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Noether theorems volume, initial import (A-07)

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{{DISPLAYTITLE:General relativity and the energy question}}

''Plain-language counterpart: [[Noether's theorems/Explainers/Why gravity has no local energy|Why gravity has no local energy]].''

[[File:NT-F-22.png|thumb|right|360px|alt=Three columns of stacked bars. Eight field variables on the left, eight field equations in the middle with the last two shaded to mark them as dependent, and six independent facts on the right. An arrow labelled minus two joins the middle column to the right. The caption explains that two free functions in the symmetry give two identities and two equations that follow from the others.|Fields against independent field equations]]

'''This page derives nothing new. It applies a criterion proved in full on
[[Noether's theorems/Second theorem/Proper and improper conservation laws|the
proper-and-improper page]].''' That is not a limitation: the criterion is the
answer, and this page's job is to apply it correctly.

== The question ==

By 1918 the problem was known and unsolved. General relativity produces no local
energy density for the gravitational field. The quantity that occupies the right
position in the equations is not a tensor: change the coordinates and it changes,
and at any chosen point coordinates exist making it zero.

[[Noether's theorems/History/Gottingen and the energy problem|The history track]]
describes how this was read at the time — as a defect awaiting a better
construction.

== The criterion applied ==

The criterion, in the form proved on the second-theorem track:

'''If the action admits the translation group, its energy relations are improper
if and only if the action is invariant under an infinite group containing the
translation group as a subgroup.'''

General relativity is invariant under all relabellings of the coordinates. That is
an infinite group. Constant translations are relabellings, so the translation
group is a subgroup of it.

The condition is met. The energy relations are improper.

By the definition on the same page, improper means the conserved quantity can be
assembled from the Lagrange expressions and their derivatives — an identity in the
costume of a conservation law, carrying no information the field equations did not
already carry.

'''There is nothing to find, and no better construction is coming.'''

== The half that is always dropped ==

The criterion is a biconditional, and the other direction is what makes it a
criterion rather than an excuse.

Noether checks it herself, in a footnote: the energy theorems of classical
mechanics, and of special relativity, are '''proper''' — because no infinite group
occurs in either. Those theories have global symmetries and no local ones, so the
condition fails and the conservation laws are genuine.

A rule that only ever returns "improper" would explain nothing. This one
separates the cases, and the separation is what makes the answer an explanation
rather than a restatement of the difficulty.

== Two things this does not say ==

'''It does not say gravity is why.''' The criterion is about the group. Noether
constructs, in the same section, an infinite group out of a logarithm with no
physical content whatever, and the improper energy relations appear exactly as the
criterion requires. Gravity satisfies a condition. It is not the reason for the
condition.

'''It does not say general relativity has no energy.''' For an isolated system —
a star with empty space stretching away — a total energy can be defined by the
behaviour far away, and it is meaningful and measurable. What the criterion
forbids is the local statement: so much here, so much there. The distinction is
the one on
[[Noether's theorems/First theorem/The Noether current and the Noether charge|the
current and charge page]], with the local statement failing and a global one
surviving by another route.

== A second, unrelated failure ==

Even where a spacetime does have a symmetry, it may have very few.
[[Noether's theorems/Mathematics/Killing vectors and conserved quantities along geodesics|The
Killing vector page]] records that a generic spacetime has none at all.

Absent symmetry and empty conservation law are different failures. General
relativity has both, and running them together makes each harder to state.

''Provenance: the criterion, the footnote on classical mechanics, and the
logarithmic example are Noether's sixth section, proved on the second-theorem
track from the public-domain German original. Nothing on this page is newly
derived. See [[Noether's theorems/Sources|Sources]].''

Noether's theorems/Applications/General relativity and the energy question - Revision history

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