# Hermann Minkowski

Hermann Minkowski (/mɪŋˈkɔːfski, -ˈkɒf-/;[1] German: [mɪŋˈkɔfski]; 22 June 1864 – 12 January 1909) was a German mathematician and 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.

Hermann Minkowski
Born22 June 1864
Died12 January 1909 (aged 44)
NationalityGerman
Alma materAlbertina University of Königsberg
Known forGeometry of numbers
Minkowski content
Minkowski diagram
Minkowski's question mark function
Minkowski space
Work on the Diophantine approximations
ChildrenLily (1898–1983), Ruth (1902–2000)
Scientific career
FieldsMathematics, Physics, Philosophy
InstitutionsUniversity of Göttingen and ETH Zurich
Doctoral advisorFerdinand von Lindemann
Doctoral studentsConstantin Carathéodory
Louis Kollros
Dénes Kőnig
Signature

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 the town of Aleksota, the Suwałki Governorate, the Kingdom of Poland, part of the Russian Empire, Lithuania, to Lewin Boruch Minkowski, a merchant who subsidized the building of the choral synagogue in Kovno,[2][3][4] and Rachel Taubmann, both of Jewish descent.[5] Hermann was a younger brother of the medical researcher Oskar (born 1858).[6] In different sources Minkowski's nationality is variously given as German,[7][8] Polish,[9][10][11] or Lithuanian-German,[12] or Russian.[13]

To escape persecution in Russia the family moved to Königsberg in 1872,[14] 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).[15]

Minkowski studied in Königsberg and taught in Bonn (1887–1894), Königsberg (1894–1896) and Zurich (1896–1902), and finally in Göttingen from 1902 until his 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.

Max Born delivered the obituary on behalf of the mathematics students at Göttingen.[16]

The main-belt asteroid 12493 Minkowski and M-matrices are named in Minkowski's honor.[17]

## Education and career

Minkowski was educated in East Prussia 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.[14]

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.[18]

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 1908 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 using the invariant interval ${\displaystyle x^{2}+y^{2}+z^{2}-c^{2}t^{2}}$ (see History of special relativity).

The mathematical basis of Minkowski space can also be found in the hyperboloid model of hyperbolic space already known in the 19th century, because isometries (or motions) in hyperbolic space can be related to Lorentz transformations, which included contributions of Wilhelm Killing (1880, 1885), Henri Poincaré (1881), Homersham Cox (1881), Alexander Macfarlane (1894) and others (see History of Lorentz transformations).

The beginning part of his address called "Space and Time" 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.

## Publications

Relativity
• 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 der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111.
• Minkowski, Hermann (1909). "Raum und Zeit" . Jahresbericht der Deutschen Mathematiker-Vereinigung: 75–88.
• Blumenthal O. (ed): Das Relativitätsprinzip, Leipzig 1913, 1923 (Teubner), Engl tr (W. Perrett & G. B. Jeffrey) The Principle of Relativity London 1923 (Methuen); reprinted New York 1952 (Dover) entitled H. A. Lorentz, Albert Einstein, Hermann Minkowski, and Hermann Weyl, The Principle of Relativity: A Collection of Original Memoirs.
• Space and Time – Minkowski's Papers on Relativity, Minkowski Institute Press, 2012 ISBN 978-0-9879871-3-6 (free ebook).
Diophantine approximations
Mathematical (posthumous)
• Minkowski, Hermann (1910). "Geometrie der Zahlen". Leipzig-Berlin: B. G. Teubner Verlag. MR 0249269. Retrieved 28 February 2016. Cite journal requires `|journal=` (help)[20]
• Minkowski, Hermann (1911). Gesammelte Abhandlungen 2 vols. Leipzig-Berlin: R. G. Teubner. Retrieved 28 February 2016.[21] Reprinted in one volume New York, Chelsea 1967.

## Notes

1. А. И. Хаеш «Коробочное делопроизводство как источник сведений о жизни еврейских обществ и их персональном составе»: 1873 г. «...купец Левин Минковский подарил молитвенному обществу при Ковенском казённом еврейском училище начатую им... постройкой молитвенную школу вместе с плацем, с тем, чтобы общество это озаботилась окончанием таковой постройки. Общество, располагая средствами добровольных пожертвований, возвело уже это здание под крышу, но затем средства сии истощились...»
2. "Kaunas: dates and facts. Electronic directory".
3. "Box-Tax Paperwork Records". Archived from the original on 8 January 2015. Kovno. In 1873 the merchant kupez, Levin Minkovsky, gave (as a gift) to the prayer association of the Kovno state Jewish school a lot with an ongoing construction of a prayer school that (the construction) he had started so that the association would take care of completing the construction. The association, having some funds from voluntary contributions, had built the structure up to the roof, but then, ran out of money
4. "Minkowski biography".
5. Oskar Minkowski (1858–1931). The Jewish genealogy site JewishGen.org (Lithuania database, registration required) contains the birth record in the Kovno rabbinical books of Hermann's younger brother Tuvia in 1868 to Boruch Yakovlevich Minkovsky and his wife Rakhil Isaakovna Taubman.
6. Gregersen, Erik, ed. (2010). The Britannica Guide to Relativity and Quantum Mechanics (1st ed.). New York: Britannica Educational Pub. Association with Rosen Educational Services. p. 201. ISBN 978-1-61530-383-0.
7. Bracher, Katherine; et al., eds. (2007). Biographical Encyclopedia of Astronomers (Online ed.). New York: Springer. p. 787. ISBN 978-0-387-30400-7.
8. Hayles, N. Katherine (1984). The Cosmic Web: Scientific Field Models and Literary Strategies in the Twentieth Century. Cornell University Press. p. 46. ISBN 978-0-8014-1742-9.
9. Falconer, K. J. (2013). Fractals: A Very Short Introduction. Oxford University Press. p. 119. ISBN 978-0-19-967598-2.
10. Bardon, Adrian (2013). A Brief History of the Philosophy of Time. Oxford University Press. p. 68. ISBN 978-0-19-930108-9.
11. Safra, Jacob E.; Yeshua, Ilan (2003). Encyclopædia Britannica (New ed.). Chicago, Ill.: Encyclopædia Britannica. p. 665. ISBN 978-0-85229-961-6.
12. Encyclopedia of Earth and Physical Sciences. New York: Marshall Cavendish. 1998. p. 1203. ISBN 9780761405511.
13. "Historical note: Oskar Minkowski (1858-1931). An outstanding master of diabetes research". 2006.
14. Greenspan 2005, pp. 42–43.
15. Schmadel, Lutz D. (2007). "(12493) Minkowski". Dictionary of Minor Planet Names – (12493) Minkowski. Springer Berlin Heidelberg. p. 783. doi:10.1007/978-3-540-29925-7_8614. ISBN 978-3-540-00238-3.
16. Dickson, L. E. (1909). "Review: Diophantische Approximationen. Eine Einführung in die Zahlentheorie von Hermann Minkowski" (PDF). Bull. Amer. Math. Soc. 15 (5): 251–252. doi:10.1090/s0002-9904-1909-01753-7.
17. Dickson, L. E. (1914). "Review: Geometrie der Zahlen von Hermann Minkowski". Bull. Amer. Math. Soc. 21 (3): 131–132. doi:10.1090/s0002-9904-1914-02597-2.
18. Wilson, E. B. (1915). "Review: Gesammelte Abhandlungen von Hermann Minkowski". Bull. Amer. Math. Soc. 21 (8): 409–412. doi:10.1090/s0002-9904-1915-02658-3.