  A model is proposed, according to which the metric tensor field in the
standard gravitational Lagrangian is decomposed into a projection (generally -
with a non-zero covariant derivative) tensor field, orthogonal to an arbitrary
4-vector field and a tensor part along the same vector field. A theorem has
been used, according to which the variation and the partial derivative, when
applied to a tensor field, commute with each other if and only if the tensor
field and its variation have zero covariant derivatives, provided also the
connection variation is zero. Since the projection field obviously does not
fulfill the above requirements of the ''commutation'' theorem, the exact
expression for the (non-zero) commutator of the variation and the partial
derivative, applied to the projection tensor field, can be found from a set of
the three defining equations. The above method will be used to construct a
modified variational approach in relativistic hydrodynamics, based on variation
of the vector field and the projection field, the last one thus accounting for
the influence of the reference system of matter (characterized by the 4-vector)
on the gravitational field.
