Shadowprophet
Truthiness
The electromagnetic stress-energy tensor is a component of the Einstein field equations, so yes we already have a metric tensor equation for the magnitude of the gravitational field associated with electromagnetic energy. An interesting feature of the stress-energy-momentum tensor is that positive pressure generates ordinary gravity, whereas negative pressure (aka tension) contributes negatively to the tensor - tension is in effect an antigravity term (regardless of the nature of that tension).
They're not really forces (they're accelerations, which is a bit different because they're not dependent on mass, as force is). A magnetic field has positive energy so it creates a gravitational field (a very small field effect anyway). Interestingly, binding energy is a negative term - so for example, the sum of the gravitational fields for a magnet and an iron bar which are widely separated in spacetime, is actually higher than the gravitational field of the same magnet and iron bar when they're stuck together via the magnetic field. However, it appears that it's impossible to get an antigravitational field this way, because the magnetic field has positive energy, and even if you coupled the magnetic field in a bound state, you could only at best cancel out that field energy, leaving the positive rest mass of the material, which of course creates a positive gravitational field.
The prevailing theory of dark energy (as vacuum energy) gets around this because the energy of the vacuum doesn't have any rest mass, and it's uniform throughout spacetime, and with both of those characteristics an antigravitational field results.
It appears that no body of matter with a positive rest mass can undergo any combination of interactions that could reverse the polarity of its overall stress-energy tensor. I've heard this from experts in general relativity, but I haven't seen the proof myself yet. Next time I talk to one I'll ask them for the proof behind this. In the meantime I trust that they're correct, but I'd like to see the basis for that conclusion.
That's a very complicated question, it turns out. I've been making my way through that paper, and at this point it looks like they're talking about a model of quantum gravity that has a cut-off in the IR range that somehow results in a long-range antigravity field that isn't very noticeable within the galaxy clusters, but which overtakes the positive gravitational field in the long-range regime, resulting in a repulsion between the galaxy clusters (I really like this idea, because it assigns the field to material bodies, instead of the current model which assigns the field to empty space, which seems absurd to me). Their work is based on the concept of asymptotically safe gravity, which is a new concept to me, so I'll have to study that in more detail before I can figure out wtf they're talking about.
Oddly, no, the gravitational field isn't relative even though it's based on the special theory of relativity (SR). In SR, all motion is relative - when two observers in relative inertial motion pass by one another, they both see the clock of the other body ticking more slowly, which their clock seem to tick at the usual constant rate.
The gravitational field is different. An astronaut far from the Earth in a low-gravity region of space would see your clock here on the Earth ticking more slowly. And an observer on the Earth would see the astronaut's clock tricking more quickly. They both agree that the clock on the Earth is ticking more slowly, so the effect isn't relative.
If gravity isn't reletave, Does that rule out Quantum Gravity as well?