The problem of energy appearing out of nowhere

[PeterDonis]

The question was if an alternative universe without that feature is self consistent. And it clearly is.

Is it? That's the question. The fact that you can write down a force equation with a damping term, so that mechanical energy is not conserved, does not prove that that equation is part of a self-consistent model. I haven't been able to find a non-paywalled version of the paper you referenced, so I can't look to see if it addresses this.

 

[Roberto Pavani]

Very interesting discussion. I'd note that the arXiv preprint I referenced in a related thread (Hobson, Barker & Lasenby, arXiv:2605.19719), proving that no local gauge-invariant energy-momentum tensor exists for the linearised gravitational field, seems relevant here as well.

 

[Dale]

down a force equation with a damping term, so that mechanical energy is not conserved, does not prove that that equation is part of a self-consistent model.

Of course it does. That is the whole point of writing the equations of motion and finding solutions. If you have a system of equations and that system of equations has a solution then it is self-consistent. It may be inconsistent with other models or principles, and it may be inconsistent with experiment, but it is self consistent.

 

[Matterwave]

Very interesting discussion. I'd note that the arXiv preprint I referenced in a related thread (Hobson, Barker & Lasenby, arXiv:2605.19719), proving that no local gauge-invariant energy-momentum tensor exists for the linearised gravitational field, seems relevant here as well.

Note that in non of my discussion w/Peter did either of us try to find the "energy momentum tensor of the gravitational field". We were always talking about the stress-energy-momentum tensor of the matter fields.

The SET of the "gravitational field" is subtle.

As a tensor, the SET must be well defined at every point and cannot be "transformed away" based on some coordinate transformation. However, the metric can be made to be $\text{diag}(1,-1,-1,-1)$ (depending on signature) at a point *and* the christoffel symbols made to be 0 *at a point* given some coordinates (see: Riemann Normal Coordinates).

So the dynamical parts of gravity is exhibited only "at the second derivative (curvature) level" -- this is a consequence of the equivalence principle.

It's hard to define a purely local object (SET) on a concept that you can locally "transform away".

In particular, note that $\nabla_a T^{ab}=0$ is a purely local conservation of energy of the matter fields and does not include the contributions from gravity. At a global scale you need a timelike KVF and asymptotic flat infinity to define something like the Komar mass which *does* include contributions from gravity.

But if the above condition on the SET is taken as a general "conservation of energy in GR" statement by practicing physics, I do not want to argue it. And I already conceded that point.