^{-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 (r

^{3}/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 L

^{3}M

^{-1}T

^{-2}. The only combination m

^{α}r

^{β}G

^{γ}that has units of time is G

^{-1/2}m

^{-1/2}r

^{3/2}.

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.)

## 3 comments:

What density would a planet have to have for this time to cross the distance to exceed the time it would take light to cross the same distance?

It's easy to answer this question, but it'll take a giant supercomputer 7.5 million years to reduce the question "what's the meaning of life, universe, and everything" to it.

...times minutes.

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