You cannot actually get around this with quasi-local quantities. You have to abandon the classic EP argument that frame
transformations literally cancel first-order geometric distortions of a flat Minkowski geometry in GR.
As you know, I think you are in serious error on this specific point.
You're entitled to your opinion. But if the energy carried by the waves is determined by first-order terms in the expansion of the metric around any given point, then if there is no first-order invariant measure of deviation from Minkowski geometry, we are stuck with a spooky non-local vacuum energy density, which is a serious headache in canonical GR. As we've previously discussed at great length.
Right - but this weird classical nonlocality is accepted by the Pundits. Maybe Hagen Kleinert's math does what you want?
Gravitational energy density is not determined by the Riemann curvature in Einstein's theory. It is determined by purely first order quantities.
Basic problem is that total energy is not well defined in GR except in very special cases when there are timelike Killing vector fields. This corresponds to flat spacetime Noether's theorem where total energy is conserved when there is symmetry under time translations. That is simply not true at all in our actual accelerating expanding universe, for example.
Is Energy Conserved in General Relativity?
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[Archive] energy conservation in General relativity Special & General Relativity .
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I think everyone who matters in this field now admits that this is a major headache in GR.
But the energy carried by the waves is *first* order in the metric.
This is different from Hagen-Kleinert's multi-valued maps corresponding to real invariant topological, e.g. disclination defects that are not found in Einstein's 1916 restricted GCTs. What you were looking for Z is in Kleinert's work.
No, what I was looking for is what I found. No need for Kleinert's model to solve the problem. You just have to get over the root misconception that the LC covariant derivative is the only curved-space covariant derivative that can be legitimately defined in GR without altering the underlying intrinsic geometry.
You cannot introduce new connections without justification. It's fine to introduce torsion and non-metricity tensor additions to Levi-Civita only if they explain data anomalies like the Pioneer, flyby etc. Otherwise, ghostly connections without empirical necessity are excess metaphysical baggage in my opinion - they are less with more.