# A Short History of Hyperspace

Hyperspace by any other name would sound as sweet, conjuring to the mind’s eye images of hypercubes and tesseracts, manifolds and wormholes, Klein bottles and Calabi Yau quintics.  Forget the dimension of time—that may be the most mysterious of all—but consider the extra spatial dimensions that challenge the mind and open the door to dreams of going beyond the bounds of today’s physics.

The geometry of n dimensions studies reality; no one doubts that. Bodies in hyperspace are subject to precise definition, just like bodies in ordinary space; and while we cannot draw pictures of them, we can imagine and study them.

(Poincare 1895)

Here is a short history of hyperspace.  It begins with advances by Möbius and Liouville and Jacobi who never truly realized what they had invented, until Cayley and Grassmann and Riemann made it explicit.  They opened Pandora’s box, and multiple dimensions burst upon the world never to be put back again, giving us today the manifolds of string theory and infinite-dimensional Hilbert spaces.

## August Möbius (1827)

Although he is most famous for the single-surface strip that bears his name, one of the early contributions of August Möbius was the idea of barycentric coordinates [1] , for instance using three coordinates to express the locations of points in a two-dimensional simplex—the triangle. Barycentric coordinates are used routinely today in metallurgy to describe the alloy composition in ternary alloys.

Möbius’ work was one of the first to hint that tuples of numbers could stand in for higher dimensional space, and they were an early example of homogeneous coordinates that could be used for higher-dimensional representations. However, he was too early to use any language of multidimensional geometry.

## Carl Jacobi (1834)

Carl Jacobi was a master at manipulating multiple variables, leading to his development of the theory of matrices. In this context, he came to study (n-1)-fold integrals over multiple continuous-valued variables. From our modern viewpoint, he was evaluating surface integrals of hyperspheres.

In 1834, Jacobi found explicit solutions to these integrals and published them in a paper with the imposing title “De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant; una cum variis theorematis de transformatione et determinatione integralium multiplicium” [2]. The resulting (n-1)-fold integrals are

when the space dimension is even or odd, respectively. These are the surface areas of the manifolds called (n-1)-spheres in n-dimensional space. For instance, the 2-sphere is the ordinary surface 4πr2 of a sphere on our 3D space.

Despite the fact that we recognize these as surface areas of hyperspheres, Jacobi used no geometric language in his paper. He was still too early, and mathematicians had not yet woken up to the analogy of extending spatial dimensions beyond 3D.

## Joseph Liouville (1838)

Joseph Liouville’s name is attached to a theorem that lies at the core of mechanical systems—Liouville’s Theorem that proves that volumes in high-dimensional phase space are incompressible. Surprisingly, Liouville had no conception of high dimensional space, to say nothing of abstract phase space. The story of the convoluted path that led Liouville’s name to be attached to his theorem is told in Chapter 6, “The Tangled Tale of Phase Space”, in Galileo Unbound (Oxford University Press, 2018).

Nonetheless, Liouville did publish a pure-mathematics paper in 1838 in Crelle’s Journal [3] that identified an invariant quantity that stayed constant during the differential change of multiple variables when certain criteria were satisfied. It was only later that Jacobi, as he was developing a new mechanical theory based on William R. Hamilton’s work, realized that the criteria needed for Liouville’s invariant quantity to hold were satisfied by conservative mechanical systems. Even then, neither Liouville nor Jacobi used the language of multidimensional geometry, but that was about to change in a quick succession of papers and books by three mathematicians who, unknown to each other, were all thinking along the same lines.

## Arthur Cayley (1843)

Arthur Cayley was the first to take the bold step to call the emerging geometry of multiple variables to be actual space. His seminal paper “Chapters in the Analytic Theory of n-Dimensions” was published in 1843 in the Philosophical Magazine [4]. Here, for the first time, Cayley recognized that the domain of multiple variables behaved identically to multidimensional space. He used little of the language of geometry in the paper, which was mostly analysis rather than geometry, but his bold declaration for spaces of n-dimensions opened the door to a changing mindset that would soon sweep through geometric reasoning.

