# Is the Hilbert-Felber model of repulsive gravity correct?

This theory states that at relative velocities exceeding 3^-0.5 c, gravitational repulsion ensues. The relevant papers are on arXiv by Franklin Felber. Here's one of them http://arxiv.org/abs/0910.1084

I have seen two interesting critiques thus far. The first is that the effect is an artifact of the choice of coordinates: http://arxiv.org/abs/1102.2870 and rebutted by Felber here http://arxiv.org/abs/1111.6564 This answer also seems germane http://physics.stackexchange.com/a/83175/11633

The second critique involves the feasibility of the proposed LHC experiment: http://arxiv.org/abs/0912.1323v1

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This is more of an extended comment than an answer, since I haven't had time to look up all the papers referenced in Felber's article. Hopefully this discussion will be a useful clarification if nothing else.

It should be well known to any GR enthusiast (whether a formal student or not) that as viewed by a distant observer it takes an infinite time for any infalling object to reach the event horizon. This has been dicussed to death in many questions on this site, including the question you linked. So if you drop an object into the black hole the radial velocity will start at zero, increase as the object accelerates towards the black hole, then decrease again as the object nears the event horizon. If the radial velocity is decreasing then that means the object is decelerating, and therefore that it is being repelled by the black hole.

If you take an object that is already moving at the speed of light, i.e. a light beam, then it can't accelerate because it's already moving at $c$. So if you shine a light ray at a black hole you will measure its velocity to decrease smoothly as the radial distance decreases. As it happens I've just discussed this in my answer to Speed of light originating from a star with gravitational pull close to black-hole strength?. This means the light is being repelled by the black hole.

The result obtained by Hilbert is that below an initial speed of $c/\sqrt{3}$ an object's radial speed first increases then decreases, while above an initial speed of $c/\sqrt{3}$ the radial velocity decreases smoothly. Given the points I've made above it shouldn't be surprising that somewhere between an initial speed of zero and an initial speed of $c$ the behaviour switches from attractive then repulsive to purely repulsive - that switchover velocity happens to be $c/\sqrt{3}$.

How real you consider the decrease in velocity, and therefore the repulsion, depends on your point of view. It is certainly not some mathematical trick - the time dilation that is involved in it is quite real and if you leave for a black hole, spend some time near it then return, you'd find you friends had aged more than you. However I would be reluctant to describe this as a repulsion between the infalling object and the black hole.

Suppose you watch a rocket accelerating away from you. As the rocket nears the speed of light its acceleration will decrease asymptotically to zero. Does this mean there is some repulsive force opposing the rocket's motor? I think most of us would agree that there is no such force and the reduction in acceleration measured in our inertial frame is a consequence of the time dilation experienced by the rocket. The effect described above is analogous to this.

What I haven't grasped at a quick read through of Felber's paper is how this can be used as a method of propulsion. The radial infall and outfall trajectories are symmetric, so it isn't obvious how the initial and final velocities of the object can differ after the speeding black hole has passed by. But I must emphasise that I am not doubting the papers - I'm just saying I haven't had a chance to read them thoroughly yet.

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Yes John; no quibbles with any of that. But where this gets interesting is the weak field case where a massive object is approaching from afar at above the critical speed relative to a test mass. The test mass will experience repulsion during the entire approach, the strength of which goes (to 1st order) with the usual inverse square dependence and thus increases over time. The boost goes something like (1 - 3*b^2)*(1 - b^2)^-3/2, where b=v/c. Cf. eqn 8 of the LHC paper (simplified for p=>0). – Andrew Palfreyman Apr 2 '14 at 19:07
@AndrewPalfreyman: yes, but after the black hole has passed, why isn't the effect reversed leaving the object displaced but stationary? – John Rennie Apr 2 '14 at 19:26
Aaah... I wonder if, in the rest frame of the distant observer, the object never passes the black hole but gets continually pushed along in front of it and therefore accelerated to the same speed. If so I'm not sure how you'd get off the train. – John Rennie Apr 2 '14 at 19:30
Complex answer I believe, and I don't yet have a complete picture as I crawl through the LHC paper. Some notes however: a) The test mass is able to attain a velocity greater than the incoming ("driver") mass, before the driver has arrived; b) Were the driver mass to pass the test mass, there is a repulsive field behind the driver, but this is hugely weaker than the forward repulsive force was; c) the approximation p=0 fails at close approach and we are in the strong field regime, which behaves differently. But if a) be true, then b) & c) don't apply, and we get useful propulsion. – Andrew Palfreyman Apr 2 '14 at 19:32
And note that even if the driver does pass by (in scenarios with finite impact parameter), because of b) we still get net propulsion. – Andrew Palfreyman Apr 2 '14 at 20:05

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.

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