Noether's theorems/Explainers/An identity hiding inside Einstein's equations

Mathematical counterpart: General relativity and the Bianchi identity.

Einstein's field equations are ten equations. Six of them are doing the work. This page says how the second theorem knows that in advance.

The count, before any calculation

General relativity's local symmetry is the freedom to label the points of spacetime however you like — and differently here than there. Spacetime has four dimensions, so that is four free labels, four free shapes.

Feed that into the counting result and out comes: four identities among the field equations, and four of them following from the rest.

Notice what was not needed. Nothing about curvature. Nothing about how gravity works. Nothing about the equations themselves beyond the fact that they come from a scoring rule with that freedom. The number four was fixed the moment somebody said "you may relabel the coordinates".

Noether gives this very case as an example in her opening section, before proving anything.

What the four identities turn out to be

Unchecked. In the literature they are the contracted Bianchi identities: the statement that a particular combination built out of the curvature has vanishing divergence, automatically, for any spacetime whatever. This volume does not derive that, because deriving it needs the symmetries of the curvature tensor and this volume does not set them up.

Three things that follow from the count alone

Four of the ten equations are not independent. If you count Einstein's equations and conclude the theory is overdetermined, you have counted wrong. It is not.

Whatever sits on the right must be divergence-free. The left-hand side of the field equations has vanishing divergence identically. The right-hand side is the energy and momentum of matter. So the matter's energy and momentum must be divergence-free too, or the equations are simply inconsistent.

That is the same structure as charge conservation in electromagnetism, and it is not a coincidence: in both cases the local symmetry produces an identity, and the identity forces a conservation law onto whatever the theory is coupled to. Gravity cannot couple to matter that does not conserve energy and momentum. The theory refuses.

The future is not determined by the present. Four functions' worth of any solution is a choice of labels, carrying no physical content. Hilbert had spelled this out in 1916 and found it disturbing; the second theorem shows it is the generic behaviour of a theory with a local symmetry, not a peculiarity of gravity.

Who found what

Worth stating plainly, because the four identities are often attributed to Noether and were not hers.

Hilbert asserted them for one particular scoring rule, in 1915. Klein derived them in general form, before Noether's paper appeared. Hilbert's own reprint of his papers, in 1924, credits Klein with the general derivation and Noether with the general proof of the theorem that has them as one instance among many.

The identities are not her discovery. The theorem is. Which is the larger result, and the reason her name is on the subject rather than theirs.

What it is like to use

The practical consequence is stranger than it sounds, and worth spelling out.

Normally you count equations against unknowns. Ten equations, ten unknowns, one answer. That instinct is wrong here and will mislead you in a specific way.

Ten equations, four of which follow from the others, gives six real constraints on ten quantities. Four quantities' worth of the answer is undetermined — and if you go looking for the extra equations that would pin them down, you will not find them, because they do not exist and were never missing.

What you do instead is decide. Pick coordinates, write the choice down, and carry on. Two people who choose differently get descriptions that disagree everywhere and agree on every measurable thing, which is the test that they both did it right.

The four undetermined functions are not ignorance. They are the labels.

The reason to know this in advance, rather than discovering it halfway through a calculation, is that the symptom is confusing. Equations that will not close look like a mistake. The count tells you before you start exactly how much will not close, and that the amount is correct.

No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.