  In model building studies, it is important to check the energy conditions for
the corresponding energy-momentum tensor determined by the gravitational field
equations in order to single out physically reasonable models. In this process,
one often encounters a situation where the energy-momentum tensor has one
off-diagonal "space-time" component in the frame with an orthonormal basis in a
given spacetime. We derive useful criteria of energy-momentum tensors for their
Hawking-Ellis types and the standard energy conditions in such situations. As
demonstrations, we apply those criteria to four different systems.
