3.2.3  Minkowski Space

After Newton introduced his universal law of gravitation and the three laws of mechanics, as we reviewed in section 16 of chapter II, some scientists and philosophers of science criticized Newton’s definitions of absolute space and time, pointing out that they are essentially metaphysical concepts and do not have any scientific meaning. In 1902, Poincaré published detailed philosophical discussions on the relativity of space and time, and on the conventionality of distant simultaneity, in addition to many other remarks on comparing non-Euclidean and Euclidean geometry.

Poincaré attempted to reformulate space and time, but his efforts were completed in 1907 by Hermann Minkowski (1864-1909) based on the work of many previous mathematicians of the 19th century who contributed to group theory and projective geometry. Using similar methods, Minkowski succeeded in formulating a geometrical interpretation of the Lorentz transformations.

Minkowski combined time with the three-dimensional Euclidean space into a four-dimensional manifold, where the space-time interval between any two events is independent of the inertial frame of reference in which they are recorded. Minkowski initially developed his space for Maxwell’s equations of electromagnetism, but the mathematical structure was soon shown to be an immediate consequence of the two postulates of Special Relativity.

In his earlier work in 1907, Minkowski first followed Poincaré in representing space and time together in Euclidean space with the complex form:, but he noted that it is a four-dimensional non-Euclidean manifold. The new space differs from four-dimensional Euclidean space, because it treats time differently from the three spatial dimensions, so that all frames of reference agree on the total space-time interval between events, even when the individual components in the Euclidean space and time differ due to length contraction and time dilation.

In 1908, aware of the fundamental restatement of the theory which he had made, Minkowski said in his address to the 80th Assembly of German Natural Scientists and Physicians:

“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.” Petkov (2010)

In the normal three-dimensional Euclidean space, which called: Galilean space, or simply: space, the isometry group is the Euclidean group. This group is generated by rotations, reflections and translations that preserve the regular Euclidean distance. When time is amended as a fourth dimension, the further transformations of translations in time and Galilean boosts are added, and the group of all these transformations is called the Galilean group. This latter group includes transformations of the Euclidean group, in which distance is purely spatial, but then time differences are separately preserved as well. In the Minkowski space-time of Special Relativity, because space and time are interwoven, distance transformations and closely related to time differences, which produces new groups called Lorentzian group and Poincaré group, that will be described further in sections 2.4 and 2.5.

The Galilean space-time and Minkowski space-time are actually the same manifolds, but they differ in what further structures are defined on them. The former has the Euclidean distance function and time, separately, together with inertial frames whose coordinates are related by Galilean transformations, while the latter has the Minkowski metric together with inertial frames whose coordinates are related by Poincaré transformations.

In 1910, Sommerfeld applied Poincaré’s and Minkowski’s complex representation of space-time to combine non-collinear velocities by spherical geometry, in order to derive the velocity-addition formula that relates the velocities of objects in different reference frames. He replaced Minkowski’s matrix notation by an elegant vector notation, in terms of “four vector” representation. Other important contributions were made by Max von Laue (1879-1960) in 1911 and 1913. He used the space-time formalism to create a relativistic theory of deformable bodies and an elementary particle theory. He extended Minkowski’s expressions for electromagnetic processes to all possible forces, thereby clarified the concept of mass-energy equivalence that will be discussed in section 2.6. Laue also showed that non-electrical forces are needed to ensure the proper Lorentz transformation properties, and for the stability of matter. Subsequent writers, dispensed with the imaginary time coordinate, and reformulated Special Relativity in explicitly non-Euclidean form, using the concept of rapidity that was previously introduced in 1911 by Alfred Robb (1873-1936). This continued since the years before World War I, and was employed in most relativity textbooks of the 20th century.