# Hermann Minkowski

**Hermann Minkowski** (22 June 1864 – 12 January 1909) was a German-Jewish mathematician, professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number theory, mathematical physics, and the theory of relativity.

Minkowski is perhaps best known for his work in relativity, in which he showed in 1907 that his former student Albert Einstein's special theory of relativity (1905), could be understood geometrically as a theory of four-dimensional space–time, since known as the "Minkowski spacetime".

## Personal life and family

Hermann Minkowski was born in Aleksotas, a village in the Kovno Governorate of the Russian Empire (now incorporated into the city of Kaunas, Lithuania) to Lewin Boruch Minkowski, a merchant who subsidized the building of The Choral synagogue in Kovno,^{}^{}^{} and Rachel Taubmann, both of Jewish descent.^{} Hermann was a younger brother of the medical researcher, Oskar (born 1858).^{} In different sources Minkowski's nationality is variously given as German,^{} Polish,^{} Lithuanian^{} or Lithuanian-German,^{} or Russian.^{}

To escape persecution in Russia the family moved to Königsberg in 1872,^{} where the father became involved in rag export and later in manufacture of mechanical clockwork tin toys (he operated his firm *Lewin Minkowski & Son* with his eldest son Max).^{}

Minkowski studied in Königsberg and taught in Bonn (1887–1894), Königsberg (1894–1896) and Zürich (1896–1902), and finally in Göttingen from 1902 until his premature death in 1909. He married Auguste Adler in 1897 with whom he had two daughters; the electrical engineer and inventor Reinhold Rudenberg was his son-in-law.

Minkowski died suddenly of appendicitis in Göttingen on 12 January 1909. David Hilbert's obituary of Minkowski illustrates the deep friendship between the two mathematicians (translated):

- Since my student years Minkowski was my best, most dependable friend who supported me with all the depth and loyalty that was so characteristic of him. Our science, which we loved above all else, brought us together; it seemed to us a garden full of flowers. In it, we enjoyed looking for hidden pathways and discovered many a new perspective that appealed to our sense of beauty, and when one of us showed it to the other and we marveled over it together, our joy was complete. He was for me a rare gift from heaven and I must be grateful to have possessed that gift for so long. Now death has suddenly torn him from our midst. However, what death cannot take away is his noble image in our hearts and the knowledge that his spirit continues to be active in us.

The asteroid 12493 Minkowski and M-matrices are named in Minkowski's honor.

## Education and career

Minkowski was educated in Germany at the *Albertina* University of Königsberg, where he earned his doctorate in 1885 under the direction of Ferdinand von Lindemann. In 1883, while still a student at Königsberg, he was awarded the Mathematics Prize of the French Academy of Sciences for his manuscript on the theory of quadratic forms. He also became a friend of another renowned mathematician, David Hilbert. His brother, Oskar Minkowski (1858–1931), was a well-known physician and researcher.

Minkowski taught at the universities of Bonn, Göttingen, Königsberg and Zürich. At the *Eidgenössische Polytechnikum*, today the ETH Zurich, he was one of Einstein's teachers.

Minkowski explored the arithmetic of quadratic forms, especially concerning *n* variables, and his research into that topic led him to consider certain geometric properties in a space of *n* dimensions. In 1896, he presented his *geometry of numbers*, a geometrical method that solved problems in number theory. He is also the creator of the Minkowski Sausage and the Minkowski cover of a curve.^{}

In 1902, he joined the Mathematics Department of Göttingen and became a close colleague of David Hilbert, whom he first met at university in Königsberg. Constantin Carathéodory was one of his students there.

## Work on relativity

By 1907 Minkowski realized that the special theory of relativity, introduced by his former student Albert Einstein in 1905 and based on the previous work of Lorentz and Poincaré, could best be understood in a four-dimensional space, since known as the "Minkowski spacetime", in which time and space are not separated entities but intermingled in a four dimensional space–time, and in which the Lorentz geometry of special relativity can be effectively represented. The beginning part of his address delivered at the 80th *Assembly of German Natural Scientists and Physicians* (21 September 1908) is now famous:

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

Notice of Minkowski's death was communicated to the Quaternion Society in 1910 by its President who had explored hyperbolic quaternions as the "Algebra of Space":

- He devoted what proved to be the last years of his life to the scientific statement of fundamental equations of electrodynamics, a work which he accomplished by development of the Algebra of Space, or as he would prefer to call it, the Algebra of Space and Time.
^{}

Einstein at first viewed Minkowski's treatment as a mere mathematical trick, before eventually realizing that a geometrical view of space–time would be necessary in order to complete his own later work in general relativity (1915).^{}

## Publications

- Relativity papers

- Minkowski, Hermann (1915) [1907]. "Das Relativitätsprinzip".
*Annalen der Physik***352**(15): 927–938. Bibcode:1915AnP...352..927M. doi:10.1002/andp.19153521505.

- Minkowski, Hermann (1908). "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern".
*Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse*: 53–111.- English translation: The Fundamental Equations for Electromagnetic Processes in Moving Bodies. In: The Principle of Relativity (1920), Calcutta: University Press, 1–69

- Minkowski, Hermann (1909). "Raum und Zeit".
*Jahresbericht der Deutschen Mathematiker-Vereinigung*: 75–88.- Various English translations on Wikisource: Space and Time

- H. A. Lorentz, Albert Einstein, Hermann Minkowski, and Hermann Weyl, 1952.
*The Principle of Relativity: A Collection of Original Memoirs*. Dover.

- Diophantine approximations

- Minkowski, Hermann (1907).
*Diophantische Approximationen: Eine Einführung in die Zahlentheorie*. Leipzig-Berlin: R. G. Teubner.^{}

- Mathematical papers (posthumous)

- Minkowski, Hermann (1910). "Geometrie der Zahlen". Leipzig-Berlin: R. G. Teubner. MR 0249269.
^{} - Minkowski, Hermann (1911).
*Gesammelte Abhandlungen 2 vols*. Leipzig-Berlin: R. G. Teubner.^{}Reprinted in one volume New York, Chelsea 1967

## See also

- Abraham–Minkowski controversy
- Brunn–Minkowski theorem
- Hasse–Minkowski theorem
- Minkowski addition
- Minkowski–Bouligand dimension
- Minkowski (crater)
- Minkowski diagram
- Minkowski functional
- Minkowski–Hlawka theorem
- Minkowski inequality
- Minkowski problem
- Minkowski's question mark function
- Minkowski space
- Minkowski–Steiner formula
- Minkowski's theorem in geometry of numbers
- Minkowski's bound
- Separating axis theorem
- Smith–Minkowski–Siegel mass formula