## Hermann Grassmann (1844)

Grassmann’s life story, although not overly tragic, was beset by lifelong setbacks and frustrations. He was a mathematician literally 30 years ahead of his time, but because he was merely a high-school teacher, no-one took his ideas seriously.

Somehow, in nearly a complete vacuum, disconnected from the professional mathematicians of his day, he devised an entirely new type of algebra that allowed geometric objects to have orientation. These could be combined in numerous different ways obeying numerous different laws. The simplest elements were just numbers, but these could be extended to arbitrary complexity with arbitrary number of elements. He called his theory a theory of “Extension”, and he self-published a thick and difficult tome that contained all of his ideas [5]. He tried to enlist Möbius to help disseminate his ideas, but even Möbius could not recognize what Grassmann had achieved.

In fact, what Grassmann did achieve was vector algebra of arbitrarily high dimension. Perhaps more impressive for the time is that he actually recognized what he was dealing with. He did not know of Cayley’s work, but independently of Cayley he used geometric language for the first time describing geometric objects in high dimensional spaces. He said, “since this method of formation is theoretically applicable without restriction, I can define systems of arbitrarily high level by this method… geometry goes no further, but abstract science knows no limits.” [6]

Grassman was convinced that he had discovered something astonishing and new, which he had, but no one understood him. After years trying to get mathematicians to listen, he finally gave up, left mathematics behind, and actually achieved some fame within his lifetime in the field of linguistics. There is even a law of diachronic linguistics named after him. For the story of Grassmann’s struggles, see the blog on Grassmann and his Wedge Product .

## Julius Plücker (1846)

Projective geometry sounds like it ought to be a simple topic, like the projective property of perspective art as parallel lines draw together and touch at the vanishing point on the horizon of a painting. But it is far more complex than that, and it provided a separate gateway into the geometry of high dimensions.

A hint of its power comes from homogeneous coordinates of the plane. These are used to find where a point in three dimensions intersects a plane (like the plane of an artist’s canvas). Although the point on the plane is in two dimensions, it take three homogeneous coordinates to locate it. By extension, if a point is located in three dimensions, then it has four homogeneous coordinates, as if the three dimensional point were a projection onto 3D from a 4D space.

These ideas were pursued by Julius Plücker as he extended projective geometry from the work of earlier mathematicians such as Desargues and Möbius. For instance, the barycentric coordinates of Möbius are a form of homogeneous coordinates. What Plücker discovered is that space does not need to be defined by a dense set of points, but a dense set of lines can be used just as well. The set of lines is represented as a four-dimensional manifold. Plücker reported his findings in a book in 1846 [7] and expanded on the concepts of multidimensional spaces published in 1868 [8].

## Ludwig Schläfli (1851)

After Plücker, ideas of multidimensional analysis became more common, and Ludwig Schläfli (1814 – 1895), a professor at the University of Berne in Switzerland, was one of the first to fully explore analytic geometry in higher dimensions. He described multidimsnional points that were located on hyperplanes, and he calculated the angles between intersecting hyperplanes [9]. He also investigated high-dimensional polytopes, from which are derived our modern “Schläfli notation“. However, Schläffli used his own terminology for these objects, emphasizing analytic properties without using the ordinary language of high-dimensional geometry.

## Bernhard Riemann (1854)

The person most responsible for the shift in the mindset that finally accepted the geometry of high-dimensional spaces was Bernhard Riemann. In 1854 at the university in Göttingen he presented his habilitation talk “Über die Hypothesen, welche der Geometrie zu Grunde liegen” (Over the hypotheses on which geometry is founded). A habilitation in Germany was an examination that qualified an academic to be able to advise their own students (somewhat like attaining tenure in US universities).

The habilitation candidate would suggest three topics, and it was usual for the first or second to be picked. Riemann’s three topics were: trigonometric properties of functions (he was the first to rigorously prove the convergence properties of Fourier series), aspects of electromagnetic theory, and a throw-away topic that he added at the last minute on the foundations of geometry (on which he had not actually done any serious work). Gauss was his faculty advisor and picked the third topic. Riemann had to develop the topic in a very short time period, starting from scratch. The effort exhausted him mentally and emotionally, and he had to withdraw temporarily from the university to regain his strength. After returning around Easter, he worked furiously for seven weeks to develop a first draft and then asked Gauss to set the examination date. Gauss initially thought to postpone to the Fall semester, but then at the last minute scheduled the talk for the next day. (For the story of Riemann and Gauss, see Chapter 4 “Geometry on my Mind” in the book Galileo Unbound (Oxford, 2018)).

