There are many questions on this website which ask similar questions to this, but none I have found have asked this exact question.

First, a bit of history on the origin of Noether's theorem. David Hilbert was very confused as to the nature of energy conservation in general relativity. Confused notions of energy conservation being a constraint on a possible theory led to the creation of Gustav Mie's (now long forgotten) competitor theory of gravitation and electromagnetism (known to some as the "Mie Theory," crazy stuff, look it up.) When it became clear that Einstein was right, and energy conservation seemed to be trivial in theory due to the Bianchi identity ($\nabla_\mu G^{\mu \nu} = 0 \implies \nabla_{\mu} T^{\mu \nu} = 0$) Hilbert asked Emmy Noether, a mathematician he held in high regard, to see if she could figure out why energy conservation seemed trivial. Three years later she had proven what we now call Noether's first and second theorems and reported her results in her famous paper.

Noether's first theorem states that every symmetry of an action gives a conserved quantity. Noether's second theorem states that every gauge symmetry of an action gives a "constraint" on the equations of motion, i.e. an off shell relationship involving the Euler Lagrange equations that vanish identitcally. (In other words, the equations of motion are not unique, which allows for multiple solutions to solve the equations of motion given some initial condition (which could have been guessed beforehand because, well, that's what gauge symmetries do.))

Funnily enough, Noether's first theorem, usually just called "Noether's theorem" now, has received universal acclaim in our field while Noether's second theorem is unknown to most. However, historically, it was Noether's second theorem which was more relevant to Hilbert's question and more important to the task at hand.

Anyway, Hilbert was absolutely delighted with her results. The conclusion which was drawn from these theorems, in their relation to general relativity was

  1. Usually in physics, global symmetries give conservation laws. Space/time translations usually give conservation of momentum/energy.
  2. However, the action for general relativity contains no global symmetries.* It only contains the gauge symmetry of general covariance. And, in fact, the off-shell identity given by Noether's second theorem among the equations of motion is exactly the Bianchi identity $\nabla_\mu G^{\mu \nu}$.
  3. Therefore, energy conservation is trivial in general relativity because of general covariance!

I have seen this statement, that general covariance forbids the existence of a non trivial conserved stress energy tensor, multiple times in the literature and in my education. Recently I saw if mentioned in the opening pages of these (very good) lectures by Compère and Fiorucci :

"Then we will enter into the main point we have to discuss, and motivate why we cannot define conserved currents and charges in Noether’s fashion for generally covariant theories" However, even though I can appreciate the logic I outlined in three steps, I think there there is something minor wrong with it. Indeed, in those lectures, I do not really see the authors proving a statement akin to the claimed "general covariance $\implies$ no conserved currents."

My issue comes down to this. Stress energy tensors are defined as $$T^{\mu \nu} = \frac{2}{\sqrt{-g}} \frac{\delta \mathcal{L}}{\delta g_{\mu \nu}}.$$ However, for any theory of gravity, for which $\mathcal{L}$ will be some function of the the metric $g_{\mu \nu}$ (maybe the Einstein-Hilbert action, maybe something else) we will have $T^{\mu \nu} = 0$ trivially just from the Euler Lagrange equation for $\delta g_{\mu \nu}$. So I would say that the stress energy tensor in gravity is trivial just because gravity is a theory of the metric, end of story. Meanwhile, general covariance is a property shared by many theories. It is not a special property at all. The Klein Gordon scalar field is generally covariant. However, it has a non trivial stress energy tensor anyway.

I sort of think I'm right here, I just want to see if anyone has any points to add or takes issue with my logic.

*As an aside, is it possible to rigorously prove that there are no global symmetries in the Einstein Hilbert action? I certainly can't think of one, but maybe there is a very obscure one no one has thought of.


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