Racing Around the Edge of a Black Hole

Question

I am trying to understand the physics of an interstellar race.

Say two ships are racing past a black hole as if it were a corner on a race track. One of the ships takes a hard line and comes very close to the black hole. The other ship takes a weak line, and does not get very close to the black hole.

The Ship that took a hard line will travel less distance, and likely use more fuel to escape the black hole's gravity (although I'm not certain of this).

Due to time-dilation, the hard-line ship will also age less. That is, due to the strong gravity near the black hole, time will go slower on the hard-line ship.

My question is: If the two ships converge after they pass the black hole (intersect at a point) which of these ships will actually arrive at this point first?

The hard-line ship will certainly be younger when it gets there, but maybe the weak-line ship will get there first?

Additionally: Who will use more fuel? While the hardline ship will have to deal with escaping the strong field of gravity, the weak-line ship will be aging much more and will be burning fuel for longer...

Thanks!

Answer 1

At speed of light, stable orbit is at event horizon. We are traveling at smaller speeds, aren't we?

Space Flight: The Application of Orbital Mechanics from 1994 First 5 minutes. Amazing music.

What I will explain here, will need knowledge of all Kepler laws (first 5 minutes). As you saw, III Kepler says that $\frac {(orbit\;radius)^3} {(orbit\;period)^2}$ is always const. That means, if we are orbiting closer, our way around is smaller and we are traveling faster.

So, if our start/finish point is always at the same fixed position, and our ships start with same: speed and height from start point, they will orbit a body and no one will win. But if one ship will slow down at start by few m/s he will start losing attitude. As you know, Apoapsis (Apogee for every celestial body) is a point where satellite has slowest speed. So our greatest speed will be at Periapsis. So we have eccentric orbit now. At one point we are lower and we are moving quickly, and at one we are high and moving slow. That was II Kepler law. But back to the III law, lower orbit means shorter period, higher orbit means longer period. But what if we have something between lower, and higher orbit? Something slightly smaller than high orbit period. So hard-line ship will win, if he won't be slingshoted. He will use a little fuel, compared to no fuel usage of other ship.

And back to that black hole. We can achieve stable orbit around it. But near it is area where we can stay on orbit only using engines, and then event horizon, yay. But I'm not good at relativistic physics, but we can pretend that the hard-line ship will cross finish line a little quicker, if his trajectory was slightly changed. From finish line perspective weak-line ship will be affected by the same amount of the time as observer at the finish, but hard-line ship will be affected by less of the time. But since his distance to go was smaller and velocity was greater, there is no way, that weak-line ship can win.

Answer 2

Remove this from your plot, the pilots have no choice.

First, the fuel used has no bearing on how close you come. How close you come is simply a matter of where you aim your ship. While you are far from the black hole this costs little fuel--probably no more than any other course corrections you are doing.

Second, and the reason you need to drop this, is that how close you come to the black hole has a big effect on how much you turn as you pass the black hole. For any given deflection angle there will be only one possible approach distance. A pilot who goes too deep (or not deep enough) ends up badly off course.

Update: Protector by Larry Niven has a relevant scene. Brennan is being chased by four Pak ships. He does a close flyby of a neutron star, destroying two of the ships chasing him by means of rifle fire causing flares on the neutron star. The other two have to evade, thus causing them to fail to make the turn and ended up 6 months behind in the chase.

Answer 3

The person in the nearer spaceship would corner a lot faster, but not for the reason you think. Angles aren't Euclidean close to a black hole. That's why light bends around them. Given the proper trajectory, the racer could travel in a straight line, and wind up at his starting point. That's probably what they'd be looking for.

To understand the relativistic effects of this, I recommend reading this description of how that works to change Mercury's orbit. That actually provides you with everything you need to understand the temporal effects.

To summarize, let's say you're going around the black hole counterclockwise. From the perspective of the start/finish, the black hole would be pulling you to the left as you went around it. There would be an optimal line where zero thrust would bring you right back to the finish line. If you entered slightly to the left of that trajectory (nearer the black hole), then you would spend the entire pass thrusting to the right so that you would be able to maintain a trajectory that would bring you back to the finish line.

Yes, the path would be shorter. Yes, time dilation would make it take less time from the flier's perspective, but it wouldn't actually make you fly through the geometry any faster.

The big change would be that you'd be physically moving faster through the arc to maintain that trajectory. The amount of "faster" would be dependent upon the amount of thrust you could manage to keep yourself on course.

There would exist a second course where you'd actually loop fully around the black hole before leaving it's influence, but I couldn't tell you if it would actually be faster.

Answer 4

The ship traveling the hard line course would be orbiting deeper in the gravity well. More than likely, any energy saved on approach of the black hole would be returned later. The hard line ship would not only travel a shorter distance but would also accelerate to a greater speed while deeper in the gravity well.

Not only would the ship traveling the hard line course physically reach the destination first, the passengers on board would have aged significantly less. They would think they won the race with years to spare.