We’re back to relativity, folks! We went over special relativity last time, and now we will explore general relativity. The main idea behind this theory is that gravity is geometry, so we often describe these geometries using certain models called metrics. This week, we will look at gravity using the more straightforward Schwarzschild metric. Hence, Karl is the name of our hypothetical climber today.
Karl has a one-track mind. Even though every single massive particle distorts spacetime, Karl only notices Earth’s gravity. You can picture Earth’s distortion of spacetime as a bowling ball’s distortion of a trampoline. Without the bowling ball, if you put a marble on the trampoline, the marble will just sit there where you placed it. However, with the bowling ball, the marble will fall to the surface of the bowling ball.
Karl’s project, Kruskal Scares, has a crux at the start and at the finish. If Karl falls at either crux, he’ll fall towards the center of the earth just like our marble did toward the center of the bowling ball. If you remember from classical mechanics, Karl will feel a greater gravitational force at the first crux than the second. You can imagine this as the bigger distortion of the trampoline closer to the bowling ball than farther from it. Okay, so this is consistent with classical mechanics…so what? Well, this is where it gets interesting. Remember how motion distorts spacetime and Albert’s clock ticks slower than his belayer’s while he (Albert) is moving? There is a similar effect from gravity’s distortion of spacetime.
Karl really wants to send, so he shakes out and rests before each crux. Since he’s right at the start with his belayer, Karl experiences time exactly as his belayer does at the first rest. At the second rest, at the top of Kruskal Scares, Karl’s clock ticks faster than his belayer’s. To Karl, it looks like his belayer is moving in slow motion (well, perhaps not with his human eye, but you get the idea). His rope will also look stretched and thinner at the bottom since the gravitational force at the bottom of the rope is greater than the force at his knot (this is called spaghettification).
More on this later…we will see how things get extra spooky if we put our climbers near a black hole. Kruskal Scares may not actually be the scariest route out there…