# If gravitational field has negative energy density, how does gravitational radiation carry positive energy?

The following question uses the analogy between EM (electromagnetism) and GM (gravitomagnetism).

In order to force two like electric charges nearby, some work has to be done. This implies that the following integral for the total energy increases: $$\int \frac{\epsilon_0 |E|^2}{2} dV \propto \int |E|^2 dV$$

Now, as EM's and GM's laws are analogous, the following similar integral must also increase in GM ($\vec{E}$ is just multiplied by some constant to obtain $\vec{g}$): $$\int |g|^2 dV$$

The problem is, however, that two like mass-charges attract. Thus total energy must decrease: $$\int K|g|^2 dV \propto \int |g|^2 dV$$ implying $K < 0$ - all gravitational energy is negative. I strongly suspect that the constant for gravitomagnetic field will also be negative.

Question: How can a system emit positive energy gravitational waves if the energy density of these waves should be negative-definite? I suspect the problem might have something to do with the inconsistency of GM, as I have read that like charges can't attract each other when the mediator is a spin-1 particle (gives rise to first-order tensor (vector) fields).

It should be noted that I am aware that the energy density of the curvature of metric is not that well-defined. However, I can still define some energy density in GM, even though it indeed might not be "real" energy in the general-relativistic sense. I can just treat the energy as a convenient mathematical tool without any physical interpretation, however due to analogies, theorems (such as energy conservation) still apply (so our radiating system should still gain energy).

• The exact same argument applies to EM waves, yet they carry energy. If a system of masses emits a gravitational wave that just means the potential energy of the system must decrease by an amount equal to the energy the wave has carried away. Just like EM in fact. – John Rennie Feb 13 '15 at 6:50
• @JohnRennie: The difference (well, if my logic was correct) is that GM fields have negative-definite energy density, while EM fields have positive-definite. So in GM, it seems to me that a system should instead gain energy by emitting negative energy waves (at least it should be totally impossible to lose energy). – kristjan Feb 13 '15 at 6:59
• I don't understand your claim that that GM fields have negative-definite energy. Do you mean the gravitational field self energy? If so that is not negative. – John Rennie Feb 13 '15 at 7:03
• @JohnRennie: Yes, I mean self-energy. So it is still positive? But then this implies the gravitostatic energy density should be something like $Kg^2$, where $K > 0$, just like in EM. But then how can two positive masses still attract (due to analogy with EM, they should repel)? – kristjan Feb 13 '15 at 7:12
• Ah, OK, on rereading your question I think I see what's puzzling you. You're puzzled by like charges attracting in GR while they repel in EM. That's the difference between a spin 1 and spin 2 field. There is some related info here and here. – John Rennie Feb 13 '15 at 7:48