Noether's theorems/Explainers/Equations that were never independent

Noether's theorems: map
Reading the volume How to read this volume Notation Glossary Index of symmetries Figure gallery
Reading by level Start here: the plain-language primer The first theorem, stated and proved The second theorem, stated and proved The two theorems side by side The mathematics: Lie groups and Lie algebras The applications: General relativity and the energy question The history: Göttingen and the energy problem
Sources Sources Bibliography Chronology Further reading
Related Hamilton's principle On Physical Lines of Force

Mathematical counterpart: The Noether identities.

The second theorem's output has a name in the later literature: the Noether identities. This page says what sort of thing they are.

A committee that is smaller than it looks

Five people on a committee, five votes. But one member has a standing rule: she always votes with the majority of the other four.

You have five votes and four decisions. The fifth vote is real, it is cast, it is counted — and it tells you nothing you did not already know.

That is the shape of a Noether identity. A theory has some number of equations of motion. The second theorem says a certain number of them are, in this sense, voting with the others. They are true. They are not extra.

What kind of statement it is

Compare with the first theorem's output.

The first theorem says: a certain combination of things flows without piling up anywhere. That is a statement that something is happening.

The second theorem says: a certain combination of things equals zero. That is a statement that nothing is happening, and specifically that the pieces were never free of each other in the first place.

Nothing flows. There is no current, there is no total, and there is nothing to measure. A reader who has met only the first theorem naturally expects a conserved quantity to come out of the second, and the expectation is the single most common source of error about it.

The simplest case

Draw a curve on paper and describe it by saying where your pen is at each moment.

You had a free choice about pace — fast, slow, a pause in the middle. The curve does not care. That freedom is the simplest possible local symmetry: one free shape, and the rule does not even care how quickly your choice varies.

The theorem promises, for that case, exactly one relation among the equations, of the simplest kind. And that is what turns up: a single statement tying the equations directly to one another, with no rates of change involved.

The practical meaning is exactly what you would guess. A description of a curve that never commits to a pace cannot determine its own pace. One of its equations is therefore redundant, and the theorem tells you in advance that exactly one will be.

That is not a flaw in the description. It is the cost of the freedom you wanted. You asked to describe a curve without committing to how it is traced, and the redundancy is the bill.

The case that started everything

In general relativity the free shape is your choice of coordinate labels, one shape for each dimension of spacetime, so four in the world we live in. And the rule cares not only about your labels but about how quickly they change from place to place.

So the theorem promises four relations among the field equations and their rates of change.

Those four relations were already known before Noether's paper. Hilbert had asserted them for one particular scoring rule, and Klein had derived them generally, and there was an argument at Göttingen about what they meant. The history track tells that story.

What Noether added was not the four relations. It was the theorem that has them as one instance among many — and that says, in advance, exactly how many to expect for any theory of this kind.

Both at once

A theory can have both kinds of symmetry: global knobs and local shapes together. Then both theorems apply, independently, and you get conserved quantities from one and identities from the other. Neither interferes with the other.

Noether says this in a single sentence and it is the cleanest possible statement that her two theorems are not rival readings of one result. They can both be true of the same theory at the same time, and they say different things.

A word about the name

Noether calls them dependencies. The term "Noether identities" is later, and this volume uses it because it is standard, while noting that it is not her word.

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