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Noether theorems volume, initial import (A-09)

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{{DISPLAYTITLE:Photon energy in an expanding universe}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Where the light's energy goes|Where the light's energy goes]].''

'''This page works its case,''' using the result derived on
[[Noether's theorems/Mathematics/Killing vectors and conserved quantities along geodesics|the
Killing vector page]].

== The setting ==

An expanding universe, homogeneous and isotropic, with a scale factor
<math>a(t)</math> multiplying the spatial part of the metric. Distances between
comoving points grow in proportion to <math>a</math>.

Light travels on null geodesics. The question is what it conserves.

== What is a symmetry, and what is not ==

Ask which directions the metric can be slid along without distortion.

'''Spatial directions are.''' The metric components do not depend on position, so
each coordinate direction in space satisfies Killing's equation and is a Killing
vector.

'''The time direction is not.''' The spatial part of the metric carries
<math>a(t)</math>, which depends on <math>t</math>. Sliding in time changes
distances, so time translation is not a symmetry and Killing's equation fails.

'''There is therefore no conserved energy, and no theorem has been violated in
saying so.'''

== What is conserved, worked ==

By the derivation on the Killing vector page, each Killing vector
<math>\xi^{a}</math> gives a quantity conserved along every geodesic:

<math>\xi_{a}\dot{x}^{a}=\mathrm{const}\qquad\qquad\mathrm{(1)}</math>

For a spatial Killing vector this contraction is the comoving momentum — the
momentum measured against the coordinate grid rather than against physical
distance. So

<math>p_{\text{comoving}}=\mathrm{const}\qquad\qquad\mathrm{(2)}</math>

Physical distance in a coordinate interval is that interval multiplied by
<math>a</math>, so physical momentum is the comoving momentum divided by
<math>a</math>:

<math>p_{\text{physical}}=\frac{p_{\text{comoving}}}{a}\propto\frac{1}{a}\qquad\qquad\mathrm{(3)}</math>

For light, energy is proportional to momentum. Hence

<math>E\propto\frac{1}{a}\qquad\qquad\mathrm{(4)}</math>

The energy of a photon falls in inverse proportion to the scale factor, which is
the cosmological redshift. Its wavelength stretches with the universe.

== Where the energy goes ==

Nowhere. The question presumes a conservation law that the theorem says does not
exist here.

The usual instinct is to look for the missing energy — in the gravitational field,
in the expansion itself. That search assumes energy must be conserved and asks
where it went. The theorem inverts the question: '''energy is conserved when time
translation is a symmetry, and here it is not.'''

Two consequences worth separating.

'''This is not the improper-energy problem.''' On
[[Noether's theorems/Applications/General relativity and the energy question|that
page]] the symmetry exists but is swallowed by a local one, so the conservation law
is empty. Here the symmetry is simply absent. Different failure, same theory.

'''And it is the theorem read forwards.''' No symmetry, no conserved quantity. The
result is usually presented as a puzzle about cosmology. It is the first theorem,
applied honestly to a case where its hypothesis fails.

''Provenance: (1) is derived on the Killing vector page from standard definitions.
(2) to (4) are this volume's own. No source was consulted for this page. See
[[Noether's theorems/Sources|Sources]].''

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