Hermann Minkowski’s Spacetime: The Theory that Einstein Overlooked

“Society is founded on hero worship”, wrote Thomas Carlyle (1795 – 1881) in his 1840 lecture on “Hero as Divinity”—and the society of physicists is no different.  Among physicists, the hero is the genius—the monomyth who journeys into the supernatural realm of high mathematics, engages in single combat against chaos and confusion, gains enlightenment in the mysteries of the universe, and returns home to share the new understanding.  If the hero is endowed with unusual talent and achieves greatness, then mythologies are woven, creating shadows that can grow and eclipse the truth and the work of others, bestowing upon the hero recognitions that are not entirely deserved.

      “Gentlemen! The views of space and time which I wish to lay before you … 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.”

Herman Minkowski (1908)

The greatest hero of physics of the twentieth century, without question, is Albert Einstein.  He is the person most responsible for the development of “Modern Physics” that encompasses:

  • Relativity theory (both special and general),
  • Quantum theory (he invented the quantum in 1905—see my blog),
  • Astrophysics (his field equations of general relativity were solved by Schwarzschild in 1916 to predict event horizons of black holes, and he solved his own equations to predict gravitational waves that were discovered in 2015),
  • Cosmology (his cosmological constant is now recognized as the mysterious dark energy that was discovered in 2000), and
  • Solid state physics (his explanation of the specific heat of crystals inaugurated the field of quantum matter). 

Einstein made so many seminal contributions to so many sub-fields of physics that it defies comprehension—hence he is mythologized as genius, able to see into the depths of reality with unique insight. He deserves his reputation as the greatest physicist of the twentieth century—he has my vote, and he was chosen by Time magazine in 2000 as the Man of the Century.  But as his shadow has grown, it has eclipsed and even assimilated the work of others—work that he initially criticized and dismissed, yet later embraced so whole-heartedly that he is mistakenly given credit for its discovery.

For instance, when we think of Einstein, the first thing that pops into our minds is probably “spacetime”.  He himself wrote several popular accounts of relativity that incorporated the view that spacetime is the natural geometry within which so many of the non-intuitive properties of relativity can be understood.  When we think of time being mixed with space, making it seem that position coordinates and time coordinates share an equal place in the description of relativistic physics, it is common to attribute this understanding to Einstein.  Yet Einstein initially resisted this viewpoint and even disparaged it when he first heard it! 

Spacetime was the brain-child of Hermann Minkowski.

Minkowski in Königsberg

Hermann Minkowski was born in 1864 in Russia to German parents who moved to the city of Königsberg (King’s Mountain) in East Prussia when he was eight years old.  He entered the university in Königsberg in 1880 when he was sixteen.  Within a year, when he was only seventeen years old, and while he was still a student at the University, Minkowski responded to an announcement of the Mathematics Prize of the French Academy of Sciences in 1881.  When he submitted is prize-winning memoire, he could have had no idea that it was starting him down a path that would lead him years later to revolutionary views.

A view of Königsberg in 1581. Six of the seven bridges of Königsberg—which Euler famously described in the first essay on topology—are seen in this picture. The University is in the center distance behind the castle.

The specific Prize challenge of 1881 was to find the number of representations of an integer as a sum of five squares of integers.  For instance, every integer n > 33 can be expressed as the sum of five nonzero squares.  As an example, 42 = 22 + 22 + 32 + 32 + 42,  which is the only representation for that number.  However, there are five representation for n = 53

The task of enumerating these representations draws from the theory of quadratic forms.  A quadratic form is a function of products of numbers with integer coefficients, such as ax2 + bxy + cy2 and ax2 + by2 + cz2 + dxy + exz + fyz.  In number theory, one seeks to find integer solutions for which the quadratic form equals an integer.  For instance, the Pythagorean theorem x2 + y2 = n2 for integers is a quadratic form for which there are many integer solutions (x,y,n), known as Pythagorean triplets, such as

The topic of quadratic forms gained special significance after the work of Bernhard Riemann who established the properties of metric spaces based on the metric expression

for infinitesimal distance in a D-dimensional metric space.  This is a generalization of Euclidean distance to more general non-Euclidean spaces that may have curvature.  Minkowski would later use this expression to great advantage, developing a “Geometry of Numbers” [1] as he delved ever deeper into quadratic forms and their uses in number theory.

