the extrinsic curvature of the spatial hypersurface together
with the so-called approximate Killing initial data and vanishes for stationary
spacetimes. In our earlier work [Phys.Rev.D 101 (2020)2, 024035], we gave a
reformulation of the non-stationary energy for vacuum spacetimes in the
Hamiltonian form of General Relativity written succinctly in the
Fischer-Marsden form. That formulation is relevant for merging black holes or
other compact sources. Here we extend this formulation to non-vacuum spacetimes
with a perfect fluid source. This is expected to be relevant for spacetimes
that have a compact star, say a neutron star colliding with a black hole or
another non-vacuum object.
