Q: What’s up with that “bowling ball creates a dip in a sheet” analogy of spacetime? Isn’t it gravity that makes the dip in the first place?

The original question was: … also brings up the famous Einstein analogy of a bowling ball in a mattress as bending spacetime. What confuses me is that this seems circular- using the analogy, say we put a bowling ball on a mattress and then roll a marble past it. The marble will fall in towards the bowling ball. But what’s causing it to fall in? Gravity!

Physicist: Here’s what this is about. Way back in the day a popular demonstration used to explain how the presence of matter creates gravity was to drop a heavy ball onto a sheet of some kind, and then roll a smaller ball around the inside of the indentation that is made. If you were to try this demonstration while floating around on the space station you’d be wasting time that could be better spent putting on pants two legs at a time (no gravity to pull the bowling ball and make the indentation in the first place).

In Newtonian mechanics gravity is a spooky, unexplained force. In Einstein’s General Relativity gravity is caused by the curvature and stretching of space and time. Objects move in straight lines like always, but the messed up spacetime they move through makes it appear as though they’re changing direction (that is to say: falling). What’s weird as hell is that they really are moving in straight lines locally, but not globally. If you carefully try to draw a straight line on a bowl you’ll find that it may be straight if you look at a tiny piece of it, but if you stand back it’s curved.

The bowling-ball-mattress thing is another example of how messed up geometry can create “force”. It’s just a bad metaphor. In the one case the pull toward the center is a result of the object in question following a straight line through a messed up spacetime, and in the other it’s trying to roll downhill.
Different.