Brownian motion

“ It is clear that if a body be in motion in the midst of a shower of such particles coming equally from all directions, it will receive a greater number of blows on its front surface, and a less number on its rear surface, than if it were at rest; and consequently its motion will be retarded. The only way of surmounting this difficulty is to suppose that the heavenly bodies, in relation to the gravity-gas, are practically at rest; in other words, that the velocity of the gravity-particles is practically indefinite compared with that of the heavenly bodies. ”

“ Yet, my theory strictly provides the dependency of the rest mass on acceleration for a class of motions, which corresponds – being the principally simplest accelerated motions – to the uniformly accelerated ones of the old mechanics, and which I call "hyperbolic motions", namely the rest mass proves to be constant up to enormous accelerations. Equations of motion in the form of the mechanical fundamental equations [9] which are adapted to the relativity principle, apply to this motion. ”

“ Later, this concept of the rigid body proves to be fruitful for the development of dynamics itself; because the rigid body as a continuous mass system of only six degrees of freedom, is not only kinematically of the highest simplicity, but also dynamically by allowing the composition of the forces – which are acting at its points – to "resulting" forces and moments of the same magnitude, whose knowledge is sufficient for the description of motion. ”

“ If one now moves this coordinate system along curve , then point follows the world line belonging to the parameter value . All points of such a normal (-axis) belong to the same value , thus they have the same velocity.
Fig. 4. The uniform motion of a rigid body is so constituted, that – as soon as one point is transformed to rest – all of its points are transformed to rest by the same transformation. This rest transformation is exactly (74) . The lines of same velocity = const., except at uniform motion, always have an envelope; the regularity of motion stops at this one. ”

“ We can say, that the localization of energy and momentum in the aether in the old sense, in not in agreement with the relativity principle. However, we not at all need the energy and momentum theorems (in the considered form) for the formation of the equations of motion. Rather the definition given on p. 46 completely suffices. The energy theorem is then added to the three equations of motion as a statement depending on them, that the sum of the works of the resultants of the external and the interior field are always equal to zero. ”

“ It is thus natural that we should start building up the new theory from this point. The new law of motion must be a differential law, which, in the first place, describes the motion of a point-mass under the influence of both inertia and gravity, and which, secondly, always preserves the same form, irrespective of the system of co-ordinates to which it be referred, so that no system of co-ordinates [Pg 45] enjoys a preference to any other. ”

“ There is an instantaneous axis of rotation, i.e. by a suitable Lorentz transformation one can achieve that all points of a straight line are at rest.
6) All points of the body have a constant "rest distance" with respect to a certain point located upon the instantaneous axis, the center of rotation.
Except the identification of these general properties of the definition, I have computed some special motions, among them the case that a point of the body carries out a uniformly-accelerated motion (hyperbolic motion) , and that the body is arbitrarily rotating about the direction of translation. ”

“ Hence, since the laws of Newtonian mechanics are based on inertial systems, it follows that all such systems are equivalent for the description of the laws of mechanics: no one system amongst them is unique, and we cannot define absolute motion or rest; any systems moving with mutual rectilinear uniform motion may be regarded as being at rest. Mathematically, this means that the laws of mechanics remain unchanged in form for any transformation from one set of inertial axes to another. ”

“ A mechanics of the relative motions of bodies, which is in harmony with the two fundamental postulates of continuity and relativity, can be [Pg 53] built up only on a fundamental law of motion that preserves its form independently of the kind of motion the system is undergoing. An available law of this kind is given if we raise the law of motion along a geodetic line, which, in the special theory of relativity, holds only for a body moving under no forces, to the rank of a general differential law of the motion in the gravitational field, too. ”

“ If we neglect quantities of the second order, and finally assume that, for small velocities of matter, the changes of the gravitational field with respect to time are small (i.e. that the derivatives of the s with respect to time may be neglected in comparison with the derivatives taken with regard to the space-co-ordinates) the system of equations (1) assumes the form: This would be the equation of motion of a point-mass as already given by Newton's mechanics, if be taken as representing the ordinary gravitational potential. ”

“ In the integral of a differential equation certain constants appear which are determined by the initial conditions, and are therefore the same during all the future motion of the system; obviously these constants of motion correspond to conservative properties. This reasoning can of course be at once extended. Any system, mechanical or not, whose motion is determined by [Pg 111] differential equations, will have certain conservative properties. For the systems of mechanics energy is one of the conservative functions; others are momentum and moment of momentum. ”

“ In the second case, the speed is not determined solely by the position of the particle; for example, the centers of force themselves may be moving or the particle may be in a resistant medium. Our principle pertains only to the first class of motion; in particular, even when a particle is attracted to multiple fixed centers of force, its momentum integrated along its trajectory should be minimal. ”

“ Each stage of evolution must initiate a higher stage.
§ 117. The force of aggregation acting on irregular masses of rare matter, diffused through a resisting medium, will not cause such masses to move in straight lines to their common centre of gravity; but, as before said, each will take a curvilinear path, directed to one or other side of the centre of gravity. All of them being differently conditioned, gravitation will impress on each a motion differing in direction, in velocity, and in the degree of its curvature—uniform aggregative force will be differentiated into multiform momenta. ”

