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			{{Noether's theorems map}}

	''Mathematical counterpart: [[Noether's theorems/First theorem/The Noether current and the Noether charge\|The Noether current and the Noether charge]].''		''Mathematical counterpart: [[Noether's theorems/First theorem/The Noether current and the Noether charge\|The Noether current and the Noether charge]].''
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	''No mathematics is used on this page. The mathematical treatment is on the		''No mathematics is used on this page. The mathematical treatment is on the
	counterpart page named above.''		counterpart page named above.''

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NT>NoetherVolumeBuild: Noether theorems volume, initial import (B-08)

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Noether theorems volume, initial import (B-08)

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{{DISPLAYTITLE:A flowing quantity and a standing total}}

''Mathematical counterpart: [[Noether's theorems/First theorem/The Noether current and the Noether charge|The Noether current and the Noether charge]].''

[[File:NT-F-04.png|thumb|right|360px|alt=A tube drawn in perspective with arrows flowing along it, labelled the current. A dashed elliptical cross-section partway along is labelled the charge, with a note that the amount crossing is the same at every slice. A caption states that conservation means the books balance rather than that nothing moves.|The Noether current as a flux through a tube]]

The first theorem hands you an object for each knob. This page says what that
object is.

== Flow and total ==

Think of water in a network of pipes.

At any point you can ask how fast water is moving and in which direction. That is
the '''flow'''. It is a different arrow at every point.

You can also draw a surface across the network and ask how much water crosses it.
That is a '''total'''.

If water is never created or destroyed anywhere in the network, the total crossing
one surface equals the total crossing any other. The number does not depend on
where you cut.

The first theorem produces exactly this pair. Physicists call the flow the
'''current''' and the total the '''charge''' — including when the quantity in
question is energy rather than electricity, which is a piece of vocabulary worth
bracing for.

== Where the flow comes from ==

Two contributions, and they have different origins.

Part of it comes from the things in the system changing. Part of it comes from
the frame having moved, in the sense of
[[Noether's theorems/Explainers/What happens at the edges|the picture-frame
picture]].

A treatment that never moves the frame gets only the first part. Energy comes from
moving the frame in time, so that treatment cannot produce energy. This is a
common shortcut and it quietly costs the most famous case.

== The flow is not unique ==

Here is something that surprises people and is not a defect.

Add to your flow a circulating pattern — water going round in a loop, never
accumulating anywhere. Every total across every surface is unchanged, because
whatever the loop carries across, it carries back.

So there are many flows giving the same totals, and nothing in the theorem picks
one out. Physics later found reasons to prefer particular choices, but the theorem
itself does not care, and the freedom is genuine rather than an artefact of a
clumsy derivation.

== Counting is not the same as counting facts ==

The theorem promises one result per knob, and the results are independent in a
precise technical sense.

They need not be independent in the sense you care about.

A car travelling at a steady speed gives you two true statements: its speed is
constant, and the square of its speed is constant. Those are two statements. They
are not two facts. Knowing either tells you the other.

Noether gives exactly this example herself, with a system where two properly
independent symmetries deliver two conserved quantities that turn out to be
tied together.

'''So the count is a count of relations, not a count of things you have
learned.''' Read the number that way and it will not mislead you.

== When the flow does not balance ==

One last case, which Noether puts in a footnote in 1918 and which turns out to
matter enormously later.

Suppose the system is not isolated — something outside is feeding into it. Then
the flow does not balance, and the totals across different surfaces differ. The
difference is exactly what was fed in.

She writes this down without comment. Modern physics calls the feeding term a
source, and the whole of how forces couple to matter is written in this shape.

== What a total means when the flow is through time ==

The pipe picture has one feature that needs unpacking, because it is where the
word "conserved" comes from.

In the pipe network you drew a surface and asked how much crossed it. In physics
the surfaces are usually snapshots: all of space, at one moment.

So the total is a quantity spread over the whole of space at one instant, and the
statement that the total is the same across different surfaces means it is the
same at every instant.

That is conservation, and it is worth noticing that it came from a statement about
'''flow''' rather than about staying still. Nothing is required to sit motionless.
Energy moves around constantly. What the theorem gives you is that the books
balance: whatever leaves a region has gone somewhere adjacent, never nowhere.

== Why this matters for gravity ==

The local statement — nothing is created or destroyed anywhere — is the strong one.
The total is a consequence of it.

You might hope to keep the total even if the local statement fails. In general
relativity that hope is where the trouble starts, because there the local
statement is the one that goes wrong. See
[[Noether's theorems/Explainers/Why gravity has no local energy|Why gravity has no
local energy]].

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

Noether's theorems/Explainers/A flowing quantity and a standing total - Revision history

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