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Einstein’s Field Equations: What Each Symbol Says, and Where Newton Is Hiding
The equation on the T-shirt contains Newton’s law of gravity, and a factor of one half explains its 8π.
After E = mc², the most widely reproduced equation in physics is probably the one Albert Einstein presented to the Prussian Academy of Sciences on 25 November 1915. It appears on T-shirts, mugs and lecture-hall doors, and most popular accounts of Einstein’s field equations stop at a single sentence, usually John Wheeler’s: spacetime tells matter how to move, and matter tells spacetime how to curve.
The summary is accurate, and reading the symbols behind it turns up a puzzle. Newton’s gravity, written in the form physicists use for matter spread through space, carries a factor of 4π. Einstein’s equation carries 8π, in the constant 8πG/c⁴. If gravity is now geometry, with no force and no inverse-square law anywhere in sight, it is not obvious how Newton’s law survives inside the new theory, and still less obvious why its numerical factor doubled.
Geometry on the left
Einstein’s field equations are the law of gravity in general relativity: they equate the curvature of spacetime at each point with the density and flow of energy and momentum at that same point…
