Hamilton's principle/Explainers/When symmetry is local: Difference between revisions
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This page explains Noether's second theorem, and why it is the one she wrote the | This page explains Noether's second theorem, and why it is the one she wrote the | ||
Latest revision as of 19:30, 15 October 2026
This page explains Noether's second theorem, and why it is the one she wrote the paper for. It uses no mathematics. The technical page it goes with is named above.
The same everywhere, or different at each point
Symmetry and conservation dealt with symmetries you apply the same way everywhere. Shift the whole experiment a metre left. Turn the whole apparatus. One change, applied uniformly.
Now consider a different kind. What if you could make the change independently at every point — a little here, a lot there, a different amount everywhere else?
That is a much stronger demand. It is not one symmetry. It is a separate symmetry at every point in space, all at once.
Physicists call this a local symmetry. Noether's second theorem is about what happens when a theory has one.
A different kind of answer
You might expect a stronger symmetry to give more conserved quantities.
It does not. It gives something else entirely: the equations of the theory stop being independent.
Some of them follow automatically from the others. Not as a consequence of the physics, but as a matter of arithmetic — they are true before you know anything about the system, for any conceivable behaviour.
That is a strange kind of result, and it takes some getting used to. A local symmetry does not tell you a quantity is preserved. It tells you that part of what looked like content is actually bookkeeping.
What redundancy means
The reason is that a locally symmetric description carries more information than the physics needs.
Several different descriptions correspond to exactly the same physical situation. If you can change the description at each point without changing anything real, then the description has spare parts.
Physicists call the spare parts gauge freedom.
The consequence: the equations cannot possibly determine everything, because some of what they would determine is not real. The evolution of the system is fixed only up to the freedom you built in.
That sounds like a defect and is not. It is the price of a description that keeps the symmetry visible, and the price is usually worth paying.
Where this actually happens
Two of the most successful theories in physics work this way.
Electromagnetism. The electric and magnetic fields can be built out of a quantity that is not unique — you can change it in a particular way, differently at each point, and nothing measurable changes. Demanding that this be so is enough, on its own, to force electromagnetism into existence. That is the subject of Electromagnetism from an action.
General relativity. You may choose whatever coordinates you like, and you may choose them differently in different regions. Nothing physical depends on the choice. That is a local symmetry with several independent parts, and it forces relations among Einstein's equations. That is at Gravity from an action.
In both cases the redundancy is not incidental. It is the organising principle from which the theory is built.
The question the paper answers
Now the reason Noether wrote it.
General relativity had made trouble for energy conservation. In older physics, energy conservation is a clear statement with a definite local meaning. In the new theory, the quantity that ought to be gravitational energy came out depending on which coordinates you chose — and coordinates are not physical.
Hilbert claimed this failure was characteristic of general relativity rather than a defect in anyone's attempt. It was a claim without a proof, and it was argued over in Göttingen for two years.
Noether's second theorem settles it.
Her result: a theory's energy conservation becomes an identity rather than a dynamical law exactly when the theory has this kind of local symmetry.
General relativity has one. So its energy conservation is an identity — true by construction, carrying no dynamical information. The various attempts to build a local gravitational energy fail because there is nothing there to build.
Hilbert was right. And Noether's answer is more general than his claim: the property has nothing to do with gravity in particular. Any theory with a local symmetry behaves this way. Gravity is one instance.
Why the second theorem is neglected
Almost everyone who has heard of Noether's theorem means the first one.
The reason is not hard to see. The first gives physicists a tool: find a symmetry, get a conserved quantity. Useful, quotable, immediately applicable.
The second tells them that something they might have wanted does not exist in the form expected. That is a harder result to use and an easier one to forget.
But the two are in the same paper, stated together and proved by the same method, and the second is the one that answers the question she was asked. Knowing only "symmetry implies conservation" is knowing the corollary and not the argument.
A note on the author
Noether was not allowed a paid position at Göttingen when she wrote this. She lectured under Hilbert's name, because the faculty would not appoint a woman.
She was dismissed from the university in 1933 and moved to the United States, where she taught until her death in 1935.
These facts bear on how the work travelled. A paper published from an unsalaried post, in a mathematics journal, by an author outside the ordinary channels, took decades to reach the physicists it concerned. The first English translation appeared in 1971, fifty-three years after publication.
Where to go next
The volume now returns to a confusion it has flagged several times, at Two different actions.