Riemann gave his lecture on 10 June 1854, and it was a masterpiece. He stripped away all the old notions of space and dimensions and imbued geometry with a metric structure that was fundamentally attached to coordinate transformations. He also showed how any set of coordinates could describe space of any dimension, and he generalized ideas of space to include virtually any ordered set of measurables, whether it was of temperature or color or sound or anything else. Most importantly, his new system made explicit what those before him had alluded to: Jacobi, Grassmann, Plücker and Schläfli. Ideas of Riemannian geometry began to percolate through the mathematics world, expanding into common use after Richard Dedekind edited and published Riemann’s habilitation lecture in 1868 [10].

## George Cantor and Dimension Theory (1878)

In discussions of multidimensional spaces, it is important to step back and ask what is dimension? This question is not as easy to answer as it may seem. In fact, in 1878, George Cantor proved that there is a one-to-one mapping of the plane to the line, making it seem that lines and planes are somehow the same. He was so astonished at his own results that he wrote in a letter to his friend Richard Dedekind “I see it, but I don’t believe it!”. A few decades later, Peano and Hilbert showed how to create area-filling curves so that a single continuous curve can approach any point in the plane arbitrarily closely, again casting shadows of doubt on the robustness of dimension. These questions of dimensionality would not be put to rest until the work by Karl Menger around 1926 when he provided a rigorous definition of topological dimension (see the Blog on the History of Fractals).

## Hermann Minkowski and Spacetime (1908)

Most of the earlier work on multidimensional spaces were mathematical and geometric rather than physical. One of the first examples of physical hyperspace is the spacetime of Hermann Minkowski. Although Einstein and Poincaré had noted how space and time were coupled by the Lorentz equations, they did not take the bold step of recognizing space and time as parts of a single manifold. This step was taken in 1908 [11] by Hermann Minkowski who claimed

“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)

For the story of Einstein and Minkowski, see the Blog on Minkowski’s Spacetime: The Theory that Einstein Overlooked.

## Felix Hausdorff and Fractals (1918)

No story of multiple “integer” dimensions can be complete without mentioning the existence of “fractional” dimensions, also known as fractals. The individual who is most responsible for the concepts and mathematics of fractional dimensions was Felix Hausdorff. Before being compelled to commit suicide by being jewish in Nazi Germany, he was a leading light in the intellectual life of Leipzig, Germany. By day he was a brilliant mathematician, by night he was the author Paul Mongré writing poetry and plays.

In 1918, as the war was ending, he wrote a small book “Dimension and Outer Measure” that established ways to construct sets whose measured dimensions were fractions rather than integers [12]. Benoit Mandelbrot would later popularize these sets as “fractals” in the 1980’s. For the background on a history of fractals, see the Blog A Short History of Fractals.

## The Fifth Dimension of Theodore Kaluza (1921) and Oskar Klein (1926)

The first theoretical steps to develop a theory of a physical hyperspace (in contrast to merely a geometric hyperspace) were taken by Theodore Kaluza at the University of Königsberg in Prussia. He added an additional spatial dimension to Minkowski spacetime as an attempt to unify the forces of gravity with the forces of electromagnetism. Kaluza’s paper was communicated to the journal of the Prussian Academy of Science in 1921 through Einstein who saw the unification principles as a parallel of some of his own attempts [13]. However, Kaluza’s theory was fully classical and did not include the new quantum theory that was developing at that time in the hands of Heisenberg, Bohr and Born.

