Special Relativity: Lorentz Transformations
The Lorentz Transformations
Special Relativity is founded on the Lorentz Transformations and the relationship between the motion of matter and its ellipsoidal shape. Thus it is important to hear what Lorentz has to say on the subject;
The simplest course is certainly to consider the electrons themselves as wholly immutable, as perfectly rigid spheres, with a constant uniformly distributed surface charge. .. But, unfortunately, it is at variance with our theorem. ... It is for this reason that I have examined what becomes of the theory, if the electrons themselves are considered as liable to the same changes of dimensions as the bodies in which they are contained. ... the explanation of Michelson's experimental result, ... admit, for moving bodies, only a contraction, determined by the coefficient in the direction of the line of motion. The electrons themselves become flattened ellipsoids.
This would enable us to predict that no experiment made with a terrestrial source of light will ever show us an influence of the Earth's motion.
It is clear that, since the observer is unconscious of these changes, ( the contraction of a measuring rod in the direction of motion), relying on his rod, he will not find the true shape of bodies. He will take for a sphere what really is an ellipsoid,
Attention must now be drawn to a remarkable reciprocity that has been pointed out by Albert Einstein. ... Let us now imagine that each observer and (one is moving with constant velocity relative to the other) is able to see the system to which the other belongs, ... It will be clear by what has been said that the impressions received by the two observers and would be alike in all respects. It would be impossible to tell which of them moves or stands still with respect to the ether. ... This is a point which Albert Einstein has laid particular stress on, in a theory in which he starts from what he calls the principle of relativity.
I cannot speak here of the many highly interesting applications which Albert Einstein has made of this principle. His results concerning electromagnetic and optical phenomena agree in the main with those which we have obtained in the preceding pages, the chief difference being that Albert Einstein simply postulates what we have deduced, ... from the fundamental equations of the electromagnetic field. By doing so, he may certainly take credit for making us see in the negative result of experiments like those of Michelson, Rayleigh and Brace, not a fortuitous compensation of opposing effects, but the manifestation of a general and fundamental principle.
Yet, I think, something may also be claimed in the favour of the form in which I have presented the theory. I cannot but regard the ether, which can be the seat of an electromagnetic field with its energy and its vibrations, as endowed with a certain degree of substantiality, however different it may be from all ordinary matter. (Lorentz, 1906)
Thus Lorentz explained the Null results of the Michelson Morley experiment as being due to a change in ellipsoidal shape of matter (and its fields) with motion;
In order to explain this absence of any effect of the Earth's translation, I have ventured the hypothesis, that the dimensions of a solid body undergo slight change when it moves through the ether. (Lorentz, 1906)
Most profoundly, Lorentz first deduced the foundations of Albert Einstein's Relativity from the assumption of a rigid Space (ether), and that the cause of the electromagnetic field effect that he was using was in fact vibrations in this Space/Ether.
Though Albert Einstein related relative motions of matter only to other matter and not back to an absolute Space like Lorentz did, (which is mathematically simpler) the important point is that the Logic of Relativity is founded on, and completely consistent with, an Absolute Space. (Contrary to current opinions)
From Lorentz's purely mathematical foundation Albert Einstein then developed his Theory of Relativity, which assumed that matter existed as a spherical spatially extended field which changes ellipsoidal shape with motion and thus also with acceleration (which leads to the ellipsoidal geometry which underpins General Relativity and gravitation).
Albert Einstein took one further step than Lorentz though, and assumed (like Leibniz and Mach) that all motion of matter was relative only to other matter, he writes;
It has, of course, been known since the days of the ancient Greeks that in order to describe the movement of a body, a second body is needed to which the movement of the first is referred. (Albert Einstein, 1919)
By doing this Albert Einstein effectively renounced the concept of a fundamental Space separate from matter (as a field), as he explains below;
Since the field exists even in a vacuum, should one conceive of the field as state of a 'carrier', or should it rather be endowed with an independent existence not reducible to anything else? In other words, is there an 'aether' which carries the field; the aether being considered in the undulatory state, for example, when it carries light waves? The question has a natural answer: Because one cannot dispense with the field concept, it is preferable not to introduce in addition a carrier with hypothetical properties. (Albert Einstein, 1950)
Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses its meaning.
