History of Lorentz transformations
The Lorentz transformation linked as the Galileo transformation, the coordinates of an event in a particular inertial system with the coordinates of the same event in another inertial system which is V moves in the positive x - direction at the speed relative to the first system. However, in contrast to the Galilean transformation, it includes in addition to the principle of relativity, the constancy of the speed of light in all inertial systems and thus forms the mathematical basis for the special theory of relativity.
First approximations to this transformation were published by Woldemar Voigt (1887 ) and Hendrik Lorentz (1892, 1895 ), where these authors the unprimed system was considered dormant than in the ether, and the "moving" primed system was identified with the earth. This transformation was by Joseph Larmor (1897, 1900) and Lorentz (1899, 1904) and completed by Henri Poincaré (1905 ), which gave its name to the transformation brought into its modern form. Albert Einstein ( 1905) was finally the equations derived from a few basic assumptions and showed the relationship between the transformation with fundamental changes in terms of space and time.
In this article, the historical expressions are replaced by modern, with the Lorentz transformation
And the Lorentz factor:
V is the relative velocity between the members, and c is the speed of light.
Voigt (1887 )
In a theoretical study of the Doppler effect of transverse waves in an elastic incompressible transmission medium or light ether Voigt developed (1887 ) the following transformation [A 1], which was the wave equation unchanged and in modern notation had the form:
If the right-hand sides of these equations are multiplied by a scale factor, the result is the formula of the Lorentz transformation. The reason for this is that the electromagnetic Lorentz invariant equations, not only, but also scale invariant and even konformalinvariant. The Lorentz transformation can be provided for example by the scale factor: [A 2] [A 3]
With yields the Voigt transformation, and with the Lorentz transformation. As, later, Poincaré and Einstein showed the transformations, however, are a group, what is the condition for the compatibility with the principle of relativity only at symmetrical and form. The Voigt - transformation is not symmetric and thus violated the principle of relativity. In contrast, the Lorentz transformation is valid outside of electrodynamics for all laws of nature. In some solutions, such as in the calculation of radiation phenomena in empty space, both transformations, however, lead to the same result.
As far as is known, Voigt's work was cited by 1887 in the literature before the advent of the modern theory of relativity for the last time in 1903, in the April issue of the Annals of Physics of the Viennese theoretician Emil Kohl. Lorentz explained in 1909 [A 3] and 1912, [A 4] that Voigt's transformation " equivalent" is to transform with the above scale factor in his own work of 1904, and that he, if he had known these equations it in his electron theory would use can. Hermann Minkowski praised Voigt's performance in 1908 in space and time [A 5] and in a discussion: [A 6]
" Minkowski: Historically, I will add that the transformations that play the main role in the principle of relativity, are first mathematically treated by Voigt in 1887. Voigt has then been drawn with their help inferences with respect to the Dopplersche principle. Voigt: Mr. Minkowski reminiscent of an old work of mine. It involves application of the Doppler 's principle, occur in specific parts, but not due to the electromagnetic, but due to the elastic theory of light. However, at that time have already found some of the same conclusions that are later recovered from the electromagnetic theory. "
Heaviside, Thomson, Searle (1888, 1889, 1896)
1888 studied Oliver Heaviside [A 7], the properties of moving charges in accordance with Maxwell's electrodynamics. He calculates, among other things, that anisotropies should occur in the electric field of moving charges in accordance with the following formula:
Based on these discovered Joseph John Thomson (1889 ) [A 8] a method to perform calculations for moving charges to facilitate substantial way, using the following mathematical transformation:
As a result, non-uniform electromagnetic wave equations are transformed in a Poisson equation. Finally noticed George Frederick Charles Searle (1896 ), [A 9] that Heaviside's ' expression for moving charges in a deformation of the electric field leads, which he called " Heaviside ellipsoid " with an axial ratio of designated.
