Lorentz transformation

“ The theory of Lorentz's transformation has been developed very considerably by Einstein, Planck, and Minkowski. The transformation has become a powerful instrument of research, inasmuch as it provides a means of transition from the mathematical expressions of physical quantities connected with a system at rest to the corresponding quantities for a similar system in uniform motion. ”

“ But Einstein now showed that if we define the conception of simultaneity rigorously, taking into account the postulate of the constancy of the velocity of light, the Lorentz-transformations, which had been found empirically, followed necessarily as those equations of transformation that must hold between the co-ordinates of two systems moving uniformly and rectilinearly with respect to each other. ”

“ Now, Planck and Einstein have already announced, that all ponderomotive forces must be transformed by the Lorentz transformation in the same way, as in electrodynamics. Thus it must be possible in all areas of physics, to combine the force density with its power to a four-force. ”

“ But if OM is [Pg 68] nearly as large as OP, that is to say, if the body is moving nearly as fast as light, MP becomes very small compared to OP, and the effects become very great. The apparent increase of mass in swiftly moving particles had been observed, and the right formula had been found, before Einstein invented his special theory of relativity. In fact, Lorentz had arrived at the formulæ called the “Lorentz transformation,” which embody the whole mathematical essence of the special theory of relativity. ”

“ These facts can be more concisely expressed in these words: the system of equations (I and II) as well as the system of equations (III) (IV) are covariant in all cases of Lorentz-transformation, where (ρu, iρ) is to be transformed as a space time vector of the 1st kind, (m - ie) is to be treated as a vector of the 2nd kind, or more significantly,—
(ρu, iρ) is a space time vector of the 1st kind, (m - ie) [17] is a space-time vector of the 2nd kind. ”

“ The transformation of Lorentz only enables us to pass from a system at rest to one in uniform motion, but it has been postulated that there is a more general transformation which can be applied to systems moving in a more general manner. ”

“ I need scarcely add that if, by the ordinary transformation of the theory of relativity, we pass to new co-ordinates in such a manner that in this system the rod II. is at rest, and if now we measure the lengths by the difference between the values of which correspond to a definite value of , I. will be found to be the shorter of the two.
The question arises as to how far the dimensions of a solid body will be changed when its parts have unequal velocities, when, for example, it has a rotation about a fixed axis. ”

“ These equations are capable of a remarkable transformation discovered by Lorentz and which owes its interest from the fact, that it explains why no experience is suited to show us the absolute motion of the universe. ”

“ At all events, a definite decision will be reached during the next few years. If the displacement of spectral lines towards the red by the gravitational potential does not exist, then the general theory of relativity will be untenable. On the other hand, if the cause of the displacement of spectral lines be definitely traced to the gravitational potential, then the study of this displacement will furnish us with important information as to the mass of the heavenly bodies. ”

“ We may, however, prove that no conformal transformation in this space exists which will transform every point at rest in the space (x, y, z) into a point moving with angular velocity about a fixed line in the (XYZ) space. Thus, as in material dynamics, the equations of electrodynamics will not be preserved in the same form for a set of axes rotating uniformly relatively to the space in which they are satisfied by actual phenomena.
I.
Some considerations concerning the Lorentz-Einstein Transformation. ”

“ Thus, if we assume those equations together with Lorentz' fundamental equations, to be the correct scheme of the electrodynamics of moving bodies, we arrive at the conclusion that that scheme is invariant, not only under the extension of the transformation of Lorentz and Einstein given by Mirimanoff, but also under the transformation obtained by inversion in the four dimensional space here developed, and therefore also under a transformation obtained by combining any number of such operations. ”

“ Motion of the Perihelion of Mercury According to Newtonian mechanics and Newton's law of gravitation, a planet which is revolving round the sun would describe an ellipse round the latter, or, more correctly, round the common centre of gravity of the sun and the planet. In such a system, the sun, or the common centre of gravity, lies in one of the foci of the orbital ellipse in such a manner that, in the course of a planet-year, the distance sun-planet grows from a minimum to a maximum, and then decreases again to a minimum. ”

“ If the sun (S) were not present, a star which is practically infinitely distant would be seen in the direction , as observed front the earth. But as a consequence of the deflection of light from the star by the sun, the star will be seen in the direction , i.e. at a somewhat greater distance from the centre of the sun than corresponds to its real position.
In practice, the question is tested in the following way. The stars in the neighbourhood of the sun are photographed during a solar eclipse. ”

“ But I show, in agreement with Lorentz, that it is incapable to accord with the impossibility of an experiment showing the absolute motion. As I have said, this is because is the only case for which all the Lorentz transformations form a group.
But with the hypothesis of Lorentz, the agreement between the formulas does not occur all alone; we obtain at the same time a possible explanation for the contraction of the electron, assuming that the electron, deformable and compressible, is subjected to a constant external pressure whose work is proportional to volume changes. ”

“ If instead of Newton's law we insert a somewhat different law of attraction into the calculation, we find that, according to this new law, the motion would still take place in such a manner that the distance sun-planet exhibits periodic variations ”

“ In Einstein’s theory there is no such privileged observer; for since a privileged state of rest in the ether is meaningless, all observers in Galilean frames are on the same footing. Hence, Lorentz’s distinction between the true time of the frame at rest and the distorted, local or electromagnetic times of the frames in motion through the ether loses its significance. Henceforth, all these so-called local times given by the Lorentz transformations constitute the true times lived, sensed and experienced by the respective observers in their respective Galilean frames. ”

“ As soon as a science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a theory. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the "truth" of the theory lies. ”

“ In point of fact, astronomers have found that the theory of Newton does not suffice to calculate the observed motion of Mercury with an exactness corresponding to that of the delicacy of observation attainable at the present time. ”