My impression is that the consensus view on PF recommends the same paper I do, which is "Measuring the active gravitational mass of a moving object", by Olson and Guarino, 1985. Which you can find a lot of threads on if you look. You can definitely find the abstract online which is good, the full paper is even better, you will probably able to find the full paper with google scholar (however, the site one finds this way doesn't inspire a whole lot of confidence to the wary reader.)I know this topic has come up several times throughout the years, but after reviewing a good number of the threads, I’m still not clear as to whether any kind of consensus on the matter has been reached or not, or if there may be new developments in either theory or observation. If you review the threads below, you will see a divergence of views on the matter. Generally, it’s either 1) yes, a relativistically moving body in frame B will be perceived by a lab frame A as having a greater gravitational pull related to the increase in its “relativistic mass,” 2) there is no effect at all, the relativistically moving frame B has the same gravitational pull as it would in the lab frame of A, and 3) it’s too complicated to figure out because only one of the terms in the stress energy tensor relates to “relativistic mass,” the equations are too hard to figure out, and there’s no observational evidence to compare anything against:
This paper sidesteps both the thorny issue of "forces", and "curved space-time" and instead focuses on something that's both easy to measure, easy to communicate without specialized knowledge, and relevant to the problem. This is the velocity change induced in a field of test particles "at rest" after a relativistic flyby. While you'll need GR (of course) to compute this, the good news is that underlying space-time both "before" and "after" the flyby can be considered to be flat, which simplifies the interpretation of the results enormously at the relatively small cost of only providing a rough measure of "average force".
I'll quote the conclusion right from the abstract:
Now the term "active gravitational mass" sounds as if it has some profound significance, but I'm not aware of it actually being used much. In spite of a rather grand-sounding name, it's really very specific to the above method of measurement. The main advantages of this scheme of measurement are pedagogical advantages in talking to a non-expert audience - it's easy to compute, measure, and describe, so that expert knowledge is not needed to interpret or communicate the results. The full paper makes no claim for any profound significance for this defintion of "mass" and additionally introduces the names of some of the sorts of masses that are actually used in the literature by experts. These expert defiitions, however, tend to be hard to communicate to a non-expert audience.If a heavy object with rest mass M moves past you with a velocity comparable to the speed of light, you will be attracted gravitationally towards its path as though it had an increased mass. If the relativistic increase in active gravitational mass is measured by the transverse (and longitudinal) velocities which such a moving mass induces in test particles initially at rest near its path, then we find, with this definition, that Mrel=gamma(1+beta2)M. Therefore, in the ultrarelativistic limit, the active gravitational mass of a moving body, measured in this way, is not gamma M but is approximately 2 gammaM.
The main conclusion one can easily draw from this paper is that the multiplicative factor of two clearly shoots down any of the ideas that "relativistic mass", or "invariant mass" causes gravity according to usual Newtonian formula. This idea is just plain wrong. So if you have this idea, please drop it.
It's perhaps unfortunate that fully understanding the actual source of gravity, the stresss-energy tensor, requires one to understand tensors, but I'm not aware of any way to fully appreciate the nature of the stress-energy tensor without understanding tensors first. One can get a rather vague idea of what the tensor expressions mean by saying that "energy, momentum, and pressure all cause gravity" without giving more detail.
Assuming you mean proper radius and rest mass, it's quite easy to say that the velocity of Bearth realtive to Earth makes no difference for someone living on the planet. The long thought experiment section on applying the Lorentz transform is both somewhat misguided and unneeded. All you really need is the fact that there is no way to measure one's absolute velocity to realize that the experience of Earth and Bearth will be the same, except for effects caused by lunar and solar tides. That's the uneeded part, the misguided part follows from trying to apply Lorentz transforms in non-flat space-time.So, in lieu of observational evidence, most posters in these threads come up with thought experiments to try to gain insight on the manner. I’m going to do the same here with a fresh thought experiment I didn’t see in the other threads and see if anyone can, say, pick it apart some, here goes:
Earth has a twin planet named “Bearth,” with exactly the same radius and mass in kgs. One day Bearth comes zooming through the solar system and passes Earth at 0.87c (relative to an x coordinate which lines up parallel to the equators of both planets), giving Bearth a relativistic gamma factor of 2 relative to Earth in the x direction. On the planet Bearth there are two twin sisters, Alice and Beth. Both of these girls are 6 feet tall (being twins, of course).
If you want to know the gravitational effects of the passage of Earth through the solar system on some test particles at rest relative to the sun and compare it to the effects of the passage of Bearth, you can use the Olson-Guarino paper to do that (as long as you can ignore solar gravity, that is, so you can disregard the curvature of space-time due to solar graity), and find the gravitational effects due to Bearth's passing on the test particle's velocity field are about 3.5x as big as the effects from Earth's passing.
If you are interested in coordinates that someone might use on Bearth, I would point out that near the planet, one logical choice would be Fermi-Normal coordinates. Actually computing them and extending them through the rest of the solar system though is difficult, I'm not aware of any published paper that does this for the exterior Schwarzschild metric (there are some that do it for the interior metric but this isn't what's needed). I'm pretty sure from my own calculations that you'll wind up with a series solution rather than a closed form one. But there isn't any huge conceptual difficulty (except for the difficulty of describing Fermi normal coordinates to a non-expert audience, and also explaining that these Fermi-normal coordinates won't cover all of space-time), The rest of difficulties are just a matter of calculational difficulty and the lack of ways to check one's work.
[add]Another logical choice might be to use harmonic coordinates, these are probably going to be much more calculationaly friendly than Fermi-Normal coordinates. I'm not super familiar with them, my understanding is you impose the de Donder gauge condition , and that when your goal is to get an approximate metric to some post-newtonian order (which is how we handle GR in practice in our solar system) this makes your life a whole lot easier.