  As we know, from the Einstein equations the vanishing of the four-divergence
of the (symmetric) energy-momentum tensor follows. This is the case because the
fourdivergence of the Einstein tensor (which is also symmetric) vanishes
identically. Inverselly, we find that from the vanishing of the four-divergence
of the energymomentum tensor (consisting in the general case of both symmetric
and antisymmetric parts) not only the Einstein equations follow (concerning the
symmetric part), but also the so-named anti-Einstein equations (concerning the
antisymmetric part). These equations must be considered as complementary to the
Einstein equations. Also, while from the Einstein equations the energy density
(or the pressure) can be found (without using an equation of state), from the
vanishing of the four-divergence of the (symmetric) energy-momentum tensor the
pressure (or the energy density) can also be found, without having to use an
additional (but arbitrary) equation of state.
