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At the International Congress of Mathematicians (ICM) in 1912 at Cambridge, Silberstein spoke on "Some applications of quaternions". Though the text was not published in the proceedings of the Congress, it did appear in the Philosophical Magazine of May, 1912, with the title "Quaternionic form of relativity". The following year Macmillan published The Theory of Relativity, which is now available on-line in the Internet Archive (see references). The quaternions used are actually biquaternions. The book is highly readable and well-referenced with contemporary sources in the footnotes.
Several reviews were published. Nature expressed some misgivings:
A systematic exposition of the principle of relativity necessarily consists very largely in the demonstration of invariant properties of certain mathematical relations. Hence it is bound to appear a little uninteresting to the experimentalist...little is done to remove the unfortunate impression that relativity is a fad of the mathematician, and not a thing for the every-day physicist.
In his reviewMorris R. Cohen wrote, "Dr. Silberstein is not inclined to emphasize the revolutionary character of the new ideas, but rather concerned to show their intimate connection with older ones." Another review by Maurice Solovine states that Silberstein subjected the relativity principle to an exhaustive examination in the context of, and with respect to, the principal problems of mathematical physics taken up at the time.
Synge had also been strongly influenced a few months previously [in January 1921] by a Toronto lecture series organized by J.C. McLennan on "Recent Advances in Physics", at which Silberstein gave eighteen lectures on "Special and Generalized Theories of Relativity and Gravitation, and on Spectroscopy", all from a mathematical standpoint.
In 1935, following a controversial debate with Albert Einstein, Silberstein published a solution of Einstein's field equations that appeared to describe a static, axisymmetricmetric with only two point singularities representing two point masses. Such a solution clearly violates our understanding of gravity: with nothing to support them and no kinetic energy to hold them apart, the two masses should fall towards each other due to their mutual gravity, in contrast with the static nature of Silberstein's solution. This led Silberstein to claim that A. Einstein's theory was flawed, in need of a revision. In response, Einstein and Rosen published a Letter to the Editor in which they pointed out a critical flaw in Silberstein's reasoning. Unconvinced, Silberstein took the debate to the popular press, with The Evening Telegram in Toronto publishing an article titled "Fatal blow to relativity issued here" on March 7, 1936. Nonetheless, Einstein was correct and Silberstein was wrong: as we know today, all solutions to Weyl's family of axisymmetric metrics, of which Silberstein's is one example, necessarily contain singular structures ("struts", "ropes", or "membranes") that are responsible for holding masses against the attractive force of gravity in a static configuration.
According to Martin Claussen, Ludwik Silberstein initiated a line of thought involving eddy currents in the atmosphere, or fluids generally. He says that Silberstein anticipated foundational work by Vilhelm Bjerknes (1862 - 1951).
1907: Electromagnetische Grundgleichungen in bivectorielle Behandlung, Ann. Physik 22 579-86 & 24:783-4
1912: Quaternionic form of relativity, Phil. Mag. 14 1912 790 - 809
1913: Second memoir on quaternionic relativity, Phil. Mag. 15 1913 135-144
^G., T. (1934). "Review of Causality: a Law of Nature or a Maxim of the Naturalist? Lecture delivered at the Royal York Hotel, Toronto, on May 14th, 1932, much enlarged". Nature. 133 (3355): 235. doi:10.1038/133235c0. ISSN0028-0836. S2CID4093081. The initials "T.G." might be those of the mathematician Thomas Greenwood, who wrote articles for Nature and had an interest in relativity theory. Greenwood, Thomas (1923). "The Significance of the Space-Time Continuum". Monist. 33 (4): 635-640. doi:10.5840/monist192333418. ISSN0026-9662.