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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!

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  • $\begingroup$ Personal question: Did you get the idea for this question from watching Interstellar? $\endgroup$ – JDSweetBeat Apr 6 '15 at 14:14
  • $\begingroup$ @DustinJackson - Dude, spoilers ;) $\endgroup$ – AndyD273 Apr 6 '15 at 14:21
  • $\begingroup$ @AndyD273- Dude, I gots ta know! ;) $\endgroup$ – JDSweetBeat Apr 6 '15 at 14:24
  • $\begingroup$ Any answer would HAVE to be backed by equations. This might be a good question for the (tag:hard-science) tag $\endgroup$ – JDSweetBeat Apr 6 '15 at 14:25
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    $\begingroup$ You would not use more fuel as you get accelerated by gravity as you approach, that speed then lefts you back away again after. $\endgroup$ – Tim B Apr 6 '15 at 14:33
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Light Deflection near a Black Hole

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.

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  • $\begingroup$ He's talking about a turn around the black hole, not an orbit about one. Your image even shows exactly the problem I brought up in my reply. Furthermore, the orbital period is related only to the semi-major axis, going closer has no effect on the orbital period in a Newtonian world. (Go deep enough and I'm not sure what Einsteinian effects might be.) $\endgroup$ – Loren Pechtel Apr 8 '15 at 20:08
  • $\begingroup$ But what about burn at periapsis? For example, hard-line ship is taking turn closer black hole, and his, trajectory is one of red lines. He will make prograde burn, so he can speed up and "lift" his vector (and go even faster!). If weak-line ship is taking one of black lines then he can't really speed up, if he wants stay on course. And it applies for every celestial body. Also, orbital period isn't related only to semi-major axis. It's time needed to make a full cycle, in this case distance around divided by speed. $\endgroup$ – not7CD Apr 8 '15 at 23:18
  • $\begingroup$ The Oberth effect isn't going to add much here unless you have some really powerful engines--the important point is how close you go controls the deflection angle and that angle must be right. As for the orbital period: en.wikipedia.org/wiki/… $\endgroup$ – Loren Pechtel Apr 9 '15 at 2:10
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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.

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    $\begingroup$ Wouldn't it be interesting to see the character hit the perfect deflection angle by luck or skill? $\endgroup$ – Desolationgame Apr 6 '15 at 20:12
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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.

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