General relativity and the Bianchi identity

Plain-language counterpart: An identity hiding inside Einstein's equations.

Two groups of five bars representing field equations. On the left, without a local symmetry, all five are independent. On the right, one free function makes the fifth follow from the other four, shown by a curved arrow. A caption states the counting: one arbitrary function gives one identity and one equation that was never independent. — Identities among the field equations from a local symmetry

This page derives the count and does not derive the identity. The distinction is the whole content.

The count, derived

General relativity's local symmetry is the freedom to relabel the points of spacetime. In $n$ dimensions that is $n$ arbitrary functions, entering the transformation of the metric through their first derivatives, so $ϱ = n$ and $σ = 1$ .

The second theorem then returns, with no further input, exactly $n$ identities among the Lagrange expressions of the gravitational action and their first derivatives. In four dimensions, four.

That number is a theorem, not an observation. It was fixed the moment the symmetry was named, before anything was computed about curvature.

Noether states this case herself, in her first section, as one of two examples given before any proof — the metric and a linear form subjected to the induced transformations, and the known $n$ dependencies following.

The identity, not derived

UNVERIFIED. The identities are stated in the literature to be the contracted Bianchi identities: the divergence of the Einstein tensor vanishes identically, which in index form is one relation for each free index, four in four dimensions. Deriving that requires the Riemann tensor's symmetries, which this volume does not set up.

What the count buys without the identity

Three consequences follow from the count alone.

Four of the ten field equations are not independent. The count says so directly, and this is the fact underlying the whole discussion on the counting page.

The source must be divergence-free. The field equations set a divergence-free object equal to the matter's energy and momentum, so the matter's energy and momentum must be divergence-free too, on pain of inconsistency. This is the same structure as the electromagnetic case and it is not a coincidence: both are the second theorem's identity forcing a conservation law on whatever is coupled.

Initial data does not determine the future. Four functions' worth of any solution is a choice of labels. That is what Hilbert had spelled out in 1916, and the history track gives the sequence.

Who had it first

Klein derived these identities in general form before Noether's paper, by the method of infinitesimal transformations. Hilbert had asserted them for a particular case. Hilbert's own 1924 reprint credits Klein with the general form and Noether with the general proof of the theorem that has them as an instance.

The four identities are not Noether's discovery. The theorem is.

Provenance: the count is derived above from the second theorem. The identification with the contracted Bianchi identities is flagged UNVERIFIED. The priority account rests on Hilbert's 1924 reprint, summarised from the secondary literature and not reached in full text. Noether's own example is read from the public-domain German original. See Sources.