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Why the π?

Today is March 14, otherwise known as Pi Day by math and science nerds, although a case can be made for July 22 being Pi Day since 22/7 is a better approximation for π than 3.14.

It is customary for science and math bloggers to acknowledge Pi Day in some way. I will do that by referring you to a nice post by Sean Carroll who explains why π appears in Einstein’s field equations for general relativity. This is particularly appropriate since March 14 also happens to be Einstein’s birthday.

To start with, note that Newton’s law of gravity F=GMm/r2 does not contain π.

But Einstein’s equations for gravity do. Einstein’s equations take the form:

Rμν – (1/2)Rgμν=8πGTμν

In this equation, the left hand side describes the curvature of space-time while the right hand side represents the energy-momentum. So the equation says that the energy and momentum in a region distorts space-time correspondingly and it is this distortion that spreads throughout and influences other bodies that are far away.

So why does π appear in the latter but not the former? Carroll explains that Newton’s law represents the force between two point-like particles at a distance with no explanation as to how that force is transmitted over space. When you generalize it, as Einstein’s did, to masses spread out over all space, the equation says that the force is due to a gravitational field that spreads out through all space. And it is the introduction of this field due to the spread out masses that causes π to appear.

Comments

  1. says

    How about May 13th (13/5)?

    22/7 is a horrible approximation. My favourite is one I learnt a few years ago, which was discovered in China in the 5th century. It’s the most accurate fractional approximation with a denominator less than 30,000.

    Repeat the first three odd digits twice: 1 1 3 3 5 5

    Separate them with the long division symbol and solve: 355/113 = 3.1415929203

    It’s accurate to within 0.0000000849 – not good enough for engineering, but probably good enough up to the second year of college.

  2. Félix Desrochers-Guérin says

    although a case can be made for July 22 being Pi Day since 22/7 is a better approximation for π than 3.14.

    You could even say it’s the more rational choice. :)

  3. Rob Grigjanis says

    There’s always delicious pi, if you look hard enough. Newton’s law, translated by Gauss;

    ∇.g = -4πGρ

  4. Rob Grigjanis says

    Actually, the equation in #5 is, pretty much, the static, weak field, low velocity limit of Einstein’s field equation.

  5. Rob Grigjanis says

    And of course, if I wasn’t such an ass, I’d have read Carroll’s post and seen that he said the same thing. *bray*.

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