The fascinating article recounting Einstein’s mistakes at different stages of his career goes beyond the usual focus on the cosmological constant and quantum mechanics. In particular, the discussion of Kaluza–Klein theory examines Einstein’s later attempts at a unification theory. But in the course of developing general relativity, Einstein made another assumption, which he later tried to revisit—one that future generations may come to regard as Einstein’s greatest “mistake.”
Curvature of spacetime is, of course, related by general relativity to the presence of mass-energy. This curvature, though it plays out in the arena of four-dimensional spacetime, corresponds to our intuitive understanding of geometric curvature in three dimensions. General relativity also makes a crucial assumption that another geometric object, called the torsion, vanishes. That is not the only assumption that could have been made, however, and as Einstein explored extensions of general relativity after 1915, he reevaluated his initial assumption.
In the 1920s and 1930s, Einstein collaborated 1 with the eminent French mathematician Elie Cartan, who was responsible for much of the foundation of 20th-century differential geometry. As early as 1922, Cartan tried to explain to Einstein that a different type of curvature, which could be called a total curvature and which contains the traditional curvature as a piece, vanishes. With this condition, called teleparallelism (TP), the torsion need not vanish. Einstein and Cartan explored the implications of TP for generalizing general relativity beyond the gravitational field, but ultimately abandoned that route. Unfortunately, the tools Cartan himself offered to differential geometry were insufficiently mature at that stage to be exploited by Einstein even if the physicist had been able to fully understand them. 1
Teleparallelism does offer advantages, including a greater mathematical richness than general relativity and a potential resolution of mathematical issues related to the nature of conservation laws in general relativity. 2,3 Wielding the methods of modern differential geometry that Cartan first introduced, physicists in the past couple of decades have elaborated unified theories with TP as an important component. 3,4 For instance, TP and another geometric ingredient 5 lead to the “natural” incorporation of electromagnetism in one such theory, fully within the tradition of the geometrical paradigm of Einstein. 2
TP may ultimately prove to be a better assumption for a geometric theory. If so, it would still be an extreme excess of Whiggery, to use Weinberg’s wonderful phrase, for those future generations to fault Einstein for his choice in general relativity. The very mathematical concepts, let alone the tools, behind TP did not even exist in 1915 when general relativity was unveiled to the world.
