system
Given the following post, respond as if you're commenting on Reddit:
human
What are the limits of gravitational slingshot acceleration?

If I have a spaceship with no humans aboard, is there a theoretical maximum speed that I could eventually get to by slingshotting around one star to the next?  Does slingshotting "stop working" when you get to a certain speed?  Or could one theoretically get to a reasonable fraction of the speed of light?
gpt
TL;DR: The faster you move, the closer you need to get to the celestial body you want to slingshot around. At some point, you burn up in the atmosphere, crash into the surface or get ripped apart by gravitational force differences.

When you do a gravitational slingshot, you're essentially "bouncing" on the planet, doing a 180 degree turn around the celestial body. From the celestial body's point of view, you approach it at the same speed and once the slingshot is complete you leave with the same speed. In other words, we can simply see it as a bounce with a restition coefficient of 1 (no energy lost) on the celestial body.

The key to a successful gravitational slingshot is to have the celestial body approach towards you. Let's say you have a planet hurtling towards you at 10km/sec, while you fly towards it at 2km/sec. From the planet's perspective, you are approaching the planet at 10+2=12km/sec, you'll loop around the planet and then go back in the direction you came from at 12km/sec. However, from our perspective, we approach the planet at 2km/sec, get flung around it and then fly away in the same direction as the planet at 22km/sec (very confused about the exact speed).

In essence, you're stealing some of the kinetic energy of the celestial body you slingshot around, and the effectiveness of this is solely dependent on how fast the celestial body is moving, so there's no theoretical maximum speed apart from the speed of light (which you can always keep getting closer and closer to as your kinetic energy increases).

However, there are practical problems that will either reduce the efficiency and practicality of a slingshot, or even make it downright impossible. The faster you go, the stronger gravity needs to be to be able to sling you around the celestial body. The only way to increase the force of gravity from the body is to get closer to it. This means that you get quite a few problems. If you're trying to sling around a planet or moon, you could start experiencing drag from the atmosphere, which would not only slow you down a lot but also potentially burn you up. If the planet/moon has a solid surface, you may not even be able to get close enough to the planet without crashing into it. Similarly, getting too close to a star has some obvious drawbacks.

A black hole is therefore optimal for a slingshot operation as it is neither warm nor has any significant atmosphere nor surface. You can always get a little bit closer to the event horizon to allow you to turn around it quicker, although at some point you'll get so close to the black hole that your ship is torn apart due to the different parts of the ship experiencing so different gravitational forces (the parts closest to the hole turns inwards, while the farthest parts don't turn enough to keep up with the center of the ship).