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Noether theorems volume, initial import (H-01)

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{{DISPLAYTITLE:Göttingen and the energy problem}}

[[File:NT-F-17.png|thumb|right|360px|alt=Three horizontal lanes for Hilbert, Klein and Noether, with dated events marked on each from November 1915 to July 1918. Dashed arrows between the Klein and Noether lanes are labelled mutual influence. A note records that both papers were revised until September, after presentation.|Gottingen 1915 to 1918: who wrote what, and when]]

Einstein published the field equations of general relativity in November 1915.
Within weeks the theory had a problem that no other field theory had, and the
problem was about energy.

== What conservation looks like in every other theory ==

A field theory conserves energy in a particular form. There is a quantity built
from the fields, and a statement that this quantity flows without accumulating
anywhere: what enters a region equals what leaves it, plus whatever the region
gained. The statement is local. It holds at each point, and it can be integrated
over a region to give a total that does not change.

Electromagnetism has this. Continuum mechanics has this. The form is so
consistent across theories that by 1915 it was taken as part of what it meant to
have a field theory at all.

== What general relativity has instead ==

General relativity does not produce a quantity of that kind for the
gravitational field.

Einstein constructed something that occupies the same position in the equations
and behaves the same way under integration. It is written with two indices and
appears added to the energy of matter. But it is not a tensor. Under a change of
coordinates it does not transform as the other quantities in the theory
transform, and at any chosen point a coordinate system can be found in which it
is zero.

This is not a small blemish. If the energy density of the gravitational field
can be made to vanish at a point by relabelling the coordinates, then it is not
telling you how much energy is at that point. The quantity became known as a
pseudotensor, and the name records the objection.

For several years the natural reading was that this was a defect of
construction. A better quantity would be found, or a better way of deriving the
one already in hand. The theory was three years old; its formalism was not
settled; someone would clean this up.

== Why the question landed at Göttingen ==

Göttingen was where the mathematics of the new theory was being examined most
closely, under Hilbert and Klein.

[[Noether's theorems/History/Hilbert and Klein|Hilbert had put general
relativity into variational form]] in November 1915, deriving the field
equations from an action rather than positing them, and in doing so had found
something odd about the count of those equations. Klein took up the energy
question directly and worked through the several incompatible expressions that
Einstein, Hilbert, Lorentz and Weyl had each proposed for gravitational energy,
looking for the single point of view from which all of them followed.

Both men reached the same place. There was a structural reason for the
difficulty, it had to do with the fact that general relativity is invariant
under transformations that may differ from point to point, and neither of them
could state the reason in general.

The general statement is what Noether supplied.

== What was actually being asked ==

It is worth separating two questions that are often run together.

The first is: what is the energy of the gravitational field? That question
assumes an answer exists and asks for it.

The second is: what property of a theory determines whether its energy
conservation takes the ordinary local form? That question does not assume
anything, and it is answerable.

Noether answered the second. The answer settles the first by showing that it was
the wrong question. A theory invariant under transformations that vary from
point to point, and containing the constant shifts among those transformations,
has energy relations of the degenerate kind, and has them exactly under that
condition. Nothing better is available, because the degeneracy is what the
symmetry produces.

The route from the question to that answer runs through Hilbert's two
communications and Klein's three notes, and is taken up next.

''Provenance: this page contains no verbatim quotation. Hilbert's papers were
located but the only electronic copy reached is OCR-damaged at word level, and
Klein's papers were read in a modern English translation whose terms of reuse
have not been established. Both are therefore summarised, not quoted. See
[[Noether's theorems/Sources|Sources]].''

Noether's theorems/History/Gottingen and the energy problem - Revision history

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