Yes, Hilbert’s calculation of gravitational repulsion, which he published in a journal he co-edited with Einstein, is correct.
Moreover, here is a simple way anyone can be convinced, without solving any equations at all, that a spacecraft can be accelerated from rest to relativistic speeds by a much heavier mass moving at relativistic speeds.
Suppose we are watching through our telescope a distant heavy mass moving at relativistic speed along the x axis in the +x direction and approaching a spacecraft initially parked at rest a distance b off the x axis.
Now, imagine that some other distant inertial observer is in the rest frame of the heavy mass. Through his telescope, he sees a spacecraft approaching the stationary heavy mass at relativistic speed and with impact parameter b. He is not surprised to see the spacecraft follow an unbounded orbit described by the well-known equation of motion of a particle in a Schwarzschild field (or in a Newtonian field if the field is weak).
Without solving the equation of motion, we already know two things about what this other observer sees: (1) The spacecraft will be deflected from its original path into some new direction; and (2) the spacecraft will have about the same speed after its interaction with the heavy mass as it had before, because the gravitational field has a conservative potential.
But the spacecraft motion that this other observer sees is related to the motion we see by a simple Lorentz transformation in the x direction, since we are each distant, unaccelerated observers. And since the other observer sees the spacecraft lose some velocity in the –x direction, we must see the spacecraft gain some velocity in the +x direction, as well as some velocity in the perpendicular direction, after its interaction with the heavy mass.
In a strong Schwarzschild field, 180-degree U-turn orbits are possible. Again, without solving equations, one can see that in such a case a spacecraft can be accelerated from rest to a speed greater than that of an approaching relativistic heavy mass.
Andrew, I enjoyed meeting you at the STAIF conference about eight years ago, and I wish you much success with the work you and Jim Woodward were doing on inertial propulsion.