Tuesday, March 14, 2023

Alcohol and Pi

This is International Pi Day (3-14, get it?), which personally never excited me much, although the number itself is interesting per se. 

Anyway, in an article about pi in the New York Times, a commenter noted the following mnemonic for the first 15 digits of pi:

And it's true; count the number of letters in each word: 3-1-4-1-5-9-2-6-5-3-5-8-9-7-9.

Not very exciting either, really, but there you have it. If you can remember the mnemonic. 

I've always been a little intrigued that pi appears in Einstein's field equations for general relativity, his theory of gravity that says essentially "mass tells space how to bend, and space tells matter how to move" (John Wheeler). 

where the R's and g's specify the shape of spacetime, and the T specifies the matter distribution in the spacetime. This is actually a total of 10 different equations, but that doesn't matter here.

Why does pi appear on the right-hand side? And an 8? Weird. What does gravity  have to do with a circle, as pi is defined at the ratio of the circumference of a circle to its diameter?

The 8πG/c4 is called the "Einstein constant" κ (kappa). G is Newton's gravitational constant, and c is the speed of light.

Wait, first, why does the speed of light appear here? It turns out that if you solve these field equations for a traveling wave, its speed in vacuum is that of light, c. Gravity travels at the speed of light. But it seems to me that it's more that light travels at the speed of gravity, because gravity--the very shape of space(time)--is more fundamental than electromagnetism. It's electromagnetism that exists in spacetime, so its fundamental properties are determined by spacetime.

I don't think the 8π is anything mystical or mysterious. OK, in a flat vacuum a gravitational wave would travel outward with its furthest points on a 4-dimensional sphere (3-dimensions of space and one of time). Something circular, since the circumference and surface area of a sphere also contains π. I think it's more that someone could have defined one of the items in the equation better, if they had deeper foresight. 

Which one? Maybe Newton's gravitational constant should have absorbed the 8π, so instead of G in Newton's equation, the proportionality constant would be G', where

G' = 8πG 

so Einstein's constant would be simply 


But then why the c4? Newton had no reason to deal with that; he didn't even know if the speed of light in a vacuum was a constant. So naturally he just used G for the proportionality constant in his equation that gives the force between two masses. So an 8π popped out the other end, because our view of the universe is nonrelativistic.

No doubt there are plenty of people who have some insight into this instead of my banal musings. I wonder what they would say.


Layzej said...

Only tangentially related, but this reminded me of a neat video by 3blue1brown about

why pi is found in the sum of this sequence:

sum(1/n^2) where n=1..infinity =(π^2)/6

Why is pi here? And why is it squared? A geometric answer to the Basel problem.

Layzej said...

This makes me wonder what is the most obtuse way to calculate pi? Based on my example above, you could calculate pi by completing the infinite series:

where n=1..infinity

But you can do one better with Einstein's equations. In that case, all you need to do is measure the speed of light in a vacuum, etc.

BTW, I love that idea that light moves at the speed of gravity. Insightful :)