Moving the bookkeeping without changing the total
Mathematical counterpart: The Belinfante-Rosenfeld improvement.
The first theorem hands you an account of where energy and momentum sit. The account it hands you is not the only correct one, and for gravity it is the wrong shape.
Two departments, one company
A firm has two departments sharing a building. The finance office assigns the heating bill to one of them.
Reassign it to the other. Every departmental figure changes. The company's total does not, and neither does the amount of heat used.
That is the situation exactly. What the first theorem gives you is one assignment of energy and momentum to places. Another assignment is available, it disagrees about where things are, and it agrees about every total.
Why more than one assignment is allowed
Recall from the flow and total page that you can always add a circulating pattern to a flow — one that carries things round in a loop and never accumulates anywhere. Totals are untouched.
The reassignment here is that, done with two indices instead of one. You add a term built to be antisymmetric: swap two of its slots and it changes sign.
Then the argument is one line. To see whether the reassignment disturbs the conservation law, you differentiate it twice. Differentiating twice does not care which order you do it in — that is symmetric. Your added term is antisymmetric. A symmetric thing multiplied by an antisymmetric thing and summed gives zero.
So the conservation law survives untouched, and it survives identically — not only for real motions, but always. The totals survive too, because the reassigned part only ever shows up as a term at the far boundary.
Why anyone bothers
Because general relativity is fussy about shape.
The object that tells gravity how much energy and momentum are present must be symmetric — swap its two slots and nothing changes. That is forced by how Einstein's equations are built.
What the first theorem delivers is not symmetric in general. So it cannot be the thing gravity responds to, and something has to be done.
The something is the reassignment above, chosen so the result comes out symmetric. It is called the Belinfante-Rosenfeld improvement, and the extra piece it moves around is the contribution from the spin of the fields involved.
The part usually got backwards
It is natural to read this as fixing a flaw in Noether's result. It is not.
Her proof determines the assignment up to exactly this freedom, and she says so. In the middle of proving her second theorem she produces, as a by-product, a non-zero quantity whose flow vanishes identically — which is the freedom arriving as an output of the mathematics rather than as an embarrassment to be explained.
So the improvement is a choice made inside a latitude she had already mapped. And the reason for choosing one option over another does not come from her theorem at all. It comes from general relativity, which needs a particular shape.
The non-uniqueness is a property of the theorem. The choice is physics.
Which assignment is the real one
A natural question, and the answer is less comfortable than it looks.
If two assignments disagree about how much energy is in a given region, and both are consistent with everything the theory says, is there a fact of the matter?
For the totals, yes. Every assignment agrees on the total, and the total is measurable.
For the local distribution, the theory does not settle it by itself. What settles it, in the end, is gravity — because gravity responds to energy and momentum locally, and how it responds picks out one assignment as the one nature uses. The symmetric one is the one that appears in Einstein's equations, and that is the sense in which the improvement is not merely tidier.
So the sequence is worth getting the right way round. Noether's theorem produces a family of correct assignments. General relativity, arriving later, selects from that family. Nothing about the selection was available in 1918, and nothing in her theorem anticipated it or needed to.
A latitude in a mathematical result got closed by a physical theory that did not exist when the result was proved. That happens more often in this subject than the textbook order of presentation suggests.
No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.