Gravitational potential wells

I've changed my mind on how I want to proceed with the CMB. I had a post starting to talk about general relativity, but I've decided that it is too much for this particular sequence. I'd want to really talk about special relativity and general relativity to really do it justice, therefore I've decided to skip it for now. However, that still leaves us needing to understand just what is the information encoded in the Cosmic Microwave Background, so I'll try to do a slightly different description.

Imagine a pendulum - like this one!

The pendulum oscillates back and forth, and as it does so, it traces out a well. The pendulum wants to rest at the bottom of the well, but it has too much energy, and so it continuously overshoots the bottom. The well looks like the line drawn in the still picture to the right. In physics, something like this is known as a potential well - the force of gravity is pulling the weight downward, towards the bottom of the well, but because of the string, the pendulum just bobs up and down in the well.

There are a surprising number of situations like this, and most gravitational interactions can be described in terms of potential wells. For example, the motion of the Earth around the Sun is an orbit that follows the same path as a pendulum in two dimensions. The Earth wants to go straight to the center of the Sun, just like the ball wants to rest at the bottom of the well; instead, the Earth goes around the Sun forever, unable to reach the middle (thankfully).

General relativity is a theory of gravity. Why does gravity create these potential wells? The answer can be thought of in terms of curvature. Large masses tend to curve the space around them, so that other masses will fall in towards the large one. In this framework, one can imagine the Sun as a giant bowling ball on a very smooth mattress. The mattress dips because of the mass of the Sun, and so the space around the Sun curves. Now, one can imagine rolling a bunch of marbles around the divot left by the Sun; if there were no friction, those marbles could roll around the Sun forever in an orbit, just like the planets.

In this sense, then, mass will curve the space around it to attract other masses. But those masses won't necessarily just fall straight in (although that can happen), but can oscillate, much as the Earth oscillates around the Sun, or as the pendulum above keeps going back and forth.