Hamilton's principle/Explainers/When symmetry is local - Revision history

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{{Technical|Hamilton's principle/Mathematics/Noether's second theorem}}

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 ==

[[File:OPL-HP-F-37.png|thumb|right|400px|Identities among field equations from a local symmetry.]]

[[Hamilton's principle/Explainers/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
[[Hamilton's principle/Explainers/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
[[Hamilton's principle/Explainers/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
[[Hamilton's principle/Explainers/Two different actions]].

← Older revision		Revision as of 04:58, 16 September 2026
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	[[File:OPL-HP-F-37.png\|thumb\|right\|400px\|Identities among field equations from a local symmetry.]]		[[File:OPL-HP-F-37.png\|thumb\|right\|400px\|Identities among field equations from a local symmetry.]]

	[[Hamilton's principle/Explainers/Symmetry and conservation]] dealt with		[[Hamilton's principle/Explainers/Symmetry and conservation\|Symmetry and conservation]] dealt with
	symmetries you apply the same way everywhere. Shift the whole experiment a metre		symmetries you apply the same way everywhere. Shift the whole experiment a metre
	left. Turn the whole apparatus. One change, applied uniformly.		left. Turn the whole apparatus. One change, applied uniformly.
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	enough, on its own, to force electromagnetism into existence. That is the subject		enough, on its own, to force electromagnetism into existence. That is the subject
	of		of
	[[Hamilton's principle/Explainers/Electromagnetism from an action]].		[[Hamilton's principle/Explainers/Electromagnetism from an action\|Electromagnetism from an action]].

	'''General relativity.''' You may choose whatever coordinates you like, and you may		'''General relativity.''' You may choose whatever coordinates you like, and you may
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	choice. That is a local symmetry with several independent parts, and it forces		choice. That is a local symmetry with several independent parts, and it forces
	relations among Einstein's equations. That is at		relations among Einstein's equations. That is at
	[[Hamilton's principle/Explainers/Gravity from an action]].		[[Hamilton's principle/Explainers/Gravity from an action\|Gravity from an action]].

	In both cases the redundancy is not incidental. It is the organising principle from		In both cases the redundancy is not incidental. It is the organising principle from
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	The volume now returns to a confusion it has flagged several times, at		The volume now returns to a confusion it has flagged several times, at
	[[Hamilton's principle/Explainers/Two different actions]].		[[Hamilton's principle/Explainers/Two different actions\|Two different actions]].