Max von Laue

“ In general, however, the exerted radiation of a resting body has a resulting momentum and accordingly exercises a motion drive, as is can be most easily seen by the example of a cavity (of perfectly reflecting substance) filled with radiation, having a small opening only at a single point. But if the body is isotropic, homogeneous and limited by a convex surface everywhere, it suffers a normal pressure from the radiation which is everywhere the same, not resulting in a net force. If one such body is at rest in the primed system, then it has a constant speed in relation to the unprimed system. ”

“ Now, in Newton's mechanics there is of course another definition of rigidity, which fits even better to the physical state of facts; there, one considers the rigid body as the limiting case of a deformable body with a very great elasticity coefficient. This passage to the limit can in any case also be executed in the theory of relativity, although it wouldn't lead to the rigid body, but to a body deformable as less as possible, which possibly has especially simple properties as well. ”

“ We are allowed to define the term "clock" as far as at page 108, i.e. all physical, chemical, and biological processes must show in their course the same dilatation. A life form in motion is less aging, and even the thought process of a man must occur slower, when the relativity principle shall be correct. However, he won't experience anything of this, as long as his velocity is not changing, since he is also at rest in a valid reference system. On the contrary, he will believe with some justification, that he thinks faster than the men who are moving with respect to him. ”

“ We assume, that in any field of physics there is one of those world tensors, whose components have the corresponding meaning, as the components of the mentioned electrodynamic tensor; i.e., that for example also in dynamics, the energy density is defined by , the momentum density and the energy current according to (1a) by etc., while is related to the elastic stresses. That we assume tensor as being symmetric, can appear to be arbitrary at this place; because also the divergence of an unsymmetrical world tensor would be a four-vector. ”

“ The difference against electrodynamics, which achieves the same independently from the relativity principle, is not a principal one though, but solely lies in the current state of physical research. We simply have much more complete and precise knowledge in the theory of electricity, than in mechanics. ”

“ In dispersive substances we have to fill in the value for n corresponding to the frequency in the primed system.
For the group velocity it is exactly the same, if we replace the refractive index n by the expression n + v (dn/dv) (v is frequency) .
So, according to the relativity principle, light is completely carried by the body, however, just because of this its velocity relative to an observer (who does not participate in the motion of the body) is not the same as the vector sum of its velocity relative to the body and that of the body relative to an observer. ”

“ The experiment is performed in air, which goes along with the motion of the earth. All parts of the arrangement, including air, preserve their position with each other constantly; so no influence of the motion is present, but Δτ = 0. Since in a system moving like a rigid body, all electromagnetic processes happen as if they are at rest.
Thus we see that all theories which seriously come into consideration for the optics of moving bodies - that of Hertz never belonged to them because of Fizeau's interference experiment on the entrainment of light - agree as to the effect in question. ”

“ The subject of the experiment is the question, whether a charged plate condenser experiences a torque due to its motion with Earth. Relativity and absolute theory accordingly are affirming this; of course, relativity theory does this only in the case of a non-co-moving reference system ; in the "rest system" there is no torque. With the absolute theory, we can now expect a rotation in consequence of the torque, since its dynamics is that of Newton; according to this theory, the condenser tries to locate itself, so that the direction of velocity is tangential to the plates. ”

“ Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none, against which the common sense of anyone (who is still unfamiliar with the matter) is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. ”

“ In both cases we can imagine a closed surface moving with the velocity through which passes no energy. As we can disregard the absorption, this surface always includes the same quantum of mechanical or electromagnetical energy. It has, however, always only its signification for one coordinate system. For another system the energy flows in general through the surface. ”

“ We can compare its field with an angle-lever, its material parts with the shell. Neither the electromagnetic momentum of one part, nor the mechanical momentum of the other part, is parallel to the velocity, therefore both need a torque for their translatory motion. However, since momentum as a whole is parallel to velocity, both momentum components (perpendicular to the velocity) are oppositely equal to one another, and the same holds thus for the torque. ”

“ However, it is now to be considered, that light (when it enters and leaves at two end-surfaces perpendicular to distance as in Zeeman's experiment) travels a shorter distance in air due to the motion, as when the body would be at rest. Because during time the body moves a distance . Thus if we denote by the distance between both of them and the stationary part of the experimental device being closest to the moving body, then the mentioned ray requires the time (to come from one of these part to another one) ”