Lorentz (1892, 1895)
Lorentz developed in 1892 [A 10] the main features of a model, later called the Lorentz ether theory, in which the ether is completely at rest in the ether which the speed of light in all directions has the same value. To be able to calculate the optics of moving bodies, Lorentz introduced the following auxiliary variables for the transformation of the ether system in a relatively moving to a system:
Where the Galilean transformation is. Now, while the "true" time for rest in the ether systems, time is a mathematical auxiliary variable, which is used for calculations of moving in the ether systems. A similar " local " has already been used by Voigt, Lorentz, however, later stated to have possessed no knowledge of his work at this time. Likewise, it is unknown whether it was the work of Thomson common.
1895 [A 11] he developed the Lorentz electrodynamics much more systematic further, with a fundamental concept for sizes to was the " theorem of corresponding states ." It follows from it that a moving observer in the ether almost the same observations in his " fictitious" ( electromagnetic ) field makes like a " real " in his field in the ether observer at rest. That is, as long as the rates are comparatively low relative to the ether, Maxwell's equations for all observers have the same shape. For the electrostatics of moving bodies he used the transformations that changed the dimensions of the body as follows:
As an additional and independent hypothesis claiming Lorentz ( 1892b, 1895) (without proof as he admitted ) that the intermolecular forces and thus material body in a similar way be deformed, and led to the declaration of the Michelson -Morley experiment, the length contraction one. [A 12] the same hypothesis had already been erected in 1889 by George FitzGerald, whose considerations were based on the work of Heaviside. But while for Lorentz length contraction was a real, physical effect, meant to him the local time for the time being only an agreement or useful method of calculation. In contrast, for the optics of moving bodies, he used the transformations:
With the help of local Lorentz could explain the aberration of light, the Doppler effect and the measured the Fizeau experiment depending on the speed of light in moving fluids. What is important is that Lorentz and Larmor later the transformations always formulated in two steps. First, the Galilean transformation, and then separated from it only the extension to " fictitious" electromagnetic system with the help of the Lorentz transformation. Their symmetrical shape given the equations only by Poincaré.
Larmor (1897, 1900)
Larmor knew at that time that the Michelson - Morley experiment was exactly enough to show movement-induced effects on the size, and so he sought a transformation which is valid also for these quantities. Although he followed a very similar pattern as Lorentz, he went beyond his work from 1895 and modified the equations, so that he was the first to complete Lorentz transformation was set up in 1897 [A 13] and some of the clutter 1900 [ A 14 ]:
He showed that the Maxwell equations are invariant under this two -step transformation were (although he carried out the proof only for the second order, not for all orders ). Larmor noted, moreover, that when an electric constitution of the molecules is assumed, the length contraction are a consequence of the transformation. He was also the first to notice a sort of time dilation as a consequence of the equations for periodic processes of moving objects run in the ratio slower than stationary objects.
Lorentz (1899, 1904)
Also Lorentz initiated in 1899 [A 15], the complete transformation by the extension of the theorem of corresponding states from. However, he used the indeterminate factor as a function of. How Larmor noticed a kind of Lorentz time dilation, because he realized that the vibrations of an oscillating electron, which moves relative to the ether, run slower. By further negative ether wind experiments ( Trouton -Noble experiment, experiments of Rayleigh and Brace ) was Lorentz forced to formulate his theory so that the ether wind effects remain to be undetectable in all sizes. To this end, he wrote the Lorentz transformation in the same form as Larmor, with a time being indeterminate factor:
In this context, he led in 1899 from the correct equations for the velocity dependence of the electromagnetic mass and he concluded in 1904 that this transformation must be applied to all the forces of nature, not just electric, and therefore the length contraction is a consequence of the transformation.
