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Imagine a two dimensional world where there are only two electrons. They are set right beside each other. Of course, immediately they will start to separate, being repelled. My question is, as they accelerate, they have more energy and thus a larger gravitational field, so is there a point when they will come back together again?

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    $\begingroup$ Have you limited us to two dimensions just for simplicity? Or because the gravitational force will scale inversely, rather than as the inverse-square, in two dimensions? $\endgroup$ – lemon Apr 7 '15 at 12:09
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Relativistic mass is a weird concept that creates a lot of problems. I describe what mass really is in this post of mine.

With that in mind, the mass is constant, even when a particle is accelerating. Its on this invariant mass that gravity acts on really, so in your example gravity will become weaker with separation, because the mass in reality stays constant.

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    $\begingroup$ what about the energy gained when the electrons accelerate $\endgroup$ – Jimmy360 Apr 7 '15 at 12:23
  • $\begingroup$ It assume it will negligibly increase the electrons gravitational field but not enough to be comparable with EM. Gravity is almost $10^{-36}$ times weaker than EM. $\endgroup$ – Constandinos Damalas Apr 7 '15 at 12:33
  • $\begingroup$ so the energy gained never approaches the force of electromagnetic repulsion even given infinite time $\endgroup$ – Jimmy360 Apr 7 '15 at 12:34
  • $\begingroup$ Don't forget that accelerating charges also emit Bremsstrahlung radiation which is proportional to $m^{-4}$ $\endgroup$ – Constandinos Damalas Apr 7 '15 at 12:39
  • $\begingroup$ I thought that only applied to negative acceleration $\endgroup$ – Jimmy360 Apr 7 '15 at 12:41
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I'll talk about 3D space.

If you have Newtonian Gravity, then only rest mass acts as a source, and it is an instantaneous action at a distance type force. You won't get a stronger force based on them moving faster. But they will still radiate electromagnetically, so there will be an electric force pushing them apart and a gravitational force pushing them together plus possibly other effects to steal some energy from somewhere to provide the energy of the radiating field. Technically the Schott fields also contain energy even though they fall off too quickly to carry energy to infinity, so a detailed energy balance needs energy exchange with the Schott field as well as the radiation field. But now we are getting close to the answer with General Relativity.

Let's not use Newtonian Gravity and use General Relativity instead. Now every possible energy, momentum, stress, and pressure acts as a source for gravity, including the energy, momentum, stress, and pressure of the particles as well as the energy, momentum, stress, and pressure of the electromagnetic field.

A few problems come up right away. Firstly, in a flat space the energy of the electric field of a point particle is infinite. In General Relativity things don't get better. If you concentrate a particle of mass $m_e$ into a region with surface area less than $16\pi G^2m_e^2/c^4$ then it will crush itself into a singularity. And if the black hole has the charge of an electron, then there will be no event horizon shielding it, the theory falls apart. And if you try to give it some mechanical spin to avoid that, then if you give it enough spin to represent the quantum of spin angular momentum that electrons have, then you just make it worse, that"s enough spin to get rid of the event horizon even if there were no electric charge, so it didn't help.

Well, you can avoid all that if you replace the electrons with spheres big enough to not form singularities. But you need to add some force to keep it together (so it doesn't fly apart), and those new forces will also contribute gravitationally. But you can adjust the bare mass of the shell to compensate.

OK, so you have some hypothetical shells with hypothetical mass and hypothetical charge. The gravitationally attract each other and they electrically repel each other. But the electric field still extends through the space between them. And that still has energy. So they aren't just gravitationally attracted to each other, they are also gravitationally attracted to the "empty" space around them and between them.

And here's what is important. All that kinetic energy from the acceleration, where did it come from? It came from the electric field energy itself. So when the shells speed up as they move away from each other the electric fields change in a way such that they have less energy. And it is the energy of the field right where the charge speeds up that goes down. So really the energy if the fields flows towards to the charges and then changes from field energy into kinetic energy.

So in effect there was some energy that was more spread out (in the fields) and it becomes more localized into the spheres as they speed up (gain kinetic energy). So energy didn't appear because of the electric repulsion. It just transfers from the fields to the charges.

If you simplified the analysis to just say that there a bunch of energy spread out and now it is more concentrated and treat the energy like mass for gravitational purposes (not totally accurate as momentum and stress are also sources of gravity). Then effectively moved mass from around and between them to being more in the shells than before. They now will be attracted to each other more strongly. But that limit where they get far away and that level of attraction, that's what we called the mass $m_e$ of the electron. And out there we know the electric repulsion of shell 1 is stronger than the gravitational attraction of shell 1 in that frame so shell 1 really starts to to moves away in that frame. And shell2 was already moving away in that frame. Similarly in the frame of shell2. So they really do move away from each other period. So regardless of frame, the distance between them gets larger in time. So they don't halt.

Note that we did neglect radiation effects, and in General Relativity there is gravitational radiation in addition to electromagnetic radiation.

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