Galilean relativity

“ But this attempt to found relativity of translation upon the Galilean transformation fails when applied to electromagnetic phenomena. The Maxwell-Lorentz electromagnetic equations are not co-variant with respect to the Galilean transformation. In particular, we note, by (21) , that a ray of light which referred to has a velocity , has a different velocity referred to ', depending upon its direction. The space of reference of is therefore distinguished, with respect to its physical properties, from all spaces of reference which are in motion relatively to it (quiescent æther) . ”

“ The two forms of relativity have been muddled up so often that it appears necessary to point out their essential differences. The general principle of relativity merely states that the natural laws can be thrown into a form which remains covariant to all choices of mesh-system. In other words, all observers, whether Galilean, accelerated or rotating, will observe the laws of nature under the same tensor form; so that there is no reason to follow classical science and elevate one type of observer above another. This is what Einstein originally meant by the relativity of all motion. ”

“ Obviously such a world would not be the one in which we live.
Summarising, we must say that in classical science the velocity of light is finite and that the invariant velocity is infinite. In relativity, , the velocity of light, is finite, as before, but is identical with the invariant velocity, which is therefore also finite. We also understand the deep-seated reason for the curious complicated form of the relativistic space and time transformations when we pass from one Galilean frame to another (the Lorentz-Einstein transformations) as contrasted with the classical transformations. ”

“ If with Einstein we follow this line of argument, we must assume that the large number of negative experiments prove conclusively that velocity through the stagnant ether or through space is physically meaningless, not only from a purely mechanical point of view, as was the belief of classical science, but from every point of view. If we accept this relativity of the ether and space for Galilean motion as a fundamental fact of nature, holding for all manner of experiments, our past difficulties are seen to be of our own making. ”

“ If is to be real, then cannot be greater than , i.e. is the limiting or maximum velocity in nature and has thus a universal significance.
If we imagine to be infinitely great in comparison with (and this will be the case for all ordinary velocities, such as those which occur in mechanics) , the equations of transformation degenerate into: This is the familiar Galilean transformation which holds for the [Pg 117] "mechanical" principle of relativity. We see that the Lorentz-Einstein transformation covers both mechanical and radiational phenomena. ”

“ Just as Newton’s discoveries were made possible thanks to the introduction of the inertial frame of reference attached to the stars, so was Einstein’s synthesis the natural outcome of the principle of the relativity of Galilean motion, and of the postulate of the invariance of the velocity of light in vacuo as referred to any Galilean frame. ”

“ The principle of equivalence demands that in dealing with Galilean regions we may equally well make use of non-inertial systems, that is, such co-ordinate systems as, relatively to inertial systems, are not free from acceleration and rotation. If, further, we are going to do away completely with the difficult question as to the objective reason for the preference of certain systems of co-ordinates, then we must allow the use of arbitrarily moving systems of co-ordinates. ”

“ 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 ”

“ If we desire to adhere to our "old-time" three-dimensional view of things, then we can characterise the development which is being undergone by the fundamental idea of the general theory of relativity as follows: The special theory of relativity has reference to Galileian domains, i.e. to those in which no gravitational field exists. In this connection a Galileian reference-body serves as body of reference, i.e. a rigid body the state of motion of which is so chosen that the Galileian law of the uniform rectilinear motion of "isolated" material points holds relatively to it. ”

“ Now Newton's (or Galilei's) law of motion states that a body which is sufficiently far removed from all other bodies continues in its state of rest or uniform motion in a straight line. This holds very approximately for the fixed stars. If, however, we refer the motion of the stars to a set of axes fixed to the earth, the stars describe circles of immense [Pg 108] radius; that is, for such a system of reference the law of inertia only holds approximately. Hence we are led to the definition of Galilean systems of co-ordinates. ”

“ It is then possible to choose a Galileian reference-body for this part of space (world) , relative to which points at rest remain at rest and points in motion continue permanently in uniform rectilinear motion. As reference-body let us imagine a spacious chest resembling a room with an observer inside who is equipped with apparatus. Gravitation naturally does not exist for this observer. He must fasten himself with strings to the floor, otherwise the slightest impact against the floor will cause him to rise slowly towards the ceiling of the room. ”

“ Besides this momentous episteomological argument, there is also a well-known physical fact which speaks in favour of an extension of the relativity theory. Let there be a Galiliean co-ordinate system K relative to which (at least in the four-dimensional region considered) a mass at a sufficient distance from other masses move uniformly in a line. ”

“ It expresses, therefore, merely a relationship between the motion of the observer and the structure of space-time; it does not describe the structure of absolute space-time itself. We must realise, then, that potential energy represents nothing inherent in the absolute world, and we must exclude it as a fundamental entity in the formulation of our laws.
As soon, however, as we refuse to take gravitational or potential energy into consideration, conservation breaks down in a field of force. Hence, it follows that conservation of mass, energy and momentum holds only for Galilean observers. ”

“ In all the experiments we shall have occasion to mention, the velocity through the ether which we shall seek to determine will be that of our earth; for, attached as we are to the earth’s surface, this will be the speed we shall best be in a position to determine. Unfortunately, the earth does not constitute a truly Galilean frame on account of its acceleration or rotation round its axis. However, this acceleration of the points on the earth’s surface is very small and the centrifugal force to which it gives rise is scarcely perceptible. ”

“ Well, then, get your Kirchhoff page ready. We shall see whether we can set out on it the special theory of relativity."
The totality of our experience compels us to assume that light travels with a constant velocity in empty space. Likewise, our whole experience in optics compels us to recognize that all inertial systems are equivalent; these are systems that are produced from an allowable one by means of a uniform translation. An allowable system is one in which Galilei's and Newton's Law of Inertia holds. ”

“ No one of them stands out more prominently than any other on the ground of symmetry with respect to the four-dimensional world. Only when we specify the frame of the observer will one particular space and one particular duration be allotted. Hence we may say that practical congruence exists for space and for time, as in classical science, provided the Galilean observer effects his measurements with rods and periodic mechanisms, such as clocks, which do not move about in his Galilean frame while performing measurements. ”

“ It is easy to show that if, of two clocks which go at exactly the same rate when relatively at rest in the Galilean field , one be placed at the centre of the rotating disc and one at the circumference, the latter will continually lose time as compared with the former. ”