Likewise sat Lorentz firmly that there must be at too, and showed in the further course this is only the case if is generally given - from which he concluded that the Lorentz contraction can occur only in the direction of movement. He formulated the proper Lorentz transformation, however, did not reach the covariance of the equations of transformation of charge density and speed. He therefore wrote in 1912 about his work from 1904: [A 4]
" It will be noticed that I have not quite achieved the transformation equations of Einstein's relativity theory in this paper. [ ... ] With this circumstance the Awkward depends some further considerations in this work together. "
Poincaré (1900, 1905)
Neither Lorentz nor Larmor gave a clear interpretation of the origin of the local time. 1900 [ A 16 ] [A 17] However, the local Poincaré interpreted as the result of a study carried out with light signals synchronization. He assumed that two observers moving in the ether A and B synchronize their clocks with optical signals. Because they believe to be in peace, they go from a constant speed of light in all directions so that they now take into account only the light travel times and have to cross their signals to verify the synchronization of the clocks. In contrast, from the perspective of a stationary observer in the ether runs a clock signal to the counter, and the other runs away him. The watches are so out of sync ( relativity of simultaneity ), but show for sizes of the first order in, only the local time. Since the moving observer, but no means have to decide whether they are moving or not, they will not notice anything from the error. Poincaré therefore understood in contrast to the Lorentz local time as well as the length contraction as a real physical effect. Similar observations were later also by Emil Cohn (1904 ) [A 18] and Max Abraham (1905 ) [A 19] given.
On June 5, 1905 (published on June 9 ) [A 2] simplified Poincaré equations ( which are equivalent to those of Larmor and Lorentz are ) and gave them their modern symmetrical form, which he opposed to Larmor and Lorentz, the Galilean transformation in the new transformation integrated directly. Apparently, Poincaré was the work of Larmor unknown, for he referred only to Lorentz and used therefore first the term " Lorentz transformation " ( the term " Lorentz transformation " already used in 1900 by Emil Cohn for the 1895 equations of Lorentz was ):
And vice versa:
He set the speed of light equal to 1 and as Lorentz, he showed that must be set. However, Poincaré could derive this general from the fact that all the transformations only on this condition form a symmetric group, which is necessary for the validity of the relativity principle. He further showed that Lorentz ' transformations of the application does not fully meet the principle of relativity. Poincaré, however, could fully demonstrate the Lorentz covariance of the Maxwell - Lorentz equations in addition to the demonstration of the group property of the transformation.
A significantly expanded version of this document from July 1905 (published January 1906 ) [A 20 ] was the realization that the combination is invariant; he introduced the term as a fourth coordinate of a four-dimensional space; he used this four-vectors before Minkowski; he showed that the transformations are a consequence of the principle of least action; and he demonstrated more fully than before the group property, which he coined the name Lorentz group ( " Le groupe de Lorentz "). But how Lorentz remained Poincaré continue to distinguish between "true" coordinates on the airwaves and " apparent" coordinates for moving observer.
Einstein ( 1905)
On June 30, 1905 (published September 1905 ) [A 21] Einstein presented within the framework of special relativity, a radically new interpretation and derivation of the transformation, which was based on two postulates, namely the principle of relativity and the principle of the constancy of the speed of light. While Poincaré had only derived the original Lorentz local time in 1895 by optical synchronization, Einstein was able to derive a similar Synchronisatiosmethode the entire transformation, and show here that operational considerations related to insufficient space and time, and no ether is needed for it ( whether Einstein was influenced by Poincaré's method of synchronization is not known ). In contrast to Lorentz, who regarded the local time just as mathematical trick, Einstein showed that the "effective" coordinates of the Lorentz transformation are equal coordinates of inertial frames in fact. This was in some ways already shown by Poincaré so, but the latter still differed between " true " and "apparent " time. Einstein's formal version of transformation was identical to the Poincaré wherein Einstein but not the speed of light equal to 1 sat. Likewise, Einstein could show that the transformations form a group:
From the transformations Einstein could in turn derived effects such as time dilation, length contraction, Doppler effect, aberration of light, or the relativistic velocity addition as a consequence of this new understanding of space and time, without any assumptions about the structure of matter, or of a substantial ether to have to make.
Minkowski (1907-1908)
The work on the principle of relativity of Lorentz, Einstein, Planck, along with four-dimensional Poincaré's approach was continued by Hermann Minkowski in the years 1907 to 1908. His main achievement was in the four-dimensional reformulation of electrodynamics, and the geometric representation of the Lorentz transformation using Minkowski diagrams. [A 22] [A 23] [A 5]