Can relativistic mass actually change gravitational pull? I've heard that relativistic mass can influence gravity, but this seems to create a paradox, unless I am missing something.
It seems to me that if there were two celestial bodies that are observed to be moving along approximately parallel trajectories at a relativistic speed, wouldn't the gravitational force between them also be larger than if the bodies were at rest, and therefore draw them closer together than they otherwise would be?
How would this attractive force be accounted for if one were observing the second celestial body from the reference frame of the first, where one would not otherwise see any significant relative motion to the other body and not have the apparent relativistic velocity to attribute the increased gravitational pull to?
What am I missing here?
 A: The gravitational effect arising from a particular volume of space is proportional to (among other things) the energy density in that volume of space.
In the case of a spinning flywheel the rotational kinetic energy and the stress of having to provide centripetal force is confined to a finite volume of space, hence there is an energy density.
In the case of linear motion the motion is not confined and so there is nothing to give rise to a density.
More generally, kinetic energy is inherently relative. There is no such thing as attributing kinetic energy to a single object. The minimum is two objects. For any pair of objects the relative velocity between those two objects determines the amount of energy that is available for transformation (to other forms of energy) when those two objects collide.
A: The answer is that "relativistic mass" does not produce gravity in the Newtonian sense. Newtonian gravity breaks down in the realm where velocities are relativistic, and you have to use general relativity to determine the answer. In GR gravity (i.e. curvature of spacetime) is produced by the stress-energy tensor, which is independent of coordinates. So if there's a frame of reference in which the two bodies are at rest relative to one another, then there is no additional gravitational attraction between them. This would be the case if they are moving on parallel paths in the same direction. If they were moving in opposite directions then there is no frame in which the relative kinetic energies disappear, and so that would produce an effect. But to determine the effect you would have to solve the Einstein field equations, I don't think the Newtonian approximation would work.
A: I believe what matters is the relative curvature of spacetime between them. If they are both moving at relativistic speed in the same direction relative to some third observer, according to that observer the additional curvature of spacetime that this causes is the same for both bodies. Since it doesn't change the relative curvature between them, it shouldn't alter the gravitational pull the experience relative to each other.
I'm sorry if that's a tad hand-wavy.
A: There are two effects you need to consider.
One is that gravity depends on the full energy-momentum-stress tensor. Usually for slow moving objects the biggest component of this is the energy, all the others are negligible, and so we commonly say gravity depends on the mass (=energy) alone. But energy and momentum are just individidual coordinates of an invariant 4-dimensional quantity, and if you change to a moving reference frame it's just like rotating your coordinate system (only it is a hyperbolic rotation, it preserves the length $x^2-y^2$ instead of $x^2+y^2$, meaning that as one coordinate increases so does the other). The invariant magnitude of the 4-dimensional quantity is the energy squared minus the momentum squared. If you change to a moving coordinate system, momentum increases along with energy in synchrony, leaving the magnitude the same.
This is the same answer as to the question: when you accelerate and see the whole universe start moving past you, where does the kinetic energy of the whole universe come from? The answer is that energy and momentum are complementary coordinates of a 4-dimensional quantity, which is what is conserved, and the increase in energy is cancelled by the increase in momentum. Or more accurately, you are just rotating your coordinate system, and some of the enormous rest energy of the whole universe is appearing in the spatial direction as momentum. A lot of apparent paradoxes can be resolved because of this sort of compensatory-coordinates effect.
The other effect you need to consider is time dilation. The convergence of your moving celestial bodies can be considered to be a clock - they get so many metres closer in such and such a time, they orbit around each other with such and such a period - and relativistic velocity slows this down. The acceleration you measure in the moving frame will be different.
A: In the example of the OP the force is observed from two different reference frames and just differs by a Lorentz transform. To answer the question in the title, a different thought experiment is needed. Assume one particle to be at rest and the other moving at speed v. Does the gravitational force depend on v? The answer is yes. Gravitation acts on energy, including kinetic energy. For example, the mass of a hydrogen atom consists of proton and electron mass plus binding energy, hence kinetic and potential energy. It is this total energy that feels gravitation, even in Newton gravity.
A: Well I do not understand the other answers, but as the correct answer is simple, the other answers must be wrong.
An observer at rest relative to a force observes that the force is F.
Then the observer changes her frame, so that the force moves at speed v.
Now the force is F/gamma, according to the observer.
This is the correct way to calculate the force in this case, if the masses move side by side.
If the masses move along the same line, then the motion has no effect on the force.
Now I don't want to hear: "Can't use special relativity, must use general relativity". General relativity has nothing to do with this.
Like quantum mechanics has nothing to do with the force between moving electrons.
