Max von Laue

“ The rate of a clock moving with velocity is thus slower in the ratio than for the same clock when at rest. This theorem is of course true, independent to which of the valid system we relate the expressions "at rest" or "in motion". We of course have to consider as a clock, every process whose progression can serve as a measure of time. ”

“ Every oscillating structure is a clock in our sense. By spectroscopic observations, whose precision (as we know) can be largely increased, we can measure its oscillation number and the influence upon it by the velocity, and although the difficulties of such an experiment have not been overcome yet, there is justified hope, that later a verification of the theory, which is directly related to the relativity of time measure, will be possible. ”

“ The proper time of a worldline-section is an invariant of the Lorentz transformation. This already becomes clear due to its physical meaning, since the hand positions of the clock are the same for all reference systems at the start and the end of that section; this we can also recognize by the fact, that the expression in (52) is transformed the same way, as the invariant expression . However, it is to be noticed, that that equation (52) is initially derived only for uniformly moving clocks, i.e. derived for straight worldlines. In general, the acceleration will have an influence on the rate. ”

“ A point of this manifold we call a "world point"; it represents an event taking place at location at time .
The space coordinates are functions of time for any material point; thus it determines a curve in the world, a "world-line". At uniform velocity it is a straight line whose angle with respect to the -axis is equal to , thus it is smaller than , because ; in general it is an arbitrary curved line, whose inclination with respect to the -axis, however, never reaches the given amount. ”

“ It is the great merit of the so early deceased mathematician of Göttingen, Hermann Minkowski, to have created a geometric interpretation that avoids this difficulty, and due to the elegance that it lends to the calculations of relativity theory, it forms a nearly indispensable tool for the reliable usage of it. ”

“ The Lorentz transformation. The [1] relativity principle claims: From the totality of natural phenomena, one can determine with increased approximation and increased exactitude, a reference system x, y, z, t in which the natural laws are valid in certain, mathematically simple forms. This reference system is, however, in no way uniquely defined by the phenomena. There is rather a three-times infinite manifold of equally valid systems, mutually moving with uniform velocity. ”

“ To arrive at the transformation formulas, we ask after those linear relations between and which transform the wave equation
into itself; i.e. to which the identity applies:
where is a proportionality factor, which is different from zero and dependent (in a unknown way at first) on . Because equation (22) is the mathematical formulation of the law of light propagation [2] . We choose this substitutions as linear, because otherwise the coordinates and the origin of time (at least in one of both systems) would become preferred points, which cannot be displaced without essential changes. ”

“ The volume of a body is to be transformed by the Lorentz transformation because of the same reason as the length parallel to the velocity. Since by that, in every system is equal to the volume in the co-moving system, if follows that the equation, which serves as the transformation formula for the volume of a body with velocity relative to and relative to , is:
Another consequence of contraction is, that the form of a body is differently seen in different systems. ”

“ The transfer of the driving motion starting from would occur momentarily by the elastic forces. And by that we arrive at the reason of this contradiction. The assumption of a rigid body is incompatible with relativity theory. The ruler bulges (under all circumstances) at first in and then it doesn't remain straight; the conclusion to a superluminal velocity of the intersection, however, is essentially based on its straightness. ”

“ If we imagine a material right-angled triangle with velocity displaced parallel to the first side, and introduce a pencil along the second side with velocity , then its top follows on the drawing plane the line in unit time. ”

“ For example, in canal rays the atoms of hydrogen or mercury are moving with velocities of about 1⁄100 of the speed of light. At the same time, oscillations of high regularity occur within them, of which we were informed by the sharp lines of their spectra. ”

“ The first of equations (27) () shows by (25) , that the expression is invariant with respect to a Lorentz transformation. One easily can convince himself, that instead of this, one can also use the invariance of expression as characteristic for a Lorentz transformation; namely, if one determines the coefficients in (26) for the latter requirement, then one finds the exact same values as above.
As well as the totality of all Galilean transformations, also the totality of all Lorentz transformations forms a group ”

“ For example, one shall think of two astronomers on different, mutually moving celestial bodies, which are able to exchange thoughts with each other. They would find out, that they would have essentially different time indications of all astronomic events ”

“ Quite similar relations occur with respect to measurements of the length of a moving rod. If it is at rest in and located parallel to the -axis, then its rest length is determined by the difference of the -coordinates of its endpoints, thus:
those can be determined by applying a measuring rod also at rest in . ”

“ The general Lorentz transformation now consists in this, that instead of the line as -axis and the distance as unity, and instead of the -space with sphere (50) as measuring surface, we render the line (with as unity-distance) as the -axis, and its conjugate diameter-space with the mentioned ellipsoid as measuring surface we render as the corresponding -space. The latter condition expresses, that the equation of the ellipsoid should read
For confirmation, we first change the coordinate system by a rotation executed in -space ”

“ If one squares and adds the terms in the row or the column, then one obtains the value 1; if one multiplies the corresponding terms of two different rows or columns, then one finds the value zero at the addition.
The coefficients and the sub-determinants will be used in the following in the definition of the four- and six-vectors.
In the special transformation (57) , the sub-determinants [see (70) ] are:
all others are zero. The given scheme thus has the following shape:
In this way, one easily confirms the given rules.
↑ A. Einstein, Zur Elektrodynamik bewegter Körper. Ann. d. Phys. ”