I’ve been thinking a lot about Special and General Relativity lately, and in particular, the Equivalence Principle — the main theme for Chapter 7 of my upcoming book on symmetry. In the process, I started playing around with a really simple way of “deriving” the General Relativistic result that time runs slow near massive bodies. I wish I could take the credit. Like so many things, Einstein thought of it first. In this case, he proposed the basic outline of what follows in 1907, but it’s still fun to work through it.
In writing up a draft of this section for the book, I thought it would be fun to adapt it a bit to include equations here on the blog. That’s why I’ve labeled this as “technical,” though the equations are really at the Freshman level. Only 1 simple bit of calculus.
The only other background you need is the Equivalence Principle. The Equivalence Principle, if you’ve never seen it, was Einstein’s central proposition in General Relativity. He phrased it many ways at different times, but a general way of writing it (courtesy of Schutz) goes something like this:
Einstein Equivalence Principle: Any local physical experiment not involving gravity will have te same result if performed in a freely falling inertial frame as if it were performed in the flat spacetime of special relativity.
though an equally good way of thinking about it is that General Relativity can’t distinguish between “real gravity” and “artificial accelerations.”
Comments, as always, are appreciated.
Beyond just the esoteric coolness of it all, the Equivalence Principle can serve as the basis for some rather surprising gravitational results. Einstein, himself, came up with a scenario for relating artificial to real gravity as early as 1907.
In this case, we’re going to imagine life on top of a large spinning disk. This is a lot like the 2-dimensional universes that I’ve written about before — you know, the ones we found couldn’t actually support life. We pretty much dismissed it out of hand, but that doesn’t mean that we couldn’t imagine how gravity would work in such a place.
In our universe, there are a bunch of super-intelligent ants slowly crawling around on the surface . The Queen, sitting at the center of the antworld, sits still as far as outsiders are concerned. Her royal court surrounds her in close proximity. To an outsider (you), her courtiers slowly rotate about the queen. To put this in mathematical terms, the rotational speed of an ant is:
where is the rate of rotation of the disk, and is the distance from the center.
The ants don’t know any of this, however. From their perspective, however, they feel a small tug that pulls them outward. As far as they’re concerned, “out” is “down.” This, as you may recall, is known as centrifugal force. It’s the exact same effect that pulls you to the side of a “gravitron” at a carnival. You may also recall that in order to stay moving in a circular path, you need to have just the right amount of acceleration:
The further the ants are from the queen, the faster they are spinning, and the stronger they are tugged outward. From the perspective of the ants, their antworld feels very much like a hill with the queen at the gently curved top, a hill that gets steeper the further you go out. An ant that loses its grip will roll outward – down the hill – at an ever-accelerating rate.
There’s at least one sense in which this analogy isn’t perfect. If you fall down a hill in our world, you’ll simply roll down in a straight line. An ant falling down the hill will start rolling straight down, but will then slowly start rolling around the hill as well. This is the famous “Coriolis Effect.” It’s the same thing that causes cyclones to spin counter-clockwise in the Northern hemisphere, and clockwise in the Southern.
But the Coriolis Effect becomes irrelevant if all of the ants stand still, each glued to a different point in antworld. From their perspectives, there is no dynamics whatsoever. Everyone seems to be fixed compared to everyone else.
We – standing outside the antworld – know better. The queen isn’t moving at all. Nearby ants are moving slowly. Ants further out move faster. The ants out in the hinterlands are moving fastest of all. We know something about the flow of time of moving ants. The faster they move, the slower time will appear to pass compared to the queen. The further out an ant is, the slower she’ll appear to age. As you may recall, the relationship between the clocks for the stationary observer (the queen), and everyone else is:
But we needn’t be quite so fancy. In the limit of
The latter term tells us about the fractional slowing of the clock. Plugging in our velocity relationship, we get:
But there’s another way of thinking about the same situation – from the perspective of the equivalence principle. Artificial gravity can be generated by rotating a spaceship. And in similar fashion, the rotating antworld creates an “artificial” gravity outward. One of the best ways of describing gravitational force is in terms of the potential. In 1-d (as is the case here), the relationship looks like:
Given that “down” is “out”, and knowing the relationship above, we immediately get:
The potential is zero at the center, and gets more and more negative further out. Since we fall to lower potentials, this makes sense. You may also notice that this means that from the ants’ perspectives:
The ants notice a curious effect. The further you go down, the slower time seems to be running. And the same is true in our universe. Time seems to run more slowly near massive bodies – nearer to the “down” direction – than further away.
I especially like this approach because — unlike how it’s normally done — we don’t need to introduce photon frequencies and the like to make our point. Everything comes out from a natural extension of Special Relativity.