III.1 Hyper Symmetry and Relativity

The concept of aether was used in ancient philosophy and medieval science as a thin transparent material that fills the upper spheres where planets move and revolve. This concept was invoked again in the late 18th century as a medium for the propagation of light waves, just like sound which needed a medium to propagate. Nonetheless, this concept was contradictory, because it required the existence of an invisible and infinite material that does not have any interaction with physical objects. On the one hand, aether had to be a perfect fluid in order to fill all space, but on the other hand it has to be millions of times more rigid than steel, in order to support the high frequencies of light waves. It also had to be massless and without viscosity, otherwise it would affect the motion of planets. Additionally, it had to be completely transparent, non-dispersive, incompressible, and continuous at a very small scale. Therefore, the existence of aether had been questioned and criticized by most physicists, but there was no other theory to replace it. However, many other complex experiments had been carried out in the late 19th century to try to detect the motion of the Earth through the aether, but all returned null results.

Subsequently, some alternative theories were proposed to explain aether dragging. Lorentz aether theory, for example, provided an elegant solution to how the motion of an absolute aether could be undetectable. In his theory of aether, Lorentz introduced the idea of length contraction, and Poincar later was able to express it in mathematical equations, but shortly after that, the new theory of Special Relativity generated the same equations without ever referring to aether. Therefore, the success of Special and General Relativity marked the end of aether theories, although Einstein himself said that his model could itself be thought of as an aether, since empty space in Relativity has its own physical properties. In 1951, Dirac reintroduced the concept of aether in an attempt to address the perceived deficiencies in current models, thus in 1999 one proposed model of dark energy has been named: quintessence, or the fifth fundamental force. As a scalar field, the quintessence is considered as some form of dark energy which could provide an alternative postulate to explain the observed accelerating rate of the expansion of the Universe, rather than Einstein s original postulate of cosmological constant.

In 1900, Henri Poincar (1854-1912) recognized that local time is actually indicated by moving clocks. He built on Lorentz work and formulated his new conceptions mathematically. He then introduced the Principle of Relativity and tried to harmonize it with Electrodynamics. He declared that simultaneity is only a convenient convention which depends on the speed of light, whereby the constancy of the speed of light would be a useful postulate for making the laws of nature simple and easy. In June and July of 1905, Poincar declared that the Relativity Principle is a general law of nature, including gravitation. He corrected some mistakes of Lorentz and proved the Lorentz covariance of the electromagnetic equations. However, he used the notion of an aether again as a perfectly undetectable medium, and distinguished between apparent and real time, but most historians of science argue that he failed to invent Special Relativity.

At the beginning of 1906, independently of Einstein, Poincar published a substantially extended work of his 1905 paper, the so-called: Palermo paper , in which he spoke literally of the Postulate of Relativity and showed that the transformations are a consequence of the principle of least action. He also developed the properties of the Poincar stresses, and demonstrated in more detail the group characteristics of the transformations, which he called the Lorentz group, that will be described in section III.1.1.

Poincar also noticed that Lorentz transformation is a rotation in four-dimensional space about the origin, by introducing  as a fourth imaginary coordinate, but he continued to refer to an aether as undetectable medium. He also described coordinates and phenomena as local or apparent for moving observers and real for observers at rest in the aether. By these various conceptions, Poincar anticipated much of Einstein s methods and terminology, but he did not invent Special Relativity.

Poincar 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 Poincar 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 the normal three-dimensional Euclidean space, which is called: Galilean space, or simply: space, the isometry group is theEuclidean 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 Poincar group, that will be described further in section III.1.1.

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 Poincar transformations.

In 1910, Sommerfeld applied Poincar 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 III.2. 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.