Minkowski in Göttingen

After graduating with a doctoral degree in 1885 from Königsberg, Minkowski did his habilitation at the university of Bonn and began teaching, moving back to Königsberg in 1892 and then to Zurich in 1894 (where one of his students was a somewhat lazy and unimpressive Albert Einstein).  A few years later he was given an offer that he could not refuse.

At the turn of the 20th century, the place to be in mathematics was at the University of Göttingen.  It had a long tradition of mathematical giants that included Carl Friedrich Gauss, Bernhard Riemann, Peter Dirichlet, and Felix Klein.  Under the guidance of Felix Klein, Göttingen mathematics had undergone a renaissance. For instance, Klein had attracted Hilbert from the University of Königsberg in 1895.  David Hilbert had known Minkowski when they were both students in Königsberg, and Hilbert extended an invitation to Minkowski to join him in Göttingen, which Minkowski accepted in 1902.

The University of Göttingen

A few years after Minkowski arrived at Göttingen, the relativity revolution broke, and both Minkowski and Hilbert began working on mathematical aspects of the new physics. They organized a colloquium dedicated to relativity and related topics, and on Nov. 5, 1907 Minkowski gave his first tentative address on the geometry of relativity.

Because Minkowski’s specialty was quadratic forms, and given his understanding of Riemann’s work, he was perfectly situated to apply his theory of quadratic forms and invariants to the Lorentz transformations derived by Poincaré and Einstein.  Although Poincaré had published a paper in 1906 that showed that the Lorentz transformation was a generalized rotation in four-dimensional space [2], Poincaré continued to discuss space and time as separate phenomena, as did Einstein.  For them, simultaneity was no longer an invariant, but events in time were still events in time and not somehow mixed with space-like properties. Minkowski recognized that Poincaré had missed an opportunity to define a four-dimensional vector space filled by four-vectors that captured all possible events in a single coordinate description without the need to separate out time and space. 

Minkowski’s first attempt, presented in his 1907 colloquium, at constructing velocity four-vectors was flawed because (like so many of my mechanics students when they first take a time derivative of the four-position) he had not yet understood the correct use of proper time. But the research program he outlined paved the way for the great work that was to follow.

On Feb. 21, 1908, only 3 months after his first halting steps, Minkowski delivered a thick manuscript to the printers for an article to appear in the Göttinger Nachrichten. The title “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern” (The Basic Equations for Electromagnetic Processes of Moving Bodies) belies the impact and importance of this very dense article [3]. In its 60 pages (with no figures), Minkowski presents the correct form for four-velocity by taking derivatives relative to proper time, and he formalizes his four-dimensional approach to relativity that became the standard afterwards. He introduces the terms spacelike vector, timelike vector, light cone and world line. He also presents the complete four-tensor form for the electromagnetic fields. The foundational work of Levi Cevita and Ricci-Curbastro on tensors was not yet well known, so Minkowski invents his own terminology of Traktor to describe it. Most importantly, he invents the terms spacetime (Raum-Zeit) and events (Erignisse) [4].

Minkowski’s four-dimensional formalism of relativistic electromagnetics was more than a mathematical trick—it uncovered the presence of a multitude of invariants that were obscured by the conventional mathematics of Einstein and Lorentz and Poincaré. In Minkowski’s approach, whenever a proper four-vector is contracted with itself (its inner product), an invariant emerges. Because there are many fundamental four-vectors, there are many invariants. These invariants provide the anchors from which to understand the complex relative properties amongst relatively moving frames.

Minkowski’s master work appeared in the Nachrichten on April 5, 1908. If he had thought that physicists would embrace his visionary perspective, he was about to be woefully disabused of that notion.