“ We may extend the concept to moving systems if we wish, as a matter of convenience to ourselves, but must recognize that such an extension is an invention of ours and not a reality of nature. Now we do make such an extension, and we make it in the simplest possible way, that is, we define the charge on an isolated body in motion as that which we should find on it if we reduced it to rest and made measurements according to the regular static procedure. ”

“ We have just considered the sort of problem that we encounter on ordinary scales of magnitude on going from low to high velocities; what becomes of the concepts of force and mass when we go to a very small scale? Down to the atomic scale we may at least slur over the new physical difficulties, for although we cannot of course experiment with actual atoms, we can nevertheless make measurements of the Brownian [17] movement of suspensions in liquids settling in a gravitational field, for example, and the extrapolation to the atom is not a very great one. ”

“ It is the foundation of classical mechanics, and hence also of the first or "mechanical" principle of relativity.
If we suppose a crow flying in a straight line at uniform velocity with respect to the earth diagonally over a train likewise moving uniformly and rectilinearly with respect to the earth (since motion is change of position we must specify our rigid body of reference, viz. the earth) , then an observer in the train would also see the crow flying in a straight line, but with a different uniform velocity, judged from a system of co-ordinates attached to the train. ”

“ The motion is now no longer one of the allowed ones, and the particle must in some way change its velocity; it must either slow down or speed up. In the first case it must either radiate energy, which a system of the simple mechanical properties we have supposed is not capable of doing, or else the law of conservation of energy fails, and also Newton's first law of motion during the process of acquiring the steady condition. If, on the other hand, the particle speeds up, it must increase its energy from nowhere, and again ordinary mechanics does not apply. ”

“ When similar forces are impressed upon exactly similar bodies, the velocities produced are exactly the same; but, if similar forces act on dissimilar bodies, then the velocities produced in the different bodies are not the same; yet the total motion produced on all bodies, according to the Second Law of Motion, must always be proportionate to the impressed force. So that when we compare the effect of similar forces on different bodies, we find that there are two factors involved, viz., the mass and velocity of the moving body. ”

“ When we assume Born's differential equation, then we come to a propagation velocity larger than that of light. We have a volume element that we consider as being rigid. This volume element at its motion, transformed to rest, conserves its form. This is Born's differential equation. If we keep this condition, then superluminal velocity is given. For the time being we don't know as to how the body will actually move since we can make no statement about the interior forces of the body. Born's differential equations is only one condition which must be satisfied by the motion of rigid bodies. ”

“ If a point is moving along the -axis, then its motion can be represented by a curve whose abscissa axis denotes the coordinate , and whose ordinate axis denotes the time . Two rigidly connected points (in the sense of ordinary mechanics) , which move at the -axis, provide two curves in this representation, whose distance measured along the line =const. is invariable. This condition of rigidity contradicts the relativity principle ”

“ However, to prevent misinterpretations, it shall be clearly noticed, that – under certain circumstances – a body of course can move without change of shape related to the employed co-moving system; this is indeed proven by the uniformly accelerated (hyperbolic) motion. Also in classical hydrodynamics, a particle of a fluid can occasionally be in motion as if it were rigid. ”

“ The appearance of these doublets is undoubtedly due to the small deviations from central symmetry of the field of force originating from the inner system in consequence of which the general type of motion of the external electron will possess a more complicated character than that of a simple central motion. ”

“ This force will be capable of sustaining the hyperbolic motion with acceleration , when one chooses:
If the external force field is only slightly variable, so that one can see it as constant in the interior of the electron, then it will generate a motion which only slightly deviates from hyperbolic motion. When we see equations (154) as valid also in this case, we are neglecting radiation.
Thus we obtain for slightly variable, yet arbitrary great accelerations, the following equations of motion and energy:
where and are related to the center of the electron. ”

“ However, also difficulties for the usefulness of the new definition are standing in the way of this agreement. [5]
§ 1. The rigid body of ordinary mechanics. While my first definition of rigidity seeks to analogize the property of ordinary rigid bodies, that the distance between any two points of the body is constant at motion, the new definition is similar to another property of ordinary rigid bodies.
The velocity vector of any of its points can namely be represented in the following way: [6]
There, means the vector with components
where is a certain point of the body ”

“ Our principles do not therefore corroborate Newton's third law of motion, respecting the equality of action and reaction in attracting forces; for by our theory, a body might by the agency of the fluid medium, be impelled toward another, without any reciprocal action ; which is by no means surprising if we consider attraction not to be an inherent or essential property of matter, but merely the action of a third body. ”

“ The first of these equations, is the equation of motion in an analogous form of one of the Newtonian equations of old mechanics, the second one is the energy equation. The quantity thus corresponds to the kinetic energy of old mechanics. The dependency of mass from velocity is given by Lorentz's formula (157) ; more essential than this one (which is also valid for the ordinary (non-electromagnetic) mass in the new kinematics) , is the dependency of rest mass from the magnitude of acceleration according to formula (153) . ”

“ And if any one suggests that the bodies have previously been in motion with respect to A, and that they are projected by elastic reaction from A, then he must be asked to give an explanation of elasticity from impact only, and without introducing action at a distance. ”

“ An abstract system of kinematics, under any geometrical arrangement or conception, is indeed quite irreconcilable with the actual system of dynamics found in nature and subject to our observation. All gases would under the operation of the first law of motion, tend to infinite and equable diffusion ”