A conservation law that is only wearing the costume

Mathematical counterpart: Proper and improper conservation laws.

This is the last section of Noether's paper and the reason she wrote it. It answers a question about energy in general relativity, and the answer is sharper than people usually report.

Two books that both balance

A shopkeeper's accounts balance at the end of the day. So do a bank's.

For the shopkeeper, balancing is news. It means the till was counted right and nothing walked out of the door. It could have failed.

For the bank, one particular sum balances because of how the entries are defined. Every transaction is written twice, once on each side. That sum cannot fail to balance, and a clerk who reports proudly that it did has reported nothing.

Both are true statements. Only one carries information.

Noether draws exactly this line. A conservation law whose conserved quantity is assembled out of the equations of motion themselves she calls improper. All others are proper.

An improper conservation law is an identity in fancy dress. It is true, it looks like physics, and it tells you nothing the equations did not already say.

The criterion, in both directions

Now the result, and the "exactly when" is the whole of it.

Suppose a theory's scoring rule survives an ordinary global symmetry — shifting everything in time, say — so the first theorem gives you a conservation law.

That conservation law is improper exactly when the global symmetry sits inside a local one that the rule also survives.

Both directions hold. If the global symmetry is part of a larger local freedom, the conservation law is empty. If it is not, the conservation law is genuine.

Noether checks the other side herself, in a footnote: the energy theorems of ordinary mechanics, and of special relativity, are proper — because those theories have no local symmetry for a global one to hide inside.

That footnote is what makes this a criterion rather than an observation. Without it you would have a rule that only ever says yes.

What it means for gravity

General relativity has a local symmetry: you may relabel the points of spacetime however you like, differently here than there. And that freedom contains, as a special case, the ordinary shift-everything symmetry that would otherwise give energy conservation.

So by the criterion, the energy relations of general relativity are improper.

This is the answer to a puzzle that had been troubling Göttingen since 1915. Attempts to write down the energy of the gravitational field had produced quantities that behave badly: change the coordinates and the energy density changes, and at any single point you can choose coordinates that make it zero. For years this was read as a defect — a sign that somebody would eventually find the right expression.

Noether showed that nobody will, because there is nothing to find. The degeneracy is what a local symmetry produces. It is a prediction of the structure, not a failure of anyone's algebra.

And it is not about gravity

Here is the part almost always dropped, and it is Noether's own emphasis.

The criterion says nothing about gravity in particular. It is a statement about symmetries. Any theory with the relevant local freedom has improper energy relations, and any theory without one has proper ones.

She proves this by example rather than by assertion. In that same final section she builds a local symmetry out of a logarithm — a construction with no physical content whatsoever, chosen precisely because nothing physical is attached to it — and shows the improper energy relations appearing exactly as the criterion says they must.

Gravity is one instance. It is not the reason. She closes by suggesting that the physicists' word "relativity" would be better replaced by invariance with respect to a group, endorsing a remark of Klein's, and that suggestion is the whole point compressed into a sentence.

What improper does not mean

Two clarifications, because the word invites over-reading.

It does not mean false. An improper conservation law is perfectly true. It holds, it can be written down, and it can be checked. It just does not tell you anything the equations of motion had not already said.

It does not mean gravity has no energy. For a system that is isolated — a star with empty space stretching away in every direction — a total energy can still be defined, by looking at the behaviour far away rather than by adding up a density point by point. That total is meaningful and it is measurable.

What is unavailable is the local statement: so much energy here, so much there. The criterion says the local version cannot exist, and it says nothing against the global one.

The distinction is exactly the one from the flow and total page. In an ordinary theory the total is a consequence of the local statement. Here the local statement fails and a total survives anyway, by a different route.

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