It occurred to Albert Einstein that gravity could be modeled as a geometric phenomenon. Instead of saying that a particle was deflected from a straight trajectory by the force of gravity, one might say that that gravity affects space and time in such a way as to alter the notion of what a straight trajectory is. To this end Einstein began to adapt the mathematics of Riemann’s metric geometry to describe the physical world. One night, Einstein sat in his study, grinding through pages and pages of coordinate transformations. His hand hurt from the writing because each term in an equation had to be preceded by multiple capitalsigma summation signs. “I’m so sick of these these stupid sigmas. I wish I could just leave them out,” Einstein thought. He looked back down at the page, paused for a moment, then scratched out all the summation signs. Because the terms only ever involved summation over coordinate indices, and the sums were invariably over those indices that appeared exactly twice, removing the sigmas did nothing to obscure the meaning of the equations, but did make them easier to read and write. “So much better,” Einstein said to himself. “Einstein, you’re a genius!”

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