Albert Einstein

“ Thus any change of axes will introduce new “geometrical” forces in the field which are quite independent of the nature of the body acted on. Gravitational forces also have this same remarkable property, and gravitation itself may be of essentially the same nature as these “geometrical” forces introduced by a change of axes. This leads to Einstein’s famous Principle of Equivalence. A gravitational field of force is strictly equivalent to one introduced by a transformation of co-ordinates and no possible experiment can distinguish between the two. ”

“ Let us now picture to our mind the world-line passing through any world point x, y, z, t; now if we find the world-line parallel to the radius vector OA′ of the hyperboloidal sheet, then we can introduce OA′ as a new time-axis, and then according to the new conceptions of time and space the substance will appear to be at rest in the world point concerned. We shall now introduce this fundamental axiom:—
The substance existing at any world point can always be conceived to be at rest, if we establish our time and space suitably. ”

“ As Eddington says “matter does not cause the curvature of space-time. It is the curvature. Just as light does not cause electromagnetic oscillations; it is the oscillations.”
We may look upon this from a slightly different point of view. The General Principle of Relativity asserts that all motion is merely relative motion between matter and matter, and as all movements must be referred to definite sets of co-ordinates, the ground of any possible framework must ultimately be material in character. ”

“ The equations of Newtonian mechanics show a twofold invariance, (i) their form remains unaltered when we subject the fundamental space-coordinate system to any possible change of position, (ii) when we change the system in its nature of motion, i. e., when we impress upon it any uniform motion of translation, the null-point of time plays no part. We are accustomed to look upon the axioms of geometry as settled once for all, while we seldom have the same amount of conviction regarding the axioms of mechanics, and therefore the two invariants are seldom mentioned in the same breath. ”

“ The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postulate, but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lorenz-transformation, with all the relations between moving rigid bodies and clocks.
The modification which the theory of space and time has undergone through the special relativity theory, is indeed a profound one, but a weightier point remains untouched. ”

“ The means hitherto at our disposal, for placing our co-ordinate system in the time-space continuum, in a definite way, therefore completely fail and it appears that there is no other way which will enable us to fit the co-ordinate system to the four-dimensional world in such a way, that by it we can expect to get a specially simple formulation of the laws of Nature. So that nothing remains for us but to regard all conceivable co-ordinate systems as equally suitable for the description of natural phenomena. ”

“ According to this, we shall have in the world, not one space, but many spaces,—quite analogous to the case that the three-dimensional space consists of an infinite number of planes. The three-dimensional geometry will be a chapter of four-dimensional physics. Now you perceive, why I said in the beginning that time and space shall reduce to mere shadows and we shall have a world complete in itself. ”

“ They take the place of the imaginary cause R1. Among all the conceivable spaces R1 and R2 moving in any manner relative to one another, there is a priori, no one set which can be regarded as affording greater advantages, against which the objection which was already raised from the standpoint of the theory of knowledge cannot be again revived. The laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner. We thus arrive at an extension of the relativity postulate. ”

“ We cannot do better than quote Eddington’s description of this famous experiment. “The principle of the experiment may be illustrated by considering a swimmer in a river. It is easily realized that it takes longer to swim to a point 50 yards up-stream and back than to a point 50 yards across-stream and back. If the earth is moving through the ether there is a river of ether flowing through the laboratory, and a wave of light may be compared to a swimmer travelling with constant velocity relative to the current. ”

“ This anti-sun would of course be equally large and equally bright if there is no absorption of light in free space.
In de Sitter’s theory, the existence of vast quantities of world-matter is not required. But beyond a definite distance from an observer, time itself stands still, so that to the observer nothing can ever “happen” there. All these theories are still highly speculative in character, but they have certainly extended the scope of theoretical physics to the central problem of the ultimate nature of the universe itself. ”

“ Einstein’s theory connects up the law of gravitation with the laws of motion, and serves to establish a very intimate relationship between matter and physical space-*time. Space, time and matter (or energy) were considered to be the three ultimate elements in Physics. The restricted theory fused space-time into one indissoluble whole. The generalised theory has further synthesised space-time and matter into one fundamental physical reality. Space, time and matter taken separately are more abstractions. Physical reality consists of a synthesis of all three. ”

“ Each one of these denotes a certain group of transformations for the differential equations of mechanics. We look upon the existence of the first group as a fundamental characteristics of space. We always prefer to leave off the second group to itself, and with a light heart conclude that we can never decide from physical considerations whether the space, which is supposed to be at rest, may not finally be in uniform motion. ”

“ In proportion as a theory becomes complex it loses in usefulness—a theory which is obliged to requisition a whole array of arbitrary assumptions in order to explain special facts is practically worse than useless, as it serves to disjoin, rather than to unite, the several groups of facts. The optical experiments of the last quarter of the nineteenth century showed the impossibility of constructing a simple ether theory, which would be amenable to analytic treatment and would at the same time stimulate further progress. ”

“ The introduction of a “Lightäther” will be proved to be superfluous, for according to the conceptions which will be developed, we shall introduce neither a space absolutely at rest, and endowed with special properties, nor shall we associate a velocity-vector with a point in which electro-magnetic processes take place.
3. Like every other theory in electrodynamics, the theory is based on the kinematics of rigid bodies; in the enunciation of every theory, we have to do with relations between rigid bodies (co-ordinate system) , clocks, and electromagnetic processes. ”

