08 August 2011

Dimensional analysis for gravity trains

A surprising fact: drill a straight tunnel through the earth, between any two points. Drop a burrito in at one end. Assuming that you could actually build the tunnel, and that there's no friction, the burrito comes out the other end in 42 minutes. This is called a gravity train and it's not hard to prove (the version I link to is due to Alexandre Eremenko of Purdue) that the time it takes the burrito to get from one end to the other is (3π/4)-1/2 (G ρ)-1/2, where G is the gravitational constant and ρ is the density of the Earth. Alternatively this can be written as (r3/Gm)1/2, where r is the radius of the earth and m its mass.

Everyone's so surprised, when they see this, that the time doesn't depend on the distance between the two points! And this is interesting, but as a result you don't see the more subtle fact that the time doesn't depend on the size of the planet. If I make a super-Earth that is twice the radius but made of the same stuff, so it's eight times as massive, then the density stays the same. Somewhat surprisingly you can see this using dimensional analysis. It's "obvious" that this time, if it exists, can only depend on the mass of the earth, the radius of the earth, and the gravitational constant. The mass of the earth, m, has dimension M; the radius, r, has dimension L; the gravitational constant has dimension L3 M-1 T-2. The only combination mα rβ Gγ that has units of time is G-1/2 m-1/2 r3/2.

Of course I'm making a big assumption there -- that the constant time "42 minutes" is actually a constant! It seems perfectly reasonable that it could depend on the distance between the two termini. I'll handwave that away by saying that it depends on the angle formed by the two termini and the center of the earth. And angles are of course dimensionless.

(The Alameda-Weehawken burrito tunnel, being non-fictional, uses magnets to accelerate and decelerate the foil-wrapper burritos and takes 64 minutes instead of the theoretical 42.)