Varićak, an ethnic Serb, was born on March 1, 1865, in the village of Švica near Otočac, Austrian Empire (present-day Croatia). He studied physics and mathematics at the University of Zagreb from 1883 to 1887. He made his PhD in 1889 and got his habilitation in 1895. In 1899 he became professor of mathematics in Zagreb, where he gave lectures until his death in 1942.[3]
From 1903 to 1908 he wrote on hyperbolic geometry (or Bolyai–Lobachevskian geometry). In 1910, following a 1909 publication of Sommerfeld, he applied hyperbolic geometry to the special theory of relativity.[4] Sommerfeld, using the imaginary form of Minkowski space, had shown in his 1909 paper[5] that the Einstein formula for combination of velocities is most clearly understandable as a formula for triangular addition on the surface of a sphere of imaginary radius. Varićak reinterpreted this result as showing that rapidity combines by the triangle rule in hyperbolic space. This is a fundamental result for the hyperbolic theory which was demonstrated later by other approaches by Robb (1911) and Borel (1913). The 1910 papers also dealt with several applications of the hyperbolic theory to optics. In 1911 Varićak was invited to speak to the Deutsche Mathematiker-Vereinigung in Karlsruhe on his work. He continued to develop the hyperbolic reinterpretation of Einstein's theory collecting his results in 1924 in a textbook, Darstellung der Relativitätstheorie im drei-dimensionalen Lobatschefskijschen Raume (Relativity in Three-Dimensional Lobachevski Space), now available in English. In the period 1909 to 1913 Varićak had correspondence with Albert Einstein[6] concerning rotation and length contraction where Varićak's interpretations differed from those of Einstein. Concerning length contraction Varićak said that in Einstein's interpretation the contraction is only an "apparent" or a "psychological" phenomenon due to the convention of clock measurements whereas in the Lorentz theory it was an objective phenomenon.[7] Einstein published a brief rebuttal, saying that his interpretation of the contraction was closer to Lorentz's.[8]
Walter (1999) re-examined Minkowski's non-Euclidean geometry. He begins by analysis of "the tip of a four-dimensional velocity vector" and notes Minkowski's equations where "both hypersurfaces provide a basis for a well-known model of non-Euclidean space of constant negative curvature, popularized by Helmholtz." In fact it is known as the hyperboloid model of hyperbolic geometry.[9]
Walter goes on to say:
More than any other mathematician, Varićak devoted himself to the development of the non-euclidean style [of relativity], unfolding Minkowski's image of velocity-vector relations in hyperbolic space, and recapitulating a variety of results in terms of hyperbolic functions. The use of hyperbolic trigonometry was shown by Varićak to entail significant notational advantages. For example, he relayed the interpretation put forth by Hergloz and Klein of the Lorentz transformation as a displacement in hyperbolic space, and indicated simple expressions for proper time and the aberration of light in terms of a hyperbolic argument.
Varićak made scholarly contributions on the life and work of Ruđer Bošković (1711–1787) These are listed in the biography of Kurepa (1965) cited below.
Of special interest for the history of relativity is that Varićak also edited and published a little-known 1755 paper of Boscovich in Latin entitled "On absolute motion – if it is possible to distinguish it from relative motion" ("Of Space and Time"). Varićak said that the paper "contains many remarkably clear and radical ideas regarding the relativity of space, time and motion."[11]
Varićak, V.(1924) Darstellung der Relativitatstheorie im drei=dimensionalen Lobatschefskijschen Raume, Zagreb (Narodni Novini); English translation (2007) Relativity in three dimensional Lobachevski Space, A.F. Kracklauer translator ISBN1-84753-364-7, at Amazon.com.
^Prvanović, Mileva & Blagojević, Milutin (2006), "Vladimir Varićak 1865–1942", in V. Đorđević; D. Vitorović & D. Marinković (eds.), Lives and work of the Serbian scientists, Belgrad: Serbian Academy of Sciencies and Arts
^Four papers in Physikalische Zeitschrift (see Publications below). English translations are available from Wikisource
^On the composition of velocities in the theory of relativity (in German) Phys. Z. 10 1909 826-829 English translation by Wikisource.
