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

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 current and the Noether charge.

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 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 Why gravity has no local energy.

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