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Noether theorems volume, initial import (M-05)

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{{DISPLAYTITLE:Killing vectors and conserved quantities along geodesics}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Symmetries of a curved space|Symmetries of a curved space]].''

[[File:NT-F-09.png|thumb|right|360px|alt=A sphere drawn with dashed latitude and longitude lines, a great circle marked as a geodesic, and three arrows tangent to the surface marked as Killing vectors. A side panel counts Killing vectors for a sphere, a cylinder, a flat plane and a lumpy surface, the last being zero.|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 <math>\dot{x}^{a}</math>, not
<math>u</math>, which is reserved for Noether's dependent variables.

== Killing vectors ==

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

<math>\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}=0\qquad\qquad\mathrm{(1)}</math>

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

The equation says the symmetric part of <math>\nabla_{a}\xi_{b}</math> vanishes,
so <math>\nabla_{a}\xi_{b}</math> is antisymmetric. That antisymmetry is the only
property the derivation below uses.

== The conserved quantity, derived ==

A geodesic with affine parameter <math>\tau</math> and tangent
<math>\dot{x}^{a}</math> satisfies

<math>\dot{x}^{b}\nabla_{b}\dot{x}^{a}=0\qquad\qquad\mathrm{(2)}</math>

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

<math>\frac{d}{d\tau}\left(\xi_{a}\dot{x}^{a}\right)=\dot{x}^{b}\nabla_{b}\left(\xi_{a}\dot{x}^{a}\right)=\dot{x}^{b}\dot{x}^{a}\nabla_{b}\xi_{a}+\xi_{a}\dot{x}^{b}\nabla_{b}\dot{x}^{a}\qquad\qquad\mathrm{(3)}</math>

The second term vanishes by the geodesic equation (2). The first term contracts
<math>\nabla_{b}\xi_{a}</math>, antisymmetric by (1), with
<math>\dot{x}^{b}\dot{x}^{a}</math>, symmetric — and a symmetric object contracted
with an antisymmetric one is zero. Hence

<math>\xi_{a}\dot{x}^{a}=\mathrm{const}\qquad\qquad\mathrm{(4)}</math>

'''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
[[Noether's theorems/First theorem/Energy, momentum and angular momentum|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
[[Noether's theorems/Applications/General relativity and the energy question|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 [[Noether's theorems/Sources|Sources]].''

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