The End of Time: The Next Revolution in Physics

To summarize this part of the story, in 1912 Einstein became aware of the possibilities opened up by non-Euclidean geometry and the work initiated by Gauss. He had begun to suspect that gravitational fields would make geometry non-Euclidean. He was also almost desperate to find a formalism that did not presuppose distinguished frames of reference. He found that Gauss’s method of arbitrary coordinates was tailor-made for his ambitions. He also saw that, space and time having been so thoroughly fused by Minkowski, the only natural thing to do was to make space-time into a kind of Riemannian space. The ideas of Gauss and Riemann must be applied, not to space alone, but to space and time. This is the incredibly beautiful idea that Minkowski made possible: gravity was to be explained by curvature in space and time. Einstein thus conjectured that space-time is curved by gravity, and that bodies subject only to gravity and inertia follow geodesics determined by the distance properties of space-time, which encapsulate all its geometrical properties. Einstein’s conjecture has been brilliantly confirmed to great accuracy in recent decades.

THE FINAL HURDLE

Finding the law of motion of bodies in a gravitational field was only part of Einstein’s problem. He also had to find how matter created a gravitational field. He needed to find equations for the gravitational field somewhat like those that Maxwell had found for the electromagnetic field. They would establish how matter interacted with the gravitational field, and also how the field itself varied in regions of space-time free of matter (matching the way electromagnetic radiation propagated as light through space-time). This part of the problem created immense difficulties for Einstein, mostly through very bad luck.

Much as I would like to tell the complete story, which is fascinating and full of ironies, I shall have to content myself with saying that, after three nerve-wracking years, Einstein finally found a generally covariant law that described how matter determined the curvature of space-time. It involves mathematical structures called tensors, all the properties of which had already been studied by mathematicians. In particular, for space-time free of matter, Einstein was able to show that a tensor known as the Ricci tensor (because it had been studied by the Italian mathematician Gregorio Ricci-Curbastro) must be equal to zero. Ironically, Grossmann had already suggested to Einstein in 1912 that in empty space the vanishing of the Ricci tensor might be the generally covariant law he was seeking. However, some understandable mistakes prevented them from recognizing the truth at that time.

It is a striking fact that all the mathematics Einstein needed already existed. In fact, I believe it is significant that he did not have to invent any of it. In 1915, he was immediately able to show that, to the best accuracy astronomers could achieve at that time, his theory gave identical predictions to Newtonian gravity except for a very small correction to the motion of Mercury. All planetary orbits are ellipses. A planet’s elliptical orbit itself very slowly rotates, under the gravitational influence of the other planets. This is known as the advance of the perihelion, the perihelion being the point at which the planet is closest to the Sun, marking one end of the ellipse’s longest diameter. According to Einstein’s theory, Mercury’s perihelion should advance by 43 seconds of arc per century more than was predicted by Newtonian theory. This very small effect shows up for Mercury because it is closer to the Sun than the other planets, and also has a large orbital eccentricity. For many years, the sole discrepancy in the observed motions of the planets had been precisely such a perihelion advance for Mercury of exactly that magnitude. All attempts to explain it had hitherto failed. Einstein’s theory explained it straight off.

GENERAL RELATIVITY AND TIME

Many more things could be said about general relativity and its discovery. However, what I want to do now is identify the aspects of the theory and the manner of its discovery that have the most bearing on time.