Dennis Lehmkuhl: Einstein's Six Paths to the Metric Tensor

Title: Einstein's Six Paths to the Metric Tensor --- and why he interpreted it differently than you do

Abstract: John Stachel, the first editor of the Collected Papers of Albert Einstein and the founder of what is today called Einstein scholarship, divides the creation of the general theory of relativity (GR) into a drama of three acts. The first act centers around 1907, when Einstein was overwhelmed by the epiphany of the equivalence principle, the idea that the force of gravity and the intertia of bodies were intimately connected. The second act takes place around 1912, when Einstein entered the promised land and proceeded from scalar theories of gravity to those based on a metric tensor. And the third act finishes in late November 1915, when Einstein found what we now call the Einstein field equations, the successors of Newton's law of gravity. Stachel further argued that the "missing link" between the second and the third act was Einstein's so-called rotating disc argument, which allowed him to forge a connection between gravity-inertia and non-Euclidean geometry.
In this talk, I shall argue that instead of being the protagonist in said drama, and in which the rotating disc argument is the one heureka moment that allowed Einstein to transition to a metric theory of gravity, Einstein, in the summer and autumn of 1912, was instead an adventurer walking on six different paths in parallel, all of which led him to the programme of finding a theory of gravity based on a metric tensor. And yet, I shall argue, it is Einstein's starting point, his scalar theory of gravity of early 1912, that, together with his equivalence principle, pointed him to these six paths, and determined the way he eventually saw the metric tensor. In particular, I shall argue that Einstein's work on a scalar theory of gravity, and his multi-path journey from there to the metric tensor, equipped him with many of the interpretational moves and tools that would influence his later interpretation of GR, and made him resist seeing GR as a "reduction of gravity to spacetime geometry". I shall decipher how Einstein saw the role of geometry in GR instead, what he himself meant by "geometry", and how his notion of geometry differed from his contemporaries and successors. I shall outline how all this led him to an interpretation of GR that saw the distinction of matter and spacetime geometry as something to be overcome rather than as something to be celebrated.