The Duality of Time Theory

In 1905, Albert Einstein (1879-1955) published his famous paper “Zur Elektrodynamik bewegter Koerper” (“On the Electrodynamics of Moving Bodies”), in which he demonstrated the inconsistency of Newtonian mechanics with Maxwell’s equations of electromagnetism. To correct the situation, Einstein introduced major changes to the mechanics of motion when the velocity is close to the speed of light. This later became known as the theory of Special Relativity. This theory is based on two postulates:

1.The laws of physics are invariant in all inertial frames of reference. 2.The speed of light in vacuum is the same for all observers, regardless of their motion relative to the source of light.

This theory of relativity was later denoted as “Special” because Einstein developed a more “General” version to incorporate non-inertial, or accelerated, frames of reference and gravity, as we shall discuss in section 3.

Unfortunately, Einstein’s paper contains no direct references to any of the previous works on the subject, but historians of science have tried to find out the possible theoretical or experimental research that influenced Einstein and led him to postulate his theory. Einstein himself later stated that his thinking was influenced by the empiricist philosophers David Hume (1711-1776) and Ernst Mach (1838-1916). The various negative aether drift experiments, that we have described in section 1 above, must have had a great influence on him, but he denied any significant influence of the Michelson-Morley experiment. However, one of Einstein’s friends, Maurice Solovine (1875-1958) reported that Poincaré’s Principle of Relativity, described in section 1.2, was closely studied and discussed by Einstein and his friends in the years before he published his paper in 1905.

Einstein also stated that Lorentz’s theory of 1895, known as Maxwell-Lorentz Electrodynamics, and also the Fizeau experiment, had considerable influence on his thinking. He said in 1909 and 1912 that he borrowed the Principle of the Constancy of the Speed of Light from Lorentz’s stationary aether, but he recognized that this principle, together with the Principle of Relativity, made any reference to aether unnecessary. He wrote in 1907, and in later papers, that the apparent contradiction between those principles can be resolved if it is admitted that Lorentz’s local time is not an auxiliary quantity, but can simply be defined as time, and that it is connected with signal velocity.

Before Einstein, Poincaré also developed similar physical interpretations of local time, as proposed by Lorentz, and he noticed the connection with signal velocity, but, contrary to Einstein, he continued to argue that clocks at rest in the stationary aether show the true time, while clocks in inertial motion relative to the aether show only the apparent time.

Eventually, near the end of his life in 1953, Einstein described the advantages of his theory over that of Lorentz as follows: “There is no doubt, that the special theory of relativity, if we regard its development in retrospect, was ripe for discovery in 1905. Lorentz had already recognized that the transformations named after him are essential for the analysis of Maxwell’s equations, and Poincaré deepened this insight still further. Concerning myself, I knew only Lorentz’s important work of 1895 ... but not Lorentz’s later work, nor the consecutive investigations by Poincaré. In this sense my work of 1905 was independent. ... The new feature of it was the realization of the fact that the bearing of the Lorentz transformation transcended its connection with Maxwell’s equations and was concerned with the nature of space and time in general. A further new result was that the ‘Lorentz invariance’ is a general condition for any physical theory. This was for me of particular importance because I had already previously found that Maxwell’s theory did not account for the micro-structure of radiation and could therefore have no general validity.” Born (2012)

Special Relativity led to many peculiar consequences, many of which have been experimentally verified, including: length contraction, time dilation, relativistic mass, mass-energy equivalence, a cosmological speed limit and relativity of simultaneity. The conventional notion of absolute universal time was replaced with the notion of relative time that is dependent on the frame of reference and position in space. Rather than two separate invariant temporal and spatial intervals between two events, there is one invariant space-time interval.

Therefore, Special Relativity replaced the Galilean transformations of the Newtonian Classical Mechanics, with the Lorentz transformations. Time and space cannot be defined separately from each other, rather, they are interwoven into a single continuum known as space-time, so that events that occur at the same time for one observer may occur at different times for another.

In addition to that, some strange predictions and paradoxes are associated with Special Relativity, such as the twin paradox and Ehrenfest paradox, which will be briefly reviewed in section 2.7 below.

Shortly after its publication, in 1906, Max Planck (1858-1947) was the first who publicly defended the theory, and described it as a generalization of Lorentz’s theory. Referring to it as “Lorentz-Einstein-Theory”, he gave it the name “relative theory”, but it was Alfred Bucherer (1863-1927) who later called it: “theory of relativity”. Einstein himself and many others continued to refer simply to the new method as the “relativity principle”, describing it as a “union of Lorentz’s theory and the relativity principle”, but Poincaré’s contributions were rarely mentioned in the first years after 1905. In 1915, Einstein used “special theory of relativity”, to distinguish it from the new General Relativity that he had just formulated, as we will review in section 3.

In 1909, Planck compared the implications of the relativity of time with the revolution by the Copernican system, and after its development by Minkowski into a space-time theory, most theoretical physicists accepted Special Relativity. In 1912, Wilhelm Wien (1864-1928) recommended both Lorentz and Einstein for the Nobel Prize in Physics, but the Nobel committee decided not to award the prize for Special Relativity.

Special Relativity is now considered as an application of linear algebra, which was not fully developed at the time. Arthur Cayley (1821-1895) had already used matrix notation and unified modern vector space and transformation theory as linear algebra, but it had not come into widespread use until much later. Minkowski later used Cayley’s matrix calculus notation in 1908, in formulating relativistic electrodynamics, but his work was later replaced by Arnold Sommerfeld (1868-1951) using vector notation. As Poincaré already showed before, the standard Lorentz transformations become translations in imaginary time that can be expressed in terms of hyperbolic space. As we noticed above, however, this mathematical trick of using complex numbers does not have any philosophical justification without considering space itself as the real part of time, in order that the normal time becomes imaginary or latent to it. With the genuine complex-time geometry, the Duality of Time relies on the same hyperbolic techniques to formulate both Special and General Relativity, in addition to making space-time intrinsically granular, thus allowing Quantum Gravity to be expressed in terms of simple Euclidean geometry, without invoking Riemannian manifolds, as it will be discussed in chapter V.