Why gravity has no local energy
Mathematical counterpart: General relativity and the energy question.
This is the question Noether's paper was written to answer. The answer is not that gravitational energy is hard to define. It is that the thing being looked for does not exist, and there is a rule saying exactly when it will not.
The problem as it stood in 1918
Every other field theory says how much energy sits in each small region. Gravity does not.
The quantity general relativity offers for the job behaves badly. Change your coordinate labels and it changes. Pick any point you like and there are labels making it zero there. A quantity you can make vanish by relabelling is not telling you how much of anything is present.
For three years this was read as unfinished business. The theory was young, the formalism unsettled, and somebody would eventually write down the right expression.
The rule
Noether's answer is a rule with two sides, and both sides are needed.
A conservation law is empty exactly when the symmetry it came from sits inside a symmetry you can dial differently at every point.
Empty here means the sense on the costume page: the conserved quantity is assembled out of the equations of motion, so saying it is conserved reports nothing the equations had not already said.
Now apply it. Energy conservation comes from the symmetry "shift everything in time". General relativity lets you relabel spacetime however you like, differently here than there — and shifting everything in time is one such relabelling.
So the symmetry sits inside a local one. The rule fires. The energy relations are empty, and no better expression is coming, because there is nothing to find.
The other side of the rule, which is the important half
A rule that always said "empty" would explain nothing.
Noether checks the other direction herself, in a footnote. Ordinary mechanics, and special relativity: their energy theorems are genuine. Neither theory has a local symmetry, so there is nothing for the global one to hide inside, and the condition fails.
That is what makes this an explanation rather than a restatement of the difficulty. The rule separates the cases, and it puts gravity on one side and everything else on the other for a stated reason.
Two misreadings
"Gravity is the reason." It is not. The rule is about symmetries, and any theory meeting the condition behaves this way. Noether proves the point by building a local symmetry out of a logarithm — a construction with no physics in it whatsoever — and showing the empty energy relations appear exactly as the rule requires. Gravity satisfies a condition; it does not cause one.
"General relativity has no energy." Also not. For an isolated system — a star with empty space stretching away in all directions — a total energy can be defined by watching how spacetime behaves far away, and it is meaningful and measurable. What is unavailable is the local statement: this much here, that much there.
The distinction is the one from the flow and total page. Usually the total follows from the local statement. Here the local statement fails and a total survives by a different route entirely.
And a second, unrelated problem
Even where a spacetime does have a symmetry, most spacetimes have very few, and a realistic one has none at all. That is the situation on the curved space page.
Keep the two apart. No symmetry is a shortage. A symmetry swallowed by a local one is a surplus. General relativity manages both, and describing either as "energy is a problem in general relativity" flattens two different facts into one vague one.
What the offending quantity actually is
It has a name — a pseudotensor — and the name records the complaint rather than explaining it.
A proper physical quantity in general relativity has to behave in a definite way when you change coordinate labels. Everybody must agree about it, even while describing it in different words. That is what being a tensor means, and the matter side of Einstein's equations satisfies it.
The gravitational energy expression does not. Relabel and it changes, and not in the disciplined way a real quantity changes. Two observers using different labels will not merely describe the same energy differently; they will disagree about whether there is any.
For a while this looked like a construction that had come out slightly wrong. Several people proposed alternatives — Einstein, Lorentz, Weyl and Hilbert each had one — and Klein's 1918 paper showed that all of them followed from a single identity and differed only in which term had been absorbed where.
That was the clue. A quantity that can be shuffled about that freely, with no principle selecting one version, is not a quantity that has a value. It took the criterion on this page to turn that suspicion into a proof.
No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.