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	''Mathematical counterpart: [[Noether's theorems/Applications/General relativity and the Bianchi identity\|General relativity and the Bianchi identity]].''		''Mathematical counterpart: [[Noether's theorems/Applications/General relativity and the Bianchi identity\|General relativity and the Bianchi identity]].''
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	''No mathematics is used on this page. The mathematical treatment is on the		''No mathematics is used on this page. The mathematical treatment is on the
	counterpart page named above.''		counterpart page named above.''

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NT>NoetherVolumeBuild: Noether theorems volume, initial import (B-29)

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Noether theorems volume, initial import (B-29)

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{{DISPLAYTITLE:An identity hiding inside Einstein's equations}}

''Mathematical counterpart: [[Noether's theorems/Applications/General relativity and the Bianchi identity|General relativity and the Bianchi identity]].''

Einstein's field equations are ten equations. Six of them are doing the work. This
page says how the second theorem knows that in advance.

== The count, before any calculation ==

General relativity's local symmetry is the freedom to label the points of
spacetime however you like — and differently here than there. Spacetime has four
dimensions, so that is four free labels, four free shapes.

Feed that into
[[Noether's theorems/Explainers/Counting the equations a gauge theory really has|the
counting result]] and out comes: four identities among the field equations, and
four of them following from the rest.

'''Notice what was not needed.''' Nothing about curvature. Nothing about how
gravity works. Nothing about the equations themselves beyond the fact that they
come from a scoring rule with that freedom. The number four was fixed the moment
somebody said "you may relabel the coordinates".

Noether gives this very case as an example in her opening section, before proving
anything.

== What the four identities turn out to be ==

'''Unchecked.''' In the literature they are the contracted Bianchi identities:
the statement that a particular combination built out of the curvature has
vanishing divergence, automatically, for any spacetime whatever. This volume does
not derive that, because deriving it needs the symmetries of the curvature tensor
and this volume does not set them up.

== Three things that follow from the count alone ==

'''Four of the ten equations are not independent.''' If you count Einstein's
equations and conclude the theory is overdetermined, you have counted wrong. It is
not.

'''Whatever sits on the right must be divergence-free.''' The left-hand side of
the field equations has vanishing divergence identically. The right-hand side is
the energy and momentum of matter. So the matter's energy and momentum must be
divergence-free too, or the equations are simply inconsistent.

That is the same structure as
[[Noether's theorems/Explainers/Counting the equations a gauge theory really has|charge
conservation in electromagnetism]], and it is not a coincidence: in both cases the
local symmetry produces an identity, and the identity forces a conservation law
onto whatever the theory is coupled to. '''Gravity cannot couple to matter that
does not conserve energy and momentum. The theory refuses.'''

'''The future is not determined by the present.''' Four functions' worth of any
solution is a choice of labels, carrying no physical content. Hilbert had spelled
this out in 1916 and found it disturbing; the second theorem shows it is the
generic behaviour of a theory with a local symmetry, not a peculiarity of gravity.

== Who found what ==

Worth stating plainly, because the four identities are often attributed to
Noether and were not hers.

Hilbert asserted them for one particular scoring rule, in 1915. Klein derived them
in general form, before Noether's paper appeared. Hilbert's own reprint of his
papers, in 1924, credits Klein with the general derivation and Noether with the
general proof of the theorem that has them as one instance among many.

'''The identities are not her discovery. The theorem is.''' Which is the larger
result, and the reason her name is on the subject rather than theirs.

== What it is like to use ==

The practical consequence is stranger than it sounds, and worth spelling out.

Normally you count equations against unknowns. Ten equations, ten unknowns, one
answer. That instinct is wrong here and will mislead you in a specific way.

Ten equations, four of which follow from the others, gives six real constraints on
ten quantities. Four quantities' worth of the answer is undetermined — and if you
go looking for the extra equations that would pin them down, you will not find
them, because they do not exist and were never missing.

What you do instead is decide. Pick coordinates, write the choice down, and carry
on. Two people who choose differently get descriptions that disagree everywhere
and agree on every measurable thing, which is the test that they both did it
right.

'''The four undetermined functions are not ignorance. They are the labels.'''

The reason to know this in advance, rather than discovering it halfway through a
calculation, is that the symptom is confusing. Equations that will not close look
like a mistake. The count tells you before you start exactly how much will not
close, and that the amount is correct.

''No mathematics is used on this page. The mathematical treatment is on the
counterpart page named above.''

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