Implications of the Biancchi constraints in General Relativity

The requirement of invariance under gauge transformations gives to physical gauge theories their special character. In this sense Einstein's General Theory of Relativity is not exceptional: the focus is on the Biancchi identity and on the associated Biancchi constraint. The principal equation $G \mu н = T \mu н$ requires that the right hand side satisfy the constraint $T\mu н$, $н = 0$, by virtue of the matter field equations.
This can be satisfied with limitation to potential flows, but for general flows the Navier–Stokes equation is used and it has not been possible to formulate it as a variational principle. The case of potential flows will be discussed, as well as one step towards formulating the Navier–Stokes equation in terms of a variational principle.