Oskar Klein was a Swedish physicist who was in the “second wave” of quantum physicists having studied under Bohr. Unaware of Kaluza’s work, Klein developed a quantum theory of a five-dimensional spacetime [14]. For the theory to be self-consistent, it was necessary to roll up the extra dimension into a tight cylinder. This is like a strand a spaghetti—looking at it from far away it looks like a one-dimensional string, but an ant crawling on the spaghetti can move in two dimensions—along the long direction, or looping around it in the short direction called a compact dimension. Klein’s theory was an early attempt at what would later be called string theory. For the historical background on Kaluza and Klein, see the Blog on Oskar Klein.

## John Campbell (1931): Hyperspace in Science Fiction

Art has a long history of shadowing the sciences, and the math and science of hyperspace was no exception. One of the first mentions of hyperspace in science fiction was in the story “Islands in Space’, by John Campbell [15], published in the Amazing Stories quarterly in 1931, where it was used as an extraordinary means of space travel.

In 1951, Isaac Asimov made travel through hyperspace the transportation network that connected the galaxy in his Foundation Trilogy [16].

## John von Neumann and Hilbert Space (1932)

Quantum mechanics had developed rapidly through the 1920’s, but by the early 1930’s it was in need of an overhaul, having outstripped rigorous mathematical underpinnings. These underpinnings were provided by John von Neumann in his 1932 book on quantum theory [17]. This is the book that cemented the Copenhagen interpretation of quantum mechanics, with projection measurements and wave function collapse, while also establishing the formalism of Hilbert space.

Hilbert space is an infinite dimensional vector space of orthogonal eigenfunctions into which any quantum wave function can be decomposed. The physicists of today work and sleep in Hilbert space as their natural environment, often losing sight of its infinite dimensions that don’t seem to bother anyone. Hilbert space is more than a mere geometrical space, but less than a full physical space (like five-dimensional spacetime). Few realize that what is so often ascribed to Hilbert was actually formalized by von Neumann, among his many other accomplishments like stored-program computers and game theory.

## Einstein-Rosen Bridge (1935)

One of the strangest entities inhabiting the theory of spacetime is the Einstein-Rosen Bridge. It is space folded back on itself in a way that punches a short-cut through spacetime. Einstein, working with his collaborator Nathan Rosen at Princeton’s Institute for Advanced Study, published a paper in 1935 that attempted to solve two problems [18]. The first problem was the Schwarzschild singularity at a radius r = 2M/c2 known as the Schwarzschild radius or the Event Horizon. Einstein had a distaste for such singularities in physical theory and viewed them as a problem. The second problem was how to apply the theory of general relativity (GR) to point masses like an electron. Again, the GR solution to an electron blows up at the location of the particle at r = 0.

To eliminate both problems, Einstein and Rosen (ER) began with the Schwarzschild metric in its usual form

where it is easy to see that it “blows up” when r = 2M/c2 as well as at r = 0. ER realized that they could write a new form that bypasses the singularities using the simple coordinate substitution

to yield the “wormhole” metric

It is easy to see that as the new variable u goes from -inf to +inf that this expression never blows up. The reason is simple—it removes the 1/r singularity by replacing it with 1/(r + ε). Such tricks are used routinely today in computational physics to keep computer calculations from getting too large—avoiding the divide-by-zero problem. It is also known as a form of regularization in machine learning applications. But in the hands of Einstein, this simple “bypass” is not just math, it can provide a physical solution.

It is hard to imagine that an article published in the Physical Review, especially one written about a simple variable substitution, would appear on the front page of the New York Times, even appearing “above the fold”, but such was Einstein’s fame this is exactly the response when he and Rosen published their paper. The reason for the interest was because of the interpretation of the new equation—when visualized geometrically, it was like a funnel between two separated Minkowski spaces—in other words, what was named a “wormhole” by John Wheeler in 1957. Even back in 1935, there was some sense that this new property of space might allow untold possibilities, perhaps even a form of travel through such a short cut.

As it turns out, the ER wormhole is not stable—it collapses on itself in an incredibly short time so that not even photons can get through it in time. More recent work on wormholes have shown that it can be stabilized by negative energy density, but ordinary matter cannot have negative energy density. On the other hand, the Casimir effect might have a type of negative energy density, which raises some interesting questions about quantum mechanics and the ER bridge.