Einstein’s Reaction

Despite his impressive ability to see into the foundational depths of the physical world, Einstein did not view mathematics as the root of reality. Mathematics for him was a tool to reduce physical intuition into quantitative form. In 1908 his fame was rising as the acknowledged leader in relativistic physics, and he was not impressed or pleased with the abstract mathematical form that Minkowski was trying to stuff the physics into. Einstein called it “superfluous erudition” [5], and complained “since the mathematics pounced on the relativity theory, I no longer understand it myself! [6]”

With his collaborator Jakob Laub (also a former student of Minkowski’s), Einstein objected to more than the hard-to-follow mathematics—they believed that Minkowski’s form of the pondermotive force was incorrect. They then proceeded to re-translate Minkowski’s elegant four-vector derivations back into ordinary vector analysis, publishing two papers in Annalen der Physik in the summer of 1908 that were politely critical of Minkowski’s approach [7-8]. Yet another of Minkowski’s students from Zurich, Gunnar Nordström, showed how to derive Minkowski’s field equations without any of the four-vector formalism.

One can only wonder why so many of his former students so easily dismissed Minkowski’s revolutionary work. Einstein had actually avoided Minkowski’s mathematics classes as a student at ETH [5], which may say something about Minkowski’s reputation among the students, although Einstein did appreciate the class on mechanics that he took from Minkowski. Nonetheless, Einstein missed the point! Rather than realizing the power and universality of the four-dimensional spacetime formulation, he dismissed it as obscure and irrelevant—perhaps prejudiced by his earlier dim view of his former teacher.

Raum und Zeit

It is clear that Minkowski was stung by the poor reception of his spacetime theory. It is also clear that he truly believed that he had uncovered an essential new approach to physical reality. While mathematicians were generally receptive of his work, he knew that if physicists were to adopt his new viewpoint, he needed to win them over with the elegant results.

In 1908, Minkowski presented a now-famous paper Raum und Zeit at the 80th Assembly of German Natural Scientists and Physicians (21 September 1908).  In his opening address, he stated [9]:

“Gentlemen!  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.”

To illustrate his arguments Minkowski constructed the most recognizable visual icon of relativity theory—the space-time diagram in which the trajectories of particles appear as “world lines”, as in Fig. 1.  On this diagram, one spatial dimension is plotted along the horizontal-axis, and the value ct (speed of light times time) is plotted along the vertical-axis.  In these units, a photon travels along a line oriented at 45 degrees, and the world-line (the name Minkowski gave to trajectories) of all massive particles must have slopes steeper than this.  For instance, a stationary particle, that appears to have no trajectory at all, executes a vertical trajectory on the space-time diagram as it travels forward through time.  Within this new formulation by Minkowski, space and time were mixed together in a single manifold—spacetime—and were no longer separate entities.

Fig. 1 The First “Minkowski diagram” of spacetime.

In addition to the spacetime construct, Minkowski’s great discovery was the plethora of invariants that followed from his geometry. For instance, the spacetime hyperbola

is invariant to Lorentz transformation in coordinates.  This is just a simple statement that a vector is an entity of reality that is independent of how it is described.  The length of a vector in our normal three-space does not change if we flip the coordinates around or rotate them, and the same is true for four-vectors in Minkowski space subject to Lorentz transformations. 

In relativity theory, this property of invariance becomes especially useful because part of the mental challenge of relativity is that everything looks different when viewed from different frames.  How do you get a good grip on a phenomenon if it is always changing, always relative to one frame or another?  The invariants become the anchors that we can hold on to as reference frames shift and morph about us. 

Fig. 2 Any event on an invariant hyperbola is transformed by the Lorentz transformation onto another point on the same hyperbola. Events that are simultaneous in one frame are each on a separate hyperbola. After transformation, simultaneity is lost, but each event stays on its own invariant hyperbola (Figure reprinted from [10]).

