Noether's theorems/Explainers/Symmetries of a curved space

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: Killing vectors and conserved quantities along geodesics.

Killing vectors on a sphere with a geodesic

A curved surface can have symmetries just as a flat one can, and where it does, things moving freely across it conserve something.

Sliding a shape along itself

A sphere can be turned about any axis and lands on itself exactly. Three independent ways to turn it, so three symmetries.

A cylinder can be turned about its long axis, and slid along that axis. Two symmetries.

A potato has none. Move it any way at all and it does not land on itself.

The mathematical name for a direction you can slide a space along without distorting it is a Killing vector, after the mathematician who wrote down the condition. Its defining property is that distances are unchanged by the sliding.

What free motion conserves

Something moving freely across a curved surface — no forces, just the surface — follows the straightest available path. Call that a geodesic. On a sphere, geodesics are the great circles.

Now the result. For each Killing vector, a moving object conserves the amount of its motion that lies along that direction.

The proof is three lines and its whole content is a mismatch of symmetry. The Killing condition makes one object antisymmetric — swap two of its slots and it flips sign. The motion supplies a symmetric one — swap and nothing happens. Multiply them together and add up, and every term cancels against its partner.

So the quantity does not change, and nothing about curvature entered the argument. One Killing vector, one conserved quantity, exactly the count the counting rule predicts.

The two familiar cases

If the shape of space does not change with time, the time direction is a Killing vector, and the conserved quantity is the energy of a freely falling object.

If it looks the same all the way round an axis, that rotation is a Killing vector, and the conserved quantity is the angular momentum.

These are how orbits around a star are calculated. Both conserved quantities come from symmetries of the space itself rather than from anything about the orbiting object.

The catch, and it is a large one

A spacetime picked at random has no Killing vectors whatsoever.

This is not a rare misfortune. It is the ordinary case. The condition a Killing vector has to satisfy is far more demanding than it looks, and a generic curved space satisfies it for nothing but the trivial direction that goes nowhere.

The spacetimes in textbooks — the static ones, the spherically symmetric ones — all have Killing vectors, which is exactly why they are the ones in textbooks. They can be solved. That gives a badly misleading impression of how available these conserved quantities are.

Two different ways of failing

Worth keeping apart, because general relativity manages both and they get run together.

No symmetry. A lumpy spacetime has no Killing vector, so there is nothing for the theorem to work on and no conserved quantity exists.

A symmetry that gives nothing. Even where there is one, the local freedom of general relativity can make the resulting conservation law empty.

Absent and empty are not the same failure. The first is a shortage of symmetry. The second is a surplus of it.

Spaces with the most symmetry possible

There is a ceiling on how symmetric a space can be, and it is worth knowing because the spaces at the ceiling are the ones every textbook uses.

For an ordinary surface, the maximum is three: two ways of sliding and one of turning. Exactly three kinds of surface achieve it — the flat plane, the sphere, and a saddle-shaped surface that curves away in every direction at once. Nothing else comes close.

For spacetime the ceiling is ten, and the flat spacetime of special relativity sits exactly at it: one for time, three for space, three for turning, three for changing to a laboratory moving at a steady speed. That is the same ten as the count on the earlier page, and it is not a coincidence — that page counted the symmetries of flat spacetime and this one counts them again from the geometry.

Now the honest part. Once you put anything in that spacetime — a star, a galaxy, anything at all — the symmetry count collapses. A spacetime containing a single non-rotating star retains four. A realistic one, with matter distributed unevenly, retains none.

So the conserved quantities of general relativity are a feature of its idealised solutions, not of the theory. The textbook cases are chosen for solvability, and solvability is symmetry.

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