## Edward Witten’s 10+1 Dimensions (1995)

A history of hyperspace would not be complete without a mention of string theory and Edward Witten’s unification of the variously different 10-dimensional string theories into 10- or 11-dimensional M-theory. At a string theory conference at USC in 1995 he pointed out that the 5 different string theories of the day were all related through dualities. This observation launched the second superstring revolution that continues today. In this theory, 6 extra spatial dimensions are wrapped up into complex manifolds such as the Calabi-Yau manifold.

## Prospects

There is definitely something wrong with our three-plus-one dimensions of spacetime. We claim that we have achieved the pinnacle of fundamental physics with what is called the Standard Model and the Higgs boson, but dark energy and dark matter loom as giant white elephants in the room. They are giant, gaping, embarrassing and currently unsolved. By some estimates, the fraction of the energy density of the universe comprised of ordinary matter is only 5%. The other 95% is in some form unknown to physics. How can physicists claim to know anything if 95% of everything is in some unknown form?

The answer, perhaps to be uncovered sometime in this century, may be the role of extra dimensions in physical phenomena—probably not in every-day phenomena, and maybe not even in high-energy particles—but in the grand expanse of the cosmos.

By David D. Nolte, Feb. 8, 2023

## Bibliography:

M. Kaku, R. O’Keefe, Hyperspace: A scientific odyssey through parallel universes, time warps, and the tenth dimension.  (Oxford University Press, New York, 1994).

A. N. Kolmogorov, A. P. Yushkevich, Mathematics of the 19th century: Geometry, analytic function theory.  (Birkhäuser Verlag, Basel ; 1996).

## References:

[1] F. Möbius, in Möbius, F. Gesammelte Werke,, D. M. Saendig, Ed. (oHG, Wiesbaden, Germany, 1967), vol. 1, pp. 36-49.

[2] Carl Jacobi, “De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, quae solis quadratis variabilium constant; una cum variis theorematis de transformatione et determinatione integralium multiplicium” (1834)

[3] J. Liouville, Note sur la théorie de la variation des constantes arbitraires. Liouville Journal 3, 342-349 (1838).

[4] A. Cayley, Chapters in the analytical geometry of n dimensions. Collected Mathematical Papers 1, 317-326, 119-127 (1843).

[5] H. Grassmann, Die lineale Ausdehnungslehre.  (Wiegand, Leipzig, 1844).

[6] H. Grassmann quoted in D. D. Nolte, Galileo Unbound (Oxford University Press, 2018) pg. 105

[7] J. Plücker, System der Geometrie des Raumes in Neuer Analytischer Behandlungsweise, Insbesondere de Flächen Sweiter Ordnung und Klasse Enthaltend.  (Düsseldorf, 1846).

[8] J. Plücker, On a New Geometry of Space (1868).

[9] L. Schläfli, J. H. Graf, Theorie der vielfachen Kontinuität. Neue Denkschriften der Allgemeinen Schweizerischen Gesellschaft für die Gesammten Naturwissenschaften 38. ([s.n.], Zürich, 1901).

[10] B. Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen, Habilitationsvortrag. Göttinger Abhandlung 13,  (1854).

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

[12] Hausdorff, F.(1919).“Dimension und ausseres Mass,”Mathematische Annalen, 79: 157–79.

[13] Kaluza, Theodor (1921). “Zum Unitätsproblem in der Physik”. Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.): 966–972

[14] Klein, O. (1926). “Quantentheorie und fünfdimensionale Relativitätstheorie“. Zeitschrift für Physik. 37 (12): 895

[15] John W. Campbell, Jr. “Islands of Space“, Amazing Stories Quarterly (1931)

[16] Isaac Asimov, Foundation (Gnome Press, 1951)

[17] J. von Neumann, Mathematical Foundations of Quantum Mechanics.  (Princeton University Press, ed. 1996, 1932).

[18] A. Einstein and N. Rosen, “The Particle Problem in the General Theory of Relativity,” Phys. Rev. 48(73) (1935).

# 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.

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.

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]:

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.

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.

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.

By David D. Nolte, April 24, 2021

[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.