As an example of a fundamental invariant, the mass of a particle in its rest frame becomes an invariant mass, always with the same value.  In earlier relativity theory, even in Einstein’s papers, the mass of an object was a function of its speed.  How is the mass of an electron a fundamental property of physics if it is a function of how fast it is traveling?  The construction of invariant mass removes this problem, and the mass of the electron becomes an immutable property of physics, independent of the frame.  Invariant mass is just one of many invariants that emerge from Minkowski’s space-time description.  The study of relativity, where all things seem relative, became a study of invariants, where many things never change.  In this sense, the theory of relativity is a misnomer.  Ironically, relativity theory became the motivation of post-modern relativism that denies the existence of absolutes, even as relativity theory, as practiced by physicists, is all about absolutes.

Despite his audacious gambit to win over the physicists, Minkowski would not live to see the fruits of his effort. He died suddenly of a burst gall bladder on Jan. 12, 1909 at the age of 44.

Arnold Sommerfeld (who went on to play a central role in the development of quantum theory) took up Minkowski’s four vectors, and he systematized it in a way that was palatable to physicists.  Then Max von Laue extended it while he was working with Sommerfeld in Munich, publishing the first physics textbook on relativity theory in 1911, establishing the space-time formalism for future generations of German physicists.  Further support for Minkowski’s work came from his distinguished colleagues at Göttingen (Hilbert, Klein, Wiechert, Schwarzschild) as well as his former students (Born, Laue, Kaluza, Frank, Noether).  With such champions, Minkowski’s work was immortalized in the methodology (and mythology) of physics, representing one of the crowning achievements of the Göttingen mathematical community.

Einstein Relents

Already in 1907 Einstein was beginning to grapple with the role of gravity in the context of relativity theory, and he knew that the special theory was just a beginning. Yet between 1908 and 1910 Einstein’s focus was on the quantum of light as he defended and extended his unique view of the photon and prepared for the first Solvay Congress of 1911. As he returned his attention to the problem of gravitation after 1910, he began to realize that Minkowski’s formalism provided a framework from which to understand the role of accelerating frames. In 1912 Einstein wrote to Sommerfeld to say [5]

I occupy myself now exclusively with the problem of gravitation . One thing is certain that I have never before had to toil anywhere near as much, and that I have been infused with great respect for mathematics, which I had up until now in my naivety looked upon as a pure luxury in its more subtle parts. Compared to this problem. the original theory of relativity is child’s play.

By the time Einstein had finished his general theory of relativity and gravitation in 1915, he fully acknowledge his indebtedness to Minkowski’s spacetime formalism without which his general theory may never have appeared.


[1] H. Minkowski, Geometrie der Zahlen. Leipzig and Berlin: R. G. Teubner, 1910.

[2] Poincaré, H. (1906). “Sur la dynamique de l’´electron.” Rendiconti del circolo matematico di Palermo 21: 129–176.

[3] H. Minkowski, “Die Grundgleichungen für die electromagnetischen Vorgänge in bewegten Körpern,” Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, pp. 53–111, (1908)

[4] S. Walter, “Minkowski’s Modern World,” in Minkowski Spacetime: A Hundred Years Later, Petkov Ed.: Springer, 2010, ch. 2, pp. 43-61.

[5] L. Corry, “The influence of David Hilbert and Hermann Minkowski on Einstein’s views over the interrelation between physics and mathematics,” Endeavour, vol. 22, no. 3, pp. 95-97, (1998)

[6] A. Pais, Subtle is the Lord: The Science and the Life of Albert Einstein. Oxford, 2005.

[7] A. Einstein and J. Laub, “Electromagnetic basic equations for moving bodies,” Annalen Der Physik, vol. 26, no. 8, pp. 532-540, Jul (1908)

[8] A. Einstein and J. Laub, “Electromagnetic fields on quiet bodies with pondermotive energy,” Annalen Der Physik, vol. 26, no. 8, pp. 541-550, Jul (1908)

[9] Minkowski, H. (1909). “Raum und Zeit.” Jahresbericht der Deutschen Mathematikier-Vereinigung: 75-88.

[10] D. D. Nolte, Introduction to Modern Dynamics : Chaos, Networks, Space and Time, 2nd ed. Oxford: Oxford University Press, 2019.