“ Suppose an observer—who is stationed at the origin of coordinates with the clock—associates a ray of light which comes to him through space, and gives testimony to the event of which the time is to be estimated,—with the corresponding position of the hands of the clock. ”

“ The force-vector of motion is equal to the moving force-vector.
This enunciation comprises four equations for the components in the four directions, of which the fourth can be deduced from the first three, because both of the above-mentioned vectors are perpendicular to the velocity-vector. From the definition of T, we see that the fourth simply expresses the “Energy-law.” Accordingly c2-times the component of the impulse-vector in the direction of the t-axis is to be defined as the kinetic-energy of the point-mass. ”

“ He considers time itself to be one of the co-ordinate axes, and in his four-dimensional world, that is in the space-time reality, relative motion is reduced to a rotation of the axes of reference. Thus, the diversity in the measurement of lengths and temporal rates is merely due to the static difference in the “frame-work” of the different observers.
The above theory of Relativity absorbed practically the whole of the electromagnetic theory based on the Maxwell-Lorentz system of field equations. It combined all the advantages of classic Maxwellian theory together with an electronic hypothesis. ”

“ If R converges to a point (x, y, z, t) , then the quotient of this mass, and the volume of R approaches a limit μ (x, y, z, t) , which is known as the mass-density at the space-time point (x, y, z, t) .
The principle of conservation of mass says—that for an infinitely thin space-time filament, the product μdJ, where μ = mass-density at the point (x, y, z, t) of the filament (i.e., the principal line of the filament) , dJ = contents of the cross-section normal to the t axis, and passing through (x, y, z, t) , is constant along the whole filament. ”

“ Remembering this point it is clear that the full group belonging to Newtonian Mechanics is simply the group Gc, with the value of c = ∞. In this state of affairs, and since Gc is mathematically more intelligible than G∞, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena possess an invariance not only for the group G∞, but in fact also for a group Gc, where c is finite, but yet exceedingly large compared to the usual measuring units. ”

“ The observer at the common origin of co-ordinates who is able to see the clock at the periphery by means of light will see the clock at the periphery going slower than the clock beside him. Since he cannot allow the velocity of light to depend explicitly upon the time in the way under consideration he will interpret his observation by saying that the clock on the periphery actually goes slower than the clock at the origin. ”

“ We have seen before that the general relativity-postulate leads to the condition that the system of equations for Physics, must be co-variants for any possible substitution of co-ordinates x1, ... x4; we have now to see how such general co-variant equations can be obtained. We shall now turn our attention to these purely mathematical propositions. It will be shown that in the solution, the invariant ds, given in equation (3) plays a fundamental rôle, which we, following Gauss’s Theory of Surfaces, style as the line-element. ”

“ In the mechanics reformed according to the world-postulate, the disharmonies which have disturbed the relations between Newtonian mechanics and modern electrodynamics automatically disappear. I shall now consider the position of the Newtonian law of attraction to this postulate. I will assume that two point-masses m and m1 describe their world-lines; a moving force-vector is exercised by m upon m1, and the expression is just the same as in the case of the electron, only we have to write +mm1 instead -ee1. ”

“ The equations of the gravitation-field free from matter must thus be in every case satisfied when all Bρμστ vanish. But this condition is clearly one which goes too far. For it is clear that the gravitation-field generated by a material point in its own neighbourhood can never be transformed away by any choice of axes, i.e., it cannot be transformed to a case of constant gμν’s.
Therefore it is clear that, for a gravitational field free from matter, it is desirable that the symmetrical tensors Bμν deduced from the tensors Bρμστ should vanish. ”

“ If, for example, the magnet moves and the conductor is at rest, then an electric field of certain energy-value is produced in the neighbourhood of the magnet, which excites a current in those parts of the field where a conductor exists. But if the magnet be at rest and the conductor be set in motion, no electric field is produced in the neighbourhood of the magnet, but an electromotive force which corresponds to no energy in itself is produced in the conductor ”

“ It is to be remembered that only a modified mechanics will occur, in which the square root of this differential combination takes the place of time, so that cases in which the velocity is greater than c will play no part, something like imaginary coordinates in geometry.
The impulse and real cause of inducement for the assumption of the group-transformation Gc is the fact that the differential equation for the propagation of light in vacant space possesses the group-transformation Gc. On the other hand, the idea of rigid bodies has any sense only in a system mechanics with the group Ginfinity. ”

“ This does not prove that there is no such thing as an ether but certainly does render the ether entirely superfluous. Universal compensation is due to a change in local units of length and time, or rather, being merely different descriptions of the same reality, there is no compensation at all.
There was another group of observed phenomena which could scarcely be fitted into a Newtonian scheme of dynamics without doing violence to it. The experimental work of Kaufmann, in 1901, made it abundantly clear that the “mass” of an electron depended on its velocity. ”

“ It is to be noticed that the energy and the frequency of a light-complex vary according to the same law with the state of motion of the observer.
Let there be a perfectly reflecting mirror at the co-ordinate-plane ξ = 0, from which the plane-wave considered in the last paragraph is reflected. Let us now ask ourselves about the light-pressure exerted on the reflecting surface and the direction, frequency, intensity of the light after reflexion.
Let the incident light be defined by the magnitudes A cos Φ, v (referred to the system K) . ”