Thanks to all. Gravitomagnetism and the Maxwell-like approximation (which we can call "GEM" for short) is something I've found "cool" and fascinating.
Interestingly, Maxwell himself may have dabbled in this, but he never formally published it. I think one of those papers I posted on Heaviside mentions this in passing, and the reason was Maxwell had problems with the required negative energy density of the fields and couldn't resolve it to his sastification. [And this is something about GR I'm not clear on at all -- how GR "views" the energy density of the fields, positive or negative].
The negative energy density is a result of the negative sign in the force law, where like "charges" attract, rather than repel. Logically, it turns out one needs to put this negative sign on the "permittivity" (~1/G), and the corresponding "permeability" (~G/c^2). From EM, we can write the energy density of the fields as 1/2D*E and 1/2B*H, which eliminates the constants.
With gravity, the minus sign means that the corresponding flux densities, D and B, must point in the opposite direction than the corresponding field intensities (E and H). So the product D*E and B*H is negative, not positive, and you need to put a minus sign on any version of the expression using the square of either field. Since g plays the role of E, we have to put a negative sign on the g field energy density, ie Energy ~ -g^2. And likewise, the gravitomagnetic energy density would go
as ~-B_g^2. Maxwell wasn't too keen on this apparently and didn't like it.
This makes things tricky to our Maxwell EM intuition, and you've got to be careful. You'll see that Nduriri let this trip him up in trying to derive a Heaviside-like GEM -- he's got the B part backwards, making it the same sign as EM.
You can always see forces as the result of energy trying to minimize itself, "spread out". With EM, the forces always work to lower the energy density. When two opposite charges attract, you'll see the energy of the total field reduces as they get closer together. The energy stored in the field is transferred to mechanical energy (and a little bit to radiation
, which gets complex in how this fits in with energy density of the fields). When the charges touch, they cancel and there is no more external field.
When two like sign charges repel, you'll see the total energy is reduced as they move farther apart. The same goes with magnetic systems. The total B energy of two long parallel wires carrying current in the same direction is less the closer together they are. And for opposite currents, it's lower as they move apart. Same thing for two dipole current loops.
The sign change of gravity puts an interesting twist on this. Two point masses gives the same field as Coloumb's law (save for the sign), and the product g^2, is the same as E^2. So as two masses get closer together, the integral of g^2 *increases*, just like it would for two positive charges being pushed together.
And this shows why that energy has to be negative. The only way more can be less is if it's negative. The total energy of the g field does decrease, it just makes larger magnitude negative number, transferring positive mechanical energy to the masses as they fall.
And it has to be the same for the gravitomagnetic field. The two parallel "wires" carrying mass current in the same direction must repel. The magnitude of the B_g energy increases, just like it would for two electric wires being pulled apart. But that is negative, and so positive energy get transferred as the wires push each other apart.
I don't think GR has any of this negative energy business in it, and I don't know how it treats "the energy of curving space-time". In the geodesic view, when a mass falls towards another, there is no real transfer of potential to kinetic energy (field energy to mechanical energy). It's just that time and space "warp" relative to a stationary observer watching it fall, and the mass's rest energy appears to be kinetic energy. In its own frame, the rest energy remains constant. The conversion to kinetic energy is only an "illusion" of curved space-time.
But that's not quite sastifying, because that "illusion" can be transferred to other matter down in the well. Consider two masses free falling towards each other. In the free falling frame of either, nothing is happening to it, but it sees the other mass gaining speed and increasing kinetic energy as it approaches.
And, add radiation into the mix, and some of the energy can be carried away from the system by the "ripples in space-time" of the gravitational radiation, and then transferred to distant masses interacting with that radiation. In GEM, even though the energy densities were negative, a radiation field would still transfer positive energy away from the local system.
So, again, I don't know how GR actually treats the energy density.
-Richard