Of pots and holes: Einstein’s bumpy road to general relativity

Readers of this volume will notice that it contains only a few papers on general relativity. This is because most papers documenting the genesis and early development of general relativity were not published in Annalen der Physik.After Einstein took up his new prestigious position at the PrussianAcademy of Sciences in the spring of 1914, the Sitzungsberichte of the Berlin academy almost by default became the main outlet for his scientific production. Two of the more important papers on general relativity, however, did find their way into the pages of the Annalen [35,41].Although I shall discuss both papers in this essay, the main focus will be on [35], the first systematic exposition of general relativity, submitted in March 1916 and published in May of that year. Einstein’s first paper on a metric theory of gravity, co-authored with his mathematician friend Marcel Grossmann, was published as a separatum in early 1913 and was reprinted the following year in Zeitschrift für Mathematik und Physik [50,51]. Their second (and last) joint paper on the theory also appeared in this journal [52]. Most of the formalism of general relativity as we know it today was already in place in this Einstein-Grossmann theory. Still missing were the generally-covariant Einstein field equations. As is clear from research notes on gravitation from the winter of 1912–1913 preserved in the so-called “Zurich Notebook,” Einstein had considered candidate field equations of broad if not general covariance, but had found all such candidates wanting on physical grounds. In the end he had settled on equations constructed specifically to be compatible with energy-momentum conservation and with Newtonian theory in the limit of weak static fields, even though it remained unclear whether these equations would be invariant under any non-linear transformations. In view of this uncertainty, Einstein and Grossmann chose a fairly modest title for their paper: “Outline (“Entwurf”) of a Generalized Theory of Relativity and of a Theory of Gravitation.” The Einstein-Grossmann theory and its fields equations are therefore also known as the “Entwurf” theory and the “Entwurf” field equations. Much of Einstein’s subsequent work on the “Entwurf” theory went into clarifying the covariance properties of its field equations. By the following year he had convinced himself of three things. First, generallycovariant field equations are physically inadmissible since they cannot determine the metric field uniquely. This was the upshot of the so-called “hole argument” (“Lochbetrachtung”) first published in an appendix to [51]. Second, the class of transformations leaving the “Entwurf” field equations invariant was as broad ∗ E-mail: [email protected] 1 An annotated transcription of the gravitational portion of the “Zurich Notebook” is published as Doc.10 in [11]. For facsimile reproductions of these pages, a new transcription, and a running commentary, see [89]. 2 See Sect. 2 for further discussion of the hole argument.