Killing vectors and conserved quantities along geodesics

Plain-language counterpart: Symmetries of a curved space.

Killing vectors on a sphere with a geodesic

A curved space has symmetries when it looks the same after being slid along itself. This page derives what those symmetries give, and the derivation is three lines.

Notation. The tangent to a curve is written ${\dot{x}}^a$, not $u$, which is reserved for Noether's dependent variables.

Killing vectors

A vector field ${\xi }^a$ generates a flow. The metric is unchanged along that flow exactly when

${\mathrm{\nabla }}_a{\xi }_b+{\mathrm{\nabla }}_b{\xi }_a=0\mathrm{(}\mathrm{1}\mathrm{)}$

which is Killing's equation. A field satisfying it is a Killing vector.

The equation says the symmetric part of ${\mathrm{\nabla }}_a{\xi }_b$ vanishes, so ${\mathrm{\nabla }}_a{\xi }_b$ is antisymmetric. That antisymmetry is the only property the derivation below uses.

The conserved quantity, derived

A geodesic with affine parameter $\tau$ and tangent ${\dot{x}}^a$ satisfies

${\dot{x}}^b{\mathrm{\nabla }}_b{\dot{x}}^a=0\mathrm{(}\mathrm{2}\mathrm{)}$

Form the contraction of the Killing vector with the tangent and differentiate along the curve:

$\frac{d}{d\tau }\left({\xi }_a{\dot{x}}^a\right)={\dot{x}}^b{\mathrm{\nabla }}_b\left({\xi }_a{\dot{x}}^a\right)={\dot{x}}^b{\dot{x}}^a{\mathrm{\nabla }}_b{\xi }_a+{\xi }_a{\dot{x}}^b{\mathrm{\nabla }}_b{\dot{x}}^a\mathrm{(}\mathrm{3}\mathrm{)}$

The second term vanishes by the geodesic equation (2). The first term contracts ${\mathrm{\nabla }}_b{\xi }_a$, antisymmetric by (1), with ${\dot{x}}^b{\dot{x}}^a$, symmetric — and a symmetric object contracted with an antisymmetric one is zero. Hence

${\xi }_a{\dot{x}}^a=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{(}\mathrm{4}\mathrm{)}$

One Killing vector, one conserved quantity along every geodesic. The count matches the first theorem's exactly, and so it should: this is the first theorem applied to the action for a geodesic, with the symmetry of the metric supplying the invariance.

The two familiar cases

If the metric components do not depend on a coordinate, then the coordinate vector field in that direction is Killing, since (1) reduces to the vanishing of derivatives of the metric in that direction.

For a static spacetime, the time direction is Killing and (4) is the conserved energy of a freely falling particle. For an axially symmetric one, the rotation direction is Killing and (4) is the conserved angular momentum. Both are the statement on the worked-cases page transported to curved space.

The limit of the method

A generic spacetime has no Killing vectors at all.

This is not a rare pathology; it is the general case. Killing's equation is an overdetermined system, and a metric picked at random satisfies it for nothing but the zero vector.

So the conserved quantities of general relativity are not generally available. They exist for the symmetric solutions — static, spherically symmetric, stationary — and those are the solutions written down in textbooks, which gives a misleading impression of how common they are.

This is a different failure from the one on the energy question page, and the two should not be run together. Here the symmetry is absent, so the first theorem gives nothing. There the symmetry is present but sits inside a local one, so the first theorem gives something empty. Absent and empty are not the same, and general relativity manages both.

Provenance: (1) to (4) are standard and are derived above. The observation distinguishing the two failure modes is this volume's own. No source was consulted for this page. See Sources.