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

Plain-language counterpart: Why gravity has no local energy.

Fields against independent field equations

This page derives nothing new. It applies a criterion proved in full on 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.

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 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. 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 Sources.