The Ascent of Gravity: The Quest to Understand the Force that Explains Everything by Chown Marcus

Author:Chown, Marcus [Chown, Marcus]
Language: eng
Format: azw3
Publisher: Orion
Published: 2017-04-06T04:00:00+00:00

The mathematics of curved space

In order to turn his basic insight that matter warps space-time, and warped space-time is gravity, into a theory of gravity, Einstein had to wrestle with the complex mathematics of curved space. Unfortunately, he had skipped mathematics lectures while a student at the Swiss Federal Polytechnic in Zurich, preferring instead to get his hands dirty among the batteries and condensers and galvanometers of the Polytechnic’s Electrotechnical Laboratory. ‘It was a mistake I realised only later, with regret,’ said Einstein.21

Fortunately, Einstein had made a lifelong friend in Marcel Grossman, a mathematics student a year his senior at the Polytechnic. Grossman’s father had used his contacts to help secure Einstein his dream job at the Swiss Federal Patent Office in Bern. And Grossman, crucially, knew about the geometry of curved spaces. He was therefore able to teach Einstein the mathematics he needed in order to express in rigorous terms his revolutionary ideas about gravity and the warpage of space-time.

This field of mathematics had been developed by a number of mathematicians, most importantly Carl Friedrich Gauss and Bernhard Riemann in the nineteenth century. Until that time, geometry was the flat-paper geometry of the Greek mathematician Euclid (who inspired the best-ever title of a popular-science book, Here’s Looking at Euclid!).22 In his Elements, written in the third century BC, Euclid listed five self-evident statements about straight lines and angles. Using these ‘axioms’ as foundations, using logic alone he constructed a great edifice of ‘theorems’ such as ‘the internal angles of a triangle always add up to 180 degrees’.

Euclid’s fifth postulate states that parallel lines never meet. Gauss and Riemann relaxed this postulate, which immediately admitted geometries of curved surfaces such as spheres. On a sphere, for instance, two parallel lines which extend northward from the equator do not stay parallel but meet at the North Pole.

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