A Modified Variational Principle in Relativistic Hydrodynamics. I.Commutativity and Noncommutativity of the Variation Operator with the Partial Derivative

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.