New Interpretation of Minkowski’s Space and Time

Abstract

This article explores Hermann Minkowski’s groundbreaking 1908 lecture Space and Time, where he  introduced his Theory of the Absolute World. This theory, which encompasses the concept of four-dimensional spacetime, proposed a multispace paradigm, suggesting that reality consists of multiple independent spacetimes. The article examines the misinterpretation of Minkowski’s theory by the scientific community and its implications for modern physics. Key concepts such as worldpoint, worldlines, and proper time are explained. 

“While there exists an unanimous consensus on the mathematical significance of spacetime for theoretical physics, for a hundred years there has been no consensus on the nature of spacetime itself.”—Vesselin Petkov[1]

License CC BY-NC-ND 4.0

Contents

1. Introduction

Hermann Minkowski (1864–1909), a prominent German mathematician and physicist, made significant contributions to the mathematical foundations of special relativity. In 1908, he showed that Albert Einstein’s special theory of relativity (1905) could be understood geometrically as a theory of four-dimensional spacetime, now known as Minkowski spacetime. He taught at German universities and was a mathematics professor to Albert Einstein in multiple courses at the Swiss Federal Polytechnic School in Zurich.

Hermann Minkowski presented his groundbreaking theory of spacetime in his famous lecture Raum und Zeit (Space and Time) at the 80th Assembly of German Natural Scientists and Physicians in Cologne on September 21, 1908.

Minkowski’s lecture was criticized for not being very clear. The lecture, while groundbreaking in its content, was reportedly difficult for many in the audience to understand. His presentation style was highly mathematical and abstract, which made it challenging for many of the physicists and mathematicians present to follow.

The concepts he introduced were revolutionary and quite different from the prevailing understanding of space and time, which added to the difficulty. Even Albert Einstein, whose work on special relativity formed the basis for Minkowski’s spacetime concept, initially found Minkowski’s mathematical approach to be overly complicated.

However, Minkowski made his lecture intentionally unclear because it contains a cryptic but extremely important discovery: Reality is a multispace, not a spacetime continuum as we believe.

For the past century, mathematicians and physicists worldwide have upheld the belief that Minkowski’s lecture supported Einstein’s theories of relativity, and they commend him for it. Yet, contrary to popular belief, the theory that Minkowski presented in his 1908 Cologne lecture was NOT intended to support Einstein’s relativity. On the contrary.

Minkowski’s Cryptic Legacy

The Theory of the Absolute World introduced by Minkowski in the 1908 lecture diverges from Einstein’s relativity. In fact, Minkowski proposed an independent theory where reality comprises a multispace manifold with an infinite number of spacetimes. Minkowski described this shift as truly radical. Essentially, this theory transitions human understanding from the single-space paradigm we all know and understand to a multispace reality, making Einstein’s relativities outdated.

However, at the time of the lecture, the theory was not yet complete. As Minkowski explained, he had independently reached the same conclusions as Einstein but chose not to publish them, as he wanted to first develop the mathematical structure in all its splendor [2].

This affirmation is corroborated by Max Born, who had attended the seminar on electron theory co-organized by Minkowski and Hilbert in the summer semester of 1905. Many years later (in 1959), Born wrote:

“I remember that Minkowski occasionally alluded to the fact that he was engaged with the Lorentz transformations, and that he was on the track of new interrelationships.1[3]”

Intrigued about the time Minkowski could have dedicated to developing his theory before 1905, I conducted a search at the Jewish National and University Library, where are archived the manuscripts of Minkowski and Einstein. In Box 9, folder 7 (1904), I discovered four pages containing his sketches of orthogonal spaces [4].

To my surprise, it appears that Minkowski developed his theory well before 1904. To conceive the intricate three-dimensional combination of the orthogonal planes P and P’ depicted in all four sketches, he must have formulated his 2D diagram prior to grasping the concept of 3D multispace. However, he only presented this 2D diagram during the Cologne lecture in 1908.

One certainty remains: he could not have spontaneously envisioned the two orthogonal planes. Hence, he likely understood the multispace concept years before 1908. Simultaneously, we comprehend the challenge that hindered him from fully elucidating the mathematical structure in its entirety. To achieve this, he would have needed to unveil the intricate shape of the 3D Multispace Diagram, as I envisioned it, which only occurred in 2012. Refer to my design in Figure 2.

It is understandable now why Minkowski could not fully articulate the radical paradigm shift in its entirety. So, to secure his scientific priority, Minkowski opted to introduce the theory using cryptic terms, such as worldpoint (Weltpunkt), worldline (Weltlinie), and world (Welt). This approach allowed him to obscure the true significance of these terms in the multispace setup by providing accurate yet insufficient definitions.

His genius yet simple encryption performed wonders. Physicists and mathematicians worldwide, not suspecting such a radical paradigm shift, thought they grasped his theory and conducted research supporting relativity based on their own perceptions. Consequently, they deluded themselves for more than a century, rendering many of their scientific efforts futile.

2. The Cologne Lecture

When Minkowski presented his Cologne lecture in September 1908, he took care to ensure the scientific priority of his own theory while simultaneously concealing enough details to prevent other scientists from fully understanding his discovery.

Let us now examine how Minkowski crafted his cryptic theory.2

Minkowski began his lecture with a bold declaration to make it clear right from the beginning that he was going to introduce a revolutionary theory:

“Gentlemen! The views on space and time which I wish to lay before you have sprung from the soil of experimental physics. Therein lies their strength. Their tendency is radical.3”

Minkowski stated their tendency is radical, but intentionally avoided elaborating that his lecture was extremely radical. He was fully aware that the theory he was about to unveil would introduce a new multispace paradigm of reality.

