Tobias Schütz, Einstein at Work on Unified Field Theory: The Five-Dimensional Einstein-Bergmann Approach, Einstein Studies 17 (Cham, Birkhäuser/Springer, 2024)
Mathematical Reviews, MR4759174

Among the more daunting challenges faced by historians of exact science is that of historically situating stacks of bare calculations. This is so, even when the author has been identified, as in Einstein’s case, where we have roughly 1750 pages of research notes that came to light in the 1980s, and are now found at the Hebrew University Library in Jerusalem. For the most part, the notes are loose-leaf, undated and non-discursive. It is to the great credit of Tobias Schütz to have read through the stack, and to have come up with a legible account of about 170 sheets, which he has published in several papers co-authored with Tilman Sauer [5, 4], a doctoral thesis [7], and now a volume in Einstein Studies [8]. Schütz’s reading of Einstein’s research notes finds a common thread in these selected leaves: the Einstein-Bergmann unified field theory (UFT) [1].

The first fifty pages provide a rapid, factual overview of Einstein’s research on relativity and gravitation, including an assessment of his early education in geometry and mathematics, up to his comprehensive presentation of general relativity [3]. A presentation then follows of the corpus of selected research notes, based on a classification into four categories: projective geometry, generalization of Theodor Kaluza’s five-dimensional theory, developments of Einstein-Bergmann theory, and $\delta$-functions. A fifth category is designated for miscellaneous correlations between different sheets in which, for example, a common equation appears. The first four categories structure the monograph, and serve as chapter headings.

Following the introduction, the chapter on projective geometry presents most of the digital sheet reproductions (28) and redrawn figures. Typically for this chapter, there is a detailed summary and discussion of the contents of the reproduction. Page summaries include a transcription of selected equations and remarks in the original German, with English translation in footnotes. The reader can verify the transcription directly, which is easily done in most cases, thanks to the quality of the reproductions and Einstein’s usually-legible hand. The task of following Einstein’s computational train of thought from reproductions alone is more demanding, and here, the commentary shines, as it spots Einstein’s mistakes and unexpected notational changes. The author also fills in gaps in reasoning, and confirms proofs. At some point, I expected to learn just how Einstein’s line drawings and calculations from projective geometry figure in his search for a UFT, or relate in some way to mathematical physics. What the author delivers instead is evidence of calculations on power series expansions at infinity, and on involutions, both being standard tools for the physicist. The connection to UFT research is at best circumstancial, as Einstein was likely aware of Schouten and van Dantzig’s observation [6] that the Einstein-Mayer UFT of 1931 [2] lends itself to their projective approach.

The third and fourth chapters take up Einstein-Bergmann theory in two forms: the published version, and a manuscript Einstein donated to the Library of Congress, which he referred to as the “Washington manuscript”. Einstein did not seek to publish this paper, which differs from Einstein-Bergmann theory in title (Einheitliche Feldtheorie), author (Einstein) and approach (axiomatic). These differences are discussed in detail, in light of several leaves of Einstein’s notes, correspondence with his assistants Peter Bergmann and Valentine Bargmann, and with Library of Congress staff.

In the fifth and final chapter, the author considers Einstein’s calculations on $\delta$-functions. From Einstein’s correspondence with his assistants, the author builds a strong case that these calculations concern Einstein-Bergmann theory, and thereby may relate to a search for particle-like solutions to the field equations. The book ends with a bibliography and an index of archival documents, omitting subject and author indexes.

In summary, Einstein at Work on Unified Field Theory presents a fascinating behind-the-scenes example of Einstein at work, almost in real time, exploring theoretical avenues with mathematical tools at a level of detail not usually found in his publications. Schütz’s book is a solid contribution to Einstein Studies, and will be of interest to Einstein scholars and mathematicians interested in the history of Einstein-Bergmann theory.