Let's imagine a asteroid that travels with 0.99999999999999999c.
(I know it's impossible).
Anyway... Relativistic mass of such object would be almost equal to earth's stationary mass.
Now let's imagine that such object passes closely to me (assuming I'm immortal etc.). According to $F_g=GMm/r^2$ (Where M is relativistic mass of this asteroid and m is my mass) such an object would attract me.
So would I gravitate after it? Would I gain it's speed constantly falling on it? Or maybe it would just pulled me for a moment, but then I would return to my ol' good non-relativistic speed in "everyday reference system"?
Or maybe something else would happen? What happens with gravity of relativistic, NON ACCELERATINNG objects? (Let's assume that there's something that makes this asteroid speed constant).