Ibix said:
I think pervect's description of the paper is that they fire an array of test particles at a Schwarzschild metric with mass ##M## with "speed at infinity" ##\beta## and look at the angles between the asymptotes of the orbit.
But if we are going to translate that to this scenario, we would have to
combine this solution with the metric of the solar system. And the question is how much doing that would
change the final metric from the one we already have for the solar system. Our intuitive answer in this thread has been "not very much". But if we take the formula in Olson & Guarino's abstract at face value, as I said before, that would not be correct.
The problem I have with that face value answer, however, is that if we switch to the shuttle's rest frame, it is obvious, as you pointed out, that the shuttle's effect on the overall spacetime geometry is miniscule. And that can't change just because we changed frames--this is just another version of the answer (given in a number of previous PF threads) to the common question of why an object doesn't turn into a black hole if it goes fast enough.
Furthermore, again if we look at things in the shuttle's rest frame, what effect would we expect on the shuttle due to the solar system? Say due to the Sun, to make things simpler. Olson & Guarino's formula, taken at face value, says we should expect the Sun's effect to be ##\gamma M_S \left( 1 + \beta^2 \right)##--in other words, a
huge effect compared to the Sun's effect when at rest. At the gamma factor we are talking about, the result of that formula is something like ##10^{20}## solar masses, i.e., comparable to the total mass in our observable universe. Is that really the correct answer? It can't be that simple, because all that mass concentrated into a volume like that of the Sun would indeed be a black hole, but we already know
that is not the correct answer.
As I say, I haven't looked at the detailed math, either in the Olson & Guarino paper or on my own, so all this is just intuitive. But it seems to me that there must be a disconnect somewhere since what the Olson & Guarino paper appears to say is so different from our intuitive answer in this thread.
One possible resolution of the disconnect might come from asking, where did all the energy
come from to boost the shuttle to such a huge gamma factor? Suppose, for example, that we detonated a 2 solar mass star in such a way that it boosted two shuttles, each to the same gamma factor, in opposite directions (so total momentum in the original star's rest frame remains zero). We would have expected the original star to have a significant effect on the spacetime geometry around it, so it seems like we should also expect the two shuttles thrown off in the explosion to have a significant effect on the spacetime geometry surrounding
them.
Then the real issue might be the specific
form of the effect. The fact that the shuttles are each moving at a tiny smidgen less than the speed of light, in the original star's rest frame (which means, to a good approximation, in the rest frame of any other star systems they pass through) means that they are only within a short enough distance to have a significant effect for a short time--short enough that the overall effect remains small, as our intuitive answer in this thread has it. (That is one way of heuristically describing the effect of the momentum components of the stress-energy tensor that I talked about earlier.) In other words, instead of being a nice spherically symmetric Schwarzschild field as the original star's field was, we now have two narrow "world tubes" filled with stress-energy whose effects will be very different from those of the original star.