Göttingen and the energy problem
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.
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 Sources.