The field thus becomes an irreducible element of physical description, irreducible in the same sense as the concept of matter (particles) in the theory of Newton. (Albert Einstein, 1954)
Albert Einstein had many valid reasons for asserting that matter is spherically spatially extended, and thus to reject the concept of the particle (which always had the problem of explaining action-at-a-distance);
According to general relativity, the concept of space detached from any physical content (matter, objects) does not exist. The physical reality of space is represented by a field whose components are continuous functions of four independent variables - the co-ordinates of space and time. Since the theory of general relatively implies the representation of physical reality by a continuous field, the concept of particles or material points cannot play a fundament part, nor can the concept of motion. The particle can only appear as a limited region in space in which the field strength or the energy density are particularly high. (Albert Einstein, 1950)
Though most of Albert Einstein's discussion of Space is in terms of matter interactions described by fields, it is important to realise that Albert Einstein actually knew that Space must somehow exist and have properties that caused these force fields, he writes;
Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. (Albert Einstein, Leiden Lecture, 1920)
In ending this summary of Special Relativity, it is important to acknowledge the great power of this mathematical theory, as Albert Einstein explains (for it leads directly to Albert Einstein's famous E=mc^2).
The heuristic method of the special theory of relativity is characterized by the following principle: only those equations are admissible as an expression of natural laws which do not change their form when the co-ordinates are changed by means of the Lorentz transformation (covariance of equations with respect to the Lorentz transformations). This method led to the discovery of the necessary connection between momentum and energy, between electric and magnetic field strength, electrostatic and electrodynamic forces, inert mass and energy; thus the number of independent concepts and fundamental equations was reduced. (Albert Einstein, 1934)
The Two Postulates of Special Relativity (Albert Einstein, 1905)1. ... the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.
A co-ordinate system that is moved uniformly and in a straight line relative to an inertial system is likewise an inertial system. By the 'special principle of relativity' is meant the generalization of this definition to include any natural event whatever: thus, every universal law of nature which is valid in relation to a co-ordinate system C must also be valid, as it stands, in relation to a co-ordinate system C' which is in uniform translatory motion relative to C. (Albert Einstein, 1954)
And therefore the Velocity of Light (as one of the laws of electrodynamics) has the same measured value in all inertial (non-accelerated) reference frames.
2. The second principle, on which the special theory of relativity rests, is the 'principle of constant velocity of light in vacuo.' This principle asserts that light in vacuo always has a definite velocity of propagation (independent of the state of motion of the observer or of the source of the light). The confidence which physicists place in this principle springs from the successes achieved by the electrodynamics of Maxwell and Lorentz. (Albert Einstein, 1954)
Albert Einstein (1905) cleverly combined the work of Faraday, Maxwell and Lorentz to propose the 'Theory of Special Relativity' which described the effects of relative Motion (inertial or non-accelerated) on the properties of matter. His famous postulate being that the laws of nature (mechanics and electrodynamics) are the same for all observers irrespective of their motion (non-accelerated), which leads to the further postulate that the velocity of light must always be measured to be the same irrespective of motion.
What these two postulates logically say is that if you measure the velocity of light c to have a particular value, then irrespective of which inertial (non-accelerated) reference frame you are moving in, you will always measure the velocity of light c to have the same value. This same measurement for the velocity of light is an experimental fact. But this does not mean that the velocity of light in Space is constant, only that it is always measured to be the same (the constant velocity of light is a further theoretical assumption).
Fig: 1. Pythagoras' Theorem is Caused by the Spherical shape of Matter. Further, three dimensional space and spherical space are equivalent, as it takes three variables to describe the surface of a sphere. In fact the cause of three dimensional space is simply that matter interacts spherically.
Einstein correctly realized that matter was spherically spatially extended, and thus interacted with other matter spherically (this being the cause of Pythagoras' Theorem).
From the latest results of the theory of relativity it is probable that our three dimensional space is also approximately spherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry. (Einstein, 1954)
But Einstein did not actually know how matter existed in Space;
The theory of relativity leads to the same law of motion without requiring any special hypothesis whatsoever as to the structure and behavior of the electron. (Einstein, 1954)
His theory is empirically (a posteriori) founded from observation of how matter 'pushes' other matter around, thus his 'representation' of matter as spherical force fields.
Albert Einstein's Metric equation is simply Pythagoras' Theorem applied to the three spatial co-ordinates, and equating them to the displacement of a ray of light ct.
Special relativity is still based directly on an empirical law, that of the constancy of the velocity of light.
dx^2 + dy^2 + dz^2 =(ct)^2 where ct is the distance traveled by light c in time t.
The fact that such a metric is called Euclidean is connected with the following. The postulation of such a metric in a three dimensional continuum is fully equivalent to the postulation of the axioms of Euclidean Geometry. The defining equation of the metric is then nothing but the Pythagorean theorem applied to the differentials of the co-ordinates. (Albert Einstein, 1934)
In the special theory of relativity those co-ordinate changes (by transformation) are permitted for which also in the new co-ordinate system the quantity (ct)^2 (fundamental invariant dS^2) equals the sum of the squares of the co-ordinate differentials. Such transformations are called Lorentz transformations. (Albert Einstein, 1934)
1. http://www.Spaceandmotion.com/Physics-Albert-Einstein-Theory-Relativity.htm - Philosophy and Metaphysics of Einstein's Special and General Relativity