A Dirac's "comprehensive action principle" -- independent equations

[Kostik]

TL;DR Summary

Dirac employs a general action principle to obtain Einstein's equation and the other dynamical equations for the various "coordinates" in the combined action. But these equations are not all independent, because.....

In Dirac's "General Theory of Relativity", he develops the "comprehensive action principle" in chapter 30. Simply put, he writes a combined action for the gravitational field and all other matter-energy fields $I=I_g+I'$. Varying this: $$\delta(I_g+I')=\int ( p^{\mu\nu}\delta g_{\mu\nu} + \sum_n \chi^n \delta \phi_n ) \sqrt{-g} \, d^4 x$$ The function $p^{\mu\nu}$ picks up a term $-(16\pi)^{-1} \left( R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R \right)$ from the gravitational field, plus any other terms in $\delta g_{\mu\nu}$ coming from the other matter-energy fields, which Dirac denotes by $N^{\mu\nu}$.

Setting $p^{\mu\nu}=0$ and $\chi^n = 0$ gives the equations for the "coordinates" $g_{\mu\nu}$ and $\phi_n$. Thus, for example, if $I'$ includes the action for the electromagnetic field and a distribution of charged matter, we get Einstein's equation plus Maxwell's (inhomogeneous) equations and the Lorentz force equation.

In chapter 30, Dirac writes $$p^{\mu\nu}= -(16\pi)^{-1} \left( R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R \right) + N^{\mu\nu}$$ and shows that ${N^{\mu\nu}}_{;\nu}=0$. He concludes that "the equations $p^{\mu\nu}=0$ and $\chi^n = 0$ "are not all independent."

How does he conclude that?

Of course, because of the contracted Bianci relation, we have ${p^{\mu\nu}}_{;\nu} = {N^{\mu\nu}}_{;\nu}=0$. But this doesn't seem particularly interesting, since we are setting $p^{\mu\nu}=0$ as one of the equations, and $p^{\mu\nu}$ (or $N^{\mu\nu}$) does not involve the $\chi^n$.

 

[anuttarasammyak]

N consists of various fields, e.g., matter, EM field. Each fileld could be independent but there is a constraint that covariant divergent of their sum, N, should be zero.