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Discussions about global conservation of energy in GR often invoke the use of the stress-energy-momentum pseudo-tensor to offer up a sort of generalization of the concept of energy defined in a way that it will be conserved, but this is somewhat controversial to say the least. Rather than dwell on the pseudo-tensor formalism, I was wondering if anyone had any thoughts on the following approach.

Prior to Noether's theorem, conservation of energy wasn't just a truism despite the fact that new forms of energy were often invented to keep it in place. This is because there was a still a condition the laws of nature had to satisfy, and in many ways this condition is a precursor to Noether's theorem. The condition is simply that it had to be possible to define this new energy such that no time evolution of the system would bring the system to a state where all other energy levels were the same, but this new energy had changed, or conversely, a state where this new energy remained the same, but the sum of the remaining energy had changed. For example, if I wanted to define gravitational potential energy as $GMm/r$, the laws of gravity could not allow a situation where, after some time evolution, a particle in orbit around a planet had returned to its exact position (thus maintaining the same potential) but its kinetic energy had somehow increased. This is of course just saying that it physics must be such that there can exist a conserved scalar quantity, which later in time we found out to mean time invariant laws of physics.

The reason I formulated this in a convoluted manner instead is the following. In GR we invoke pseudo-tensors because Noether's theorem tells us energy should not be conserved given that the metric is not static, but we try to recover some semblance of energy conservation by asking if the dynamics of that metric do not somehow encode a quantity somehow resembling energy. Couldn't we ask instead if some sort of equivalent to energy was conserved in GR by asking :

In GR, do there exist initial conditions $\rho(t_0,x,y,z)$ and $g_{\mu \nu}(t_0,x,y,z)$) such that after some time we recover the same $\rho(t,x,y,z) = \rho(t_0,x,y,z)$ but with a different $g_{\mu \nu}$, or conversely, we recover the same $g_{\mu \nu}$ but with different $\rho$ ?

Obviously here we only ask about $\rho$ and not the entire $T_{\mu \nu}$ tensor. For starters, setting $T_{\mu \nu}$ immediately determines $g_{\mu \nu}$ so what I'm asking would be impossible. This isn't a problem because we're looking to generalize the concept of a conserved scalar, so it makes more sense to ask about the scalar field $\rho(t,x,y,z)$ than it does to ask about the tensor field $T_{\mu \nu}$. What this means is that you'd have to play around with how the momenta change in order to find a situation in which the answer is yes.

For the Friedmann equations, the answer is obviously no so some semblance of energy conservation still exists ($\rho \propto a^{-3(1+w)}$, therefore the $ \rho$s will return to the same values if the universe ever collapses in on itself and returns to previous values of $a$, though I'm not sure if this is necessarily the case for time varying values of $w$). Can this be shown to generally be the case for General Relativity ?

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