“ Because the conditions of the body with respect to and are only distinguished by the velocity, which it has with respect to , but not with respect to , then this means: A body brought by us into motion from a state of rest, contracts in the direction of motion. So far as the relativity principle (as it is pronounced by the Lorentz transformation) holds in reality, all bodies must show this behavior. ”

“ If one arrives by a different choice at different results, then all laws wouldn't be invariable with respect to the same transformation, thus the relativity principle would be wrong. Evidently, such laws related to physical processes in empty space, are the most simple ones. Since in the other ones, the velocity of matter relative to the references system is included in its mathematical formulation, and must be co-transformed by a transformation; in vacuum, however, were we know nothing about moving things, this complication vanishes. ”

“ 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. ”

“ All three proposals have the (quite justified) idea in common, that a rigid body, contrary to the infinitely many degrees of freedom of the deformable ones, has only a finite number of them (there, the body is of course imagined as a continuum) . We now want to show, that the relativity principle excludes this possibility due to dynamical causes, so that any attempt in that direction is hopeless from the outset. ”

“ The older theory needs a force whose work covers a portion of the energy radiation to maintain uniform motion. According to the theory of relativity, however, it completely stems from the energy of the light source.
The given values for the energy and momentum radiation are valid, since they depend only on the acceleration of the moving electron, even if it doesn't oscillate together with a dipole, but is moving in any other way. ”

“ First – although this is without practical importance in the light of the present precision of Harress's experiment, but it is of principal importance –, Harzer overlooks that also the parts of the light path traversed in the rotational apparatus, which are lying in the air, are contributing to the phase difference at the interference. Then Harzer with full justification considers that the light ray must be curved in the rotating body. Yet the approach, from which he finds out its shape, [6] is not self-evident in our view, although it can surely be demonstrated from relativity theory. ”

“ 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. ”

“ Therefore I will adduce an instance which shows independently of every special theory, that the velocity of the energy cannot be transformed in the same way as the velocity of a material point. We consider three coordinate systems, moving uniformly relative to one another; the latter two will have the velocity relative to . A body subjected to a tension (negative pressure) equal in all directions is in rest relative to . In the system it has the velocity , in the velocity . ”

“ Imagine the plane of the experiment at any point on the earth tangentially to it. The time dt required by light to traverse the distance ds (measured on earth) is extended by , if ds is moving with velocity . Consequently, the propagation time of the positively rotating ray is longer than in the case of vanishing rotation, and smaller for the negatively rotating ray by the same amount. ”

“ Because also Newton's dynamics – maybe in the clearest way in the principle of d’Alembert – says that the inertial force resisting the increase of mechanical momentum, exactly compensates the force exerted by the elastic stresses – other forces don't exist at all in pure dynamics. In this way, one has as the fundamental equation of dynamics which is invariant against Lorentz transformation and encloses the momentum and energy theorem, the relation
where the components of have the given physical meaning. ”

“ Since Einstein's electrodynamics, [1] which is based on the principle of relativity, is equivalent with the older theory of Lorentz as long one restricts himself to the first power of the relations of all body velocities to the speed of light, it is obvious that it allows to calculate Fresnel's dragging coefficient as a first approximation. ”

“ The motion of the body at instant then has surely degrees of freedom. Because a part of it which is not yet belonging to any of these intersections, is still at rest in , while in the parts represented by the intersections, a motion dominates that is independent from the other disturbances, and which naturally has at least one degree of freedom. The number of disturbance points, however, can be arbitrarily increased; thus the number of kinematic degrees of freedom of a body has no upper limit. ”

“ The totality doesn't require any torque, it rather represents (as in the model consisting of angle-lever and shell) a "completely static system" [10] , which obeys the dynamics of the mass-point for stationary and quasi-stationary translatory motions.
That the momentum of the angle-lever has not the direction of its velocity, is also given by another consequence, which appears to be paradox in Newton's dynamics. ”

