Why does gravity have to be non-linear?

[Q-reeus]

The way I interpret that discussion and other sources is that fully resolved issues are:

1) gravitational binding energy is negative
2) Several positive energy theorems are proven (slightly different assumptions and generality)

It is unsolved how (or even whether it is meaningfully possible) to make statements about the energy content of a specific region of spacetime with no matter or E/M fields.

OK well thanks for that frank response. Evidently GR is not a simple topic!

 

[Jonathan Scott]

From the point of view of semi-Newtonian static approximations, there is a simple self-consistent model for the location of conserved gravitational energy as follows:

1. Placing a test object in a potential decreases its energy by the time dilation factor. It also decreases the energy of the source of the potential by exactly the same amount because of the potential due to the test object acting on the source.

2. There is an additional positive energy density of $(1/{8\pi G}) \, \mathbf{g}^2$ where $\mathbf{g}$ is the field, like the Coulomb energy density in an electric field, and with the same sign. Integrated over all space, this gives a positive term equal to the potential energy in part 1, so the overall potential energy of the system comes out correct.

These two effects seem closely related to the "non-linear" illusion in my first post, except for irritating factors of 2.

 

[Q-reeus]

From the point of view of semi-Newtonian static approximations, there is a simple self-consistent model for the location of conserved gravitational energy as follows:

1. Placing a test object in a potential decreases its energy by the time dilation factor. It also decreases the energy of the source of the potential by exactly the same amount because of the potential due to the test object acting on the source.

2. There is an additional positive energy density of $(1/{8\pi G}) \, \mathbf{g}^2$ where $\mathbf{g}$ is the field, like the Coulomb energy density in an electric field, and with the same sign. Integrated over all space, this gives a positive term equal to the potential energy in part 1, so the overall potential energy of the system comes out correct.

These two effects seem closely related to the "non-linear" illusion in my first post, except for irritating factors of 2.

This makes sense - the net field energy is positive and precisely half the net decrease in matter potential energy. Hence the system binding energy is equal and opposite to the total field energy - correct? Not sure if this is 'widely accepted' but then amazingly from a recent thread even whether a uniformly accelerated charge radiates is controversial after more than a century of debate! Anyway so the sign of field energy density is positive in both the static field and GW cases - the latter seemingly confirmed by match between Hulse-Taylor binary neutron star data and Einsteins quadrupole formula. A nice consistency here imo.
One more piece of this gravitational energy triangle is still problematic for me at least. It seems most cosmologists subscribe to the view the universe has zero total energy: http://en.wikipedia.org/wiki/Zero-energy_universe. Given the enormous positive energy tied up in matter+radiation+maybe dark energy, the assumption is 'gravity acts as a kind of negative energy' (prevalent and annoying use of vague language), exactly cancelling it all out to zero. So this ADFW (or similar acronym) universe is clearly non-static, but is that the 'magical ingredient' that allows a negative sign for gravitational energy in this case?

 

[Jonathan Scott]

This makes sense - the net field energy is positive and precisely half the net decrease in matter potential energy. Hence the system binding energy is equal and opposite to the total field energy - correct?

Yes, that's right.

The fact that this model works doesn't necessarily mean that the energy is "really" in the field in that way.

If anyone knows what the Landau-Lifgarbagez pseudotensor gives as the effective energy density in this simple case, I'd be interested to know. I don't have any symbolic maths software. I tried using pencil and paper on a couple of occasions but was confused by the conventions in the sources, and the answer typically came out 8 or 16 times more than I had been hoping. I know more about tensors now, but I have less patience!