Nonstationary energy of perfect fluid sources in general relativity

The Arnowitt-Deser-Misner (ADM) energy for asymptotically flat spacetimes or its generalizations to asymptotically nonflat 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 nonstationary spacetimes, there is a refinement of the ADM energy, the so-called Dain's invariant that measures the nonstationary 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 [E. Altas and B. Tekin, Nonstationary energy in general relativity, Phys. Rev. D 101, 024035 (2020)], we gave a reformulation of the nonstationary 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 nonvacuum 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 nonvacuum object.