“ What is unsatisfactory for some in this way of conclusion, is the fact that it only gives a reason for its recognition, but not a cause; it doesn't follow the causal relation of reality. Before we attempt to fill this gap, we still want to emphasize that such a deviation of the line of reasoning from the causal interconnection is nothing unusual in physics. On the contrary, its value for science (especially as regards the most general natural laws) rests on the fact, that it relieves us from the effort, to follow the often less vivid causal-connections of nature in all details. ”

“ And when Wiechert writes [5] : "Yet, as soon as the validity of the relativity principle will be concluded from the observations, so that (as previously described) a natural limitation of the ordinary states of motion must be assumed, is is impossible to avoid the conclusion, that just not all strides are equivalent for the world process", then this conclusion is totally inadmissible in the system of the unconditional relativity principle. Because this view necessarily must deny the occurrence of superluminal velocities for matter. ”

“ General relativity would of course be capable of giving some statements about it, and we want to show at first that no noticeable influences of acceleration are expected according to it. The occurring centrifugal forces are namely of a magnitude of at most 10 to 100 times as great as Earth's attraction upon the same body. Now, since gravitational acceleration itself has no noticeable influence upon the optical processes even in respect to much mightier celestial bodies than Earth, then such one isn't expected in Harress's experiment as well. ”

“ The experiment is aimed to bring a decision to the question (related to Stokes' theory) whether the aether next to earth joins its rotation or not. In the first case, no influence of rotation on the location of the middle interference fringe of same phase is expected, but in the second case we have to expect it. We want to see what can be expected as regards the various electrodynamic theories of moving bodies, whether the experiment could bring a decision between them. ”

“ All of those who show no current (and thus also no momentum) in a valid system (the rest system [17] ) . To those belong: heat [18] , chemical energy, elastic energy, inner energy of atoms, and maybe also new and unknown energy forms. Electromagnetic energy has to be excluded in general, since it can also flow in the rest system . If this doesn't occur in a special case, then one can include it in as well; tensor then encloses also Maxwell's stresses besides the elastic ones, and vector also the electromagnetic momentum besides the mechanical momentum. ”

“ 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. ”

“ The second ruler, which should be momentarily brought into motion by an impulse emanating from , doesn't remain straight but bulges in . The velocity, with which the disturbance emanating in is communicated to the other points of the ruler, is itself at most equal to that of light; the distant parts at first remain at rest, and only after a certain time they are also affected by the disturbance. ”

“ Until now, we only have spoken about the restricted relativity theory. The general theory denies the existence of coordinate systems, which have by themselves the preferred state of an inertial system. The attempt by Lange should be without result, if all bodies, except the test bodies, are taken away. Just because those large masses of the fixed stars are on the whole at rest in the astronomical coordinate system, it has something in advantage before the others. Nevertheless, this only inessentially changes the view on Lorentz contraction previously presented. ”

“ If both kinds of forces are transformed in the same way to other reference frames, then this resultant remains zero in all systems, and there is no reason for a rotation. Although it's unquestionable that this answer hits the truth, it isn't entirely satisfactory in so far as it is connected to molecular theory, with which this problem has nothing to do per se.
Already with respect to Newton's mechanics it has often been point out, [9] that it is more logical to place the dynamics of continua in front of that of the mass point. ”

“ Both experiments are proving, first, that the optical processes are different in a reference system rotating relative to Earth, as in a system fixed to Earth; we are permitted to consider the latter system with sufficient approximation as a valid system in the sense of the restricted relativity theory. Regarding mechanical processes it's known that every experiment concerning centrifugal forces provides the corresponding confirmation (Sagnac's interpretation of his experiment as a confirmation of the existence of an "aether" is not decisive at all) . ”

“ 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. ”

“ According to equation (1a) for the absolute phase velocity of light relative to the reference system fixed at Earth, the component of the body's velocity perpendicular to the ray direction doesn't matter, but it is only about the velocity parallel to it. 2. If a line has the smallest distance from the rotation axis and if its direction forms with it the angle , then the component of the body's velocity with respect to this direction is equal to in any of the points, where is the angular velocity of the rotation. ”

“ Transformation of momentum, energy and stresses. The transformation formulas for momentum, energy density and the stresses in the passage from one valid reference system to another system , is definitely determined by deriving them from the components of the world tensor . ”

“ 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 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. ”

“ 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. ”