“The primary focus of physical statistics is not solely on spatial distributions but rather on their progression through time.” — Richard von Mises

In the realm of Newtonian mechanics, space and time were regarded as separate entities. The three dimensions of space—length, width, and height—formed the foundation of Euclidean geometry, which was later extended by Nicolai Lobachevsky (1793–1856) in the 1820s through hyperbolic geometry.

Time, similarly, was often viewed as a philosophical concept, distinct from the physical phenomena of the universe. It was not until the 17th and 18th centuries that scientists and physicists began to take the notion of time seriously.

Contrary to popular belief, Albert Einstein (1879–1955) was not the first to grasp the concept of a four-dimensional spacetime continuum. Initially, he struggled with this perspective and even dismissed it. Over time, however, he embraced it, leading many to mistakenly credit him with its discovery.

The notion that space and time are interconnected was first introduced by the German mathematician Hermann Minkowski (1864–1909) in 1908.

Minkowski began his academic journey at the Gymnasium in Königsberg, where he was influenced by the works of great mathematicians like Dirichlet, Dedekind, and Gauss. It was also there that he formed a lasting friendship with David Hilbert (1862–1943). During this period, Minkowski was primarily focused on issues concerning quadratic forms.

He earned his PhD in 1885 with a thesis titled “Studies on Square Shapes, Determining the Number of Different Forms Containing a Given Genus”. Following this, he took on a professorship at ETH Zürich, where Einstein attended some of his lectures.

In 1902, Minkowski moved with his family to the University of Göttingen, aided by Hilbert. This marked the beginning of his interest in mathematical physics. Hilbert's influence was significant, as Minkowski engaged in various seminars to learn about the latest developments in electrodynamics.

He began to realize that the non-Euclidean geometries formulated by Lobachevsky, Bolyai, and Riemann could better accommodate the work of Hendrik Lorentz and Einstein.

Minkowski proposed the idea of merging space and time into a single framework, which later provided a mathematical foundation for the general theory of relativity.

In 1907, mathematicians at Göttingen hosted a colloquium on relativity where Minkowski presented his insights on the geometry of relativity.

This integrated view of space and time can be illustrated through a simple example: when traveling from Rome to Paris, one navigates through physical space while simultaneously moving through time, as the journey takes a finite duration.

In another example, if friends plan to meet outdoors, selecting an appropriate location in space must be paired with choosing a suitable time. Missing either aspect would result in incomplete information. Thus, time and space are inseparable.

Furthermore, if you travel near the speed of light, the distance covered may be vast, yet only a few minutes may pass in time. Hence, the faster one travels through space, the slower time progresses.

“Mathematics, so to speak, was to be master, and physical theory could be made to bow to the master.”

Minkowski utilized his understanding of quadratic forms to elaborate on the Lorentz transformation established by Einstein and Poincaré. Poincaré, in one of his papers, indicated that the Lorentz transformation could be seen as a generalized rotation in four dimensions but did not fully explore the integration of space and time.

Conversely, Minkowski defined a four-dimensional vector space, encompassing all possible events without the need to separate time from space.

In 1908, he submitted a thorough manuscript consisting of 60 pages on a four-dimensional approach to relativity titled “The Basic Equations for Electromagnetic Processes of Moving Bodies”.

This work introduced concepts such as the light cone, world line, spacelike vector, and timelike vector. The four-dimensional framework was not merely mathematical manipulation; it revealed the existence of invariants in relativistic electromagnetics.

In September of that year, he delivered his renowned paper Raum und Zeit at the 80th Assembly of German Natural Scientists and Physicians in Cologne, beginning with a provocative statement:

“The ideas of space and time that I wish to present have emerged from experimental physics, and therein lies their strength. They are radical. From now on, space by itself, and time by itself, will fade into mere shadows, and only a union of the two will maintain an independent reality.”

Minkowski introduced spacetime diagrams for clearer representations and to bolster his arguments. He plotted spatial dimensions on the x-axis and temporal dimensions (ct) on the y-axis, referring to particle trajectories as world lines. These diagrams illustrated the position of objects over time. A stationary observer, for instance, would still have a vertical trajectory, signifying their continuous passage through time.

He demonstrated that light photons consistently travel along a 45-degree angle, termed a lightlike curve composed of null vectors. In contrast, massive particles follow a timelike curve that is steeper than 45 degrees. The area below 45 degrees represents spacelike vectors, which are unreachable since one would need to exceed the speed of light to access them. All physical phenomena can be described using this light cone.

Minkowski’s Raum und Zeit is, in essence, static, flat, and infinite.

He concluded his presentation with the following remarks:

“I believe that the unyielding validity of the world postulate [i.e., the relativity postulates] is the true nucleus of an electromagnetic worldview, revealed by Lorentz and further illuminated by Einstein, now stands fully visible.”

Morris Kline (1908–1992), an American mathematician and historian, observed in his review:

“A pivotal aspect of the paper is the differing methodologies employed by mathematical physicists compared to theoretical physicists. In a 1908 paper, Minkowski reformulated Einstein’s 1905 work by introducing four-dimensional (spacetime) non-Euclidean geometry, a step that Einstein initially underestimated.”

He further noted:

“What is more crucial is the philosophy that Minkowski, Hilbert—who collaborated with Minkowski for several years—Felix Klein, and Hermann Weyl embraced, which emphasized that mathematical elegance and harmony should govern the acceptance of new physical truths.”

“Mathematics was to be the master, and physical theory could be made to bow to the master.”

Einstein always regarded mathematics as an abstract tool, often disconnected from reality, despite his profound physical intuition.

He was notably hesitant to acknowledge Minkowski’s contributions, famously stating, “Since mathematics has invaded relativity theory, I can no longer comprehend it!” He frequently dismissed it as “superfluous erudition.”

Time, however, validated Minkowski's insights!

Minkowski greatly influenced Einstein's thought process. According to Leo Corry:

“In the early phase of his scientific journey, Albert Einstein viewed mathematics merely as a tool aiding physical intuition. Over time, he came to see mathematics as the bedrock of scientific creativity. A significant factor in this transformation was the impact of two eminent German mathematicians: David Hilbert and Hermann Minkowski.”

Minkowski's groundbreaking work not only assisted Einstein in generalizing gravity and accounting for accelerated reference frames but also contributed to various fields, including string theory and quantum gravity.

Einstein recognized Minkowski’s formalism as foundational for his general theory of relativity.