"Gentlemen!"

Perhaps there are also ladies in the audience, but very few. It is 1908, September 21. Wilhelmian Germany is near its zenith. The Eightieth Congress of Naturalists and Physicians is meeting in Cologne. At the speaker's podium: Hermann Minkowski.

He is 44 years old, holder of a chaired professorship which the University of Göttingen has created expressly for his benefit. Among professional mathematicians, he is famous for his unconventional "geometry of numbers" and his defiantly visual approach to algebra in an age which increasingly disdains visualisation. Lately, though, his interests have turned increasingly toward physics.

He has less than four months to live -- he will die of appendicitis on January 12. But had he lived to be a hundred, this moment in Cologne would still be the one for which he will always be remembered. He is about to forever change the way people think of the universe itself.

And he knows it.

"Gentlemen! The conceptions of time and space which I wish to develop here have arisen on the basis of experimental physics. Therein lies their strength. Their tendency is radical. From now on space-in-itself and time-in-itself are destined to be reduced to shadows, and only a sort of union of the two will retain an independent existence."

The sponsor of the Congress, die Gesellschaft Deutscher Naturforscher und Ärzte, has been at the heart of German Kultur since the days of Naturphilosophie. Its founder was Lorenz Oken, Romantic searcher for the Theory of Everything, whom we encountered in an earlier blog. If Oken's shade can hear Minkowski's words, it must feel that the GDNÄ's grand Transcendental mission is at last being fulfilled.

"I wish first to show how from the mechanics now generally accepted we might arrive at a change in our ideas of space and time."

Minkowski is paradoxically both a revolutionary and a traditionalist. He is among the first to recognise that Einstein's relativity, proposed in 1905, will require science to reject views of the world accepted since classical antiquity; nevertheless, he considers this rejection not an absolute break with the past but a natural extension of principles also established in ancient times. Accordingly, he begins his speech with a novel presentation of mechanics designed to show that much of relativity is already present in Newton.

"The equations of Newton's mechanics show a double invariance. Their form is maintained, first, if we subject our system of original coordinates in space to any change of position ; second, if we change its state of motion, that is to say, impart to it any uniform translation ; neither does the zero-point of time play any part."

The first invariance mentioned here is usually called "translation symmetry", and can be interpreted in two ways. In an "active" sense, the laws of physics should not depend on the position of the observer: if one moves around in the universe, one does not encounter a multitude of local jurisdictions, each with its own rules, nor is there any special place which can be called the centre of the universe -- the "relativity principle" of Copernicus.

In a "passive" sense, the absence of a unique centre of the world means that only relative distances should appear in the laws of physics. For example, the gravitational force between two objects depends on how far apart they are, say 100 meters. It does not change if one gives their coordinates as 0 meters and 100 meters instead of, say, 1,000,038 meters and 1,000,138 meters.

The second or "Galilean" invariance means that, according to Galileo and Newton, it is impossible to determine which of two uniformly-moving objects is "really" moving:

Animation posted by Luc Tremblay [3 min.]

"Invariance" is becoming an important part of mathematical physics in the years around 1900, a trend which by 1915 will result in one of the last great triumphs of Nineteenth-Century classical physics: the proof by Minkowski's former student Emmy Noether that the invariance of physics under (certain types of) transformations leads to the existence of conserved quantities: for example, translation invariance -- the laws of physics are the same here as they are over there -- implies conservation of momentum.

A similar invariance in time -- the laws of physics are the same today as they were yesterday -- implies conservation of energy. Such "time-translation invariance" is of course what Minkowski means by "neither does the zero-point of time play any part": in an active sense, the laws of physics do not change with time; in a passive sense, they do not care whether we use a calendar which begins at the birth of Christ, the Hegira, or the French Revolution.

"We are accustomed to consider the axioms of geometry as settled before we approach the axioms of mechanics, and therefore these two invariances are seldom mentioned together. Each of them represents a certain group of transformations which transform the differential equations of mechanics back into themselves."

"Group theory" gives the concept of "symmetry" or "invariance" mathematical rigour. A detailed understanding of group theory is not needed to follow Minkowski's paper; simply recognise that when he refers to a group, he means a symmetry. For those who are curious, the following video, from a mathematician who for some reason goes by "Singing Banana" (thus proving that he does not live in the Long Nineteenth-Century), gives a qualitative taste of formal group-theory:

[4 minutes]

"The existence of the first group [translational symmetry] is regarded as a fundamental property of space. It is usually preferred to treat the second group with contempt in order to pass lightly over the fact that we can never describe from physical phenomena whether the space we have assumed to be at rest is not after all in a state of uniform translation."

After all, mediæval philosophers assumed the Earth was at rest. What if the entire universe is actually moving, in a uniform and therefore -- according to Newton -- undetectable way?

"Thus the two groups have an entirely separate existence, side by side. Their quite heterogeneous character may have discouraged their combination ; but precisely this combination into one group gives us food for thought.

"We shall try to illustrate these relations graphically. Let x, y, z be rectangular coordinates of space and let t represent time. As they occur in our experience places and times are always combined. No one has ever observed a place except at a time, nor a time except in a place. But here I am still respecting the dogma that space and time have each an independent significance ... "

The translation ironically illustrates Minkowski's point. The original reads: "Ich respektiere aber noch das Dogma ... " Does he mean, "But here I am still respecting ..."? Could one not equally say "But for now I am still respecting ..."?

"But here I am still respecting the dogma that space and time have each an independent significance. I shall call a point in space at a definite time, that is, a system of values, x, y, z, t, a world-point. The multiplicity of all possible systems of values x, y, z, t, I shall call the world.

"I might boldly sketch four world-axes on the blackboard."

Unfortunately, Minkowski's sketch is not included in the published versions of his talk. Shown here is a tesseract construction by 4d44 for Wikimedia Commons.

"Even one such axis consists merely of vibrating molecules and travels with the earth in space, thus alone furnishing us with sufficient food for abstract thought ; the somewhat greater abstraction involved in the number four does not disturb the mathematician.

"In order not to have an empty void anywhere we shall assume that there is something perceptible everywhere and at all times. To avoid the terms 'matter' or 'electricity' we shall call this something substance.

"Let us direct our attention to the substance-point at the world-point x, y, z, t, and imagine that we are able to recognize this substance-point at every other time."

This is a rather Germanic way of saying that he intends to follow the life history of a particular particle, or "substance-point", as it evolves in time.

"Let the changes dx, dy, dz, of the space coordinates of this substance-point correspond to an element of time dt."

That is, as the substance-point or particle moves about in space, a time interval also passes.

"We thus obtain as a representation, so to speak, of the eternal course of the substance-point a curve in the world, a world-line whose points can be determined uniquely in terms of a parameter t from − ∞ to + ∞."

World-line of a planet in a circular orbit. Graphic by Stannerd

"The whole world stands resolved into such world-lines, and I wish at once to make the fundamental assertion that according to my opinion physical laws may find their most complete expression as mutual relations among these world lines."