According to the special relativity, when the speed of an object increases, its mass also increases. Does it mean that objects moving at speeds close to the speed of light exert higher gravitational force to the objects nearby? If yes, what would be the gravitational formula?
Can we inspect this situation in the boundaries of special relativity or do we have to switch to general relativity?
Note: I have checked suggested questions but did not come up with a similar one.
Does it mean that objects moving at speeds close to the speed of light exert higher gravitational force to the objects nearby? If yes, what would be the gravitational formula?
Special relativity cannot be used for analyzing situations where tidal gravity is substantial. General relativity must be used instead.
In General Relativity the gravitation of an object is not a function of mass, but rather it depends on the stress energy tensor. The stress energy tensor involves the energy, momentum, pressure, and stress.
In the Newtonian limit all of the other terms are negligible except for the energy. But as you consider a reference frame where the object is moving at relativistic velocities then the momentum terms are also not negligible. So the gravitation of such an object cannot simply be described as solely due to the energy. In rough terms the momentum effects roughly cancel out the energy terms such that things like orbits are consistent in all frames.