He continued by proposing a fundamental change in our views of space and time:

“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 on its own.”

This statement reflects Minkowski’s conviction that his reformulation of the concepts of space and time into a spacetime manifold was not just a mathematical exercise but was deeply rooted in physical observations, and most importantly, was completely different from the unique spacetime continuum paradigm of reality.

He continues by explaining:

“First of all, I would like to show how it might be possible, setting out from the adopted mechanics of the present day, along a purely mathematical line of thought, to arrive at changed ideas of space and time.”

Further on, Minkowski details his thinking by showing that to illustrate these relations he is using a mathematical graphic method:

“Let x, y, z be orthogonal coordinates for space, and let t denote time. The objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place. But I respect the dogma that space and time each have an independent meaning. I will call a point in space at a given time, i.e. a system of values x, y, z, t, a worldpoint (ein Weltpunkt).”

We observe that Minkowski introduced the spacetime concept by naming this manifold a worldpoint (ein Weltpunkt) to designate it as the smallest spacetime building block of reality.

Note that this simple concept is essential to the understanding of all of Minkowski’s multispace theory. The Weltpunkt is for his his theory like an atom is for matter or a brick used to build a house. Minkowski builds all of his theory beginning with it, and the Weltpunkt is the key element for anyone wanting to comprehend the existence of reality.

He continues by generalizing it to the whole of reality:

“The manifold of all possible systems of values x, y, z, t will be called the world (die Welt).”

So, the reality is what he called the world. Then, Minkowski makes an important clarification of this description of reality:

“Not to leave a yawning void anywhere, we will imagine that everywhere and at all times there is something perceptible. To avoid saying matter or electricity, I will use for this something the word substance (das Wort Substanz). We fix our attention on the substantial point which is at the worldpoint x, y, z, t, and imagine that we are able to recognize this substantial point at any other time.”

In this way, Minkowski clarifies that he does not talk about a mathematical abstraction but about a truly physical spacetime (the substantial point).

Minkowski continues by expanding the size of this point-like manifold into a real spacetime:

“Corresponding to a time element dt, we have the variations dx; dy; dz of the space coordinates of this substantial point. Then we obtain, as an image, so to speak, of the eternal career of the substantial point, a curve in the world, a worldline (eine Weltlinie), the points of which can be referred to uniquely to the parameter t running from -∞ to +∞.”

With this explanation, Minkowski establishes that a worldline is an independent spacetime manifold inside the world manifold.4

Minkowski used this last definition to announce the breakthrough discovery that reality is a multispace manifold[5], and not a unique spacetime continuum as Einstein later assessed. And this manifold consists of many spacetime manifolds:

“The whole world seems to resolve itself into such worldlines, and I would fain anticipate myself by saying that in my opinion the laws of physics might find their most perfect expression as interrelations between these worldlines.”

Minkowski’s groundbreaking declaration becomes clearer when we translate it into modern language:

“The whole reality seems to resolve itself into such independent spacetimes, and I would like to say now that, in my opinion, the laws of physics find their most perfect expression in the interrelationships between these independent spacetimes.”

Let us say it again: The reality is based on interrelationships between independent spacetimes, not between objects in the universe’s space, as we now think.

Finally, to be sure that he unequivocally transmits the right message even if he probably believed that no one would understand his unfinished theory, Minkowski ends the first part of the lecture with a conclusion that is cryptic in the present paradigm but completely understandable in the multispace paradigm:

“We would then have in the world no longer the space, but an infinite number of spaces, analogously as there are in three-dimensional space an infinite number of planes. Three-dimensional geometry becomes a chapter in four-dimensional physics. You see why I said at the outset that space and time are to fade away into shadows, and that only a world in itself will subsist.”5

Not understanding the extreme importance of Minkowski’s declaration, the vast majority of physicists did not take his hint seriously, considering it nothing more than a grandiose announcement of the mathematician’s own relativity theory.

References

Notes

1. Ich erinnere mich, daß Minkowski gelegentlich Andeutungen machte, daß er sich mit den Lorentz-Transformationen beschäftigte und neuen Zusammenhängen auf der Spur sei.

2. The saying goes: “Traduttore, traditore” (“Translator, traitor”). Following his passing, numerous English translations of his lecture emerged, each imbued with a unique essence based on the translator’s interpretation of Minkowski’s ideas. Consequently, translating his work posed a formidable challenge for all. Faced with this dilemma, I opted for a 2009 bilingual edition in German and English of the lecture to personally verify the accuracy of the translation. In certain instances, I presented both versions for clarity ([14]).

3. M. H.! Die Anschauungen über Raum und Zeit, die ich Ihnen entwickeln möchte, sind auf experimentell-physikalischem Boden erwachsen. Darin liegt ihre Stärke. Ihre Tendenz ist eine radikale.

4. Note that in the present understanding of physics, a worldline is a path that an object traces through 4-dimensional spacetime. It represents the history of an object’s location in space at each instant of time. In fact, physicists invented its meaning because they could not admit that they had no idea what Minkowski meant by it.

5. Hiernach würden wir dann in der Welt nicht mehr den Raum, sondern unendlich viele Räume haben, analog wie es im dreidimensionalen Räume unendlich viele Ebenen gibt. Die dreidimensionale Geometrie wird ein Kapitel der vierdimensionalen Physik. Sie erkennen, weshalb ich am Eingange sagte, Raum und Zeit sollen zu Schatten herabsinken und nur eine Welt an sich bestehen.