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.