Relativity: The Special and General Theory w/Figures & Formulas

by Albert Einstein

This primitive method of place specification deals only with places on the surface of rigid bodies, and is dependent on the existence of points on this surface which are distinguishable from each other.

If, relative to K, K' is a uniformly moving coordinate system devoid of rotation, then natural phenomena run their course with respect to K' according to exactly the same general laws as with respect to K. This statement is called the principle of relativity (in the restricted sense).

However, the most careful observations have never revealed such anisotropic properties in terrestrial physical space, i.e. a physical nonequivalence of different directions. This is a very powerful argument in favour of the principle of relativity.

By means of similar considerations based on observations of double stars, the Dutch astronomer De Sitter was also able to show that the velocity of propagation of light cannot depend on the velocity of motion of the body emitting the light.

Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event.

The results of the last three sections show that the apparent incompatibility of the law of propagation of light with the principle of relativity (Section VII) has been derived by means of a consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as follows: The time-interval (time) between two events is independent of the condition of motion of the body of reference. The space-interval (distance) between two points of a rigid body is independent of the condition of motion of the body of reference.

The rigid rod is thus shorter when in motion than when at rest, and the more quickly it is moving, the shorter is the rod.

From this we conclude that in the theory of relativity the velocity c plays the part of a limiting velocity, which can neither be reached nor exceeded by any real body.

As a consequence of its motion the clock goes more slowly than when at rest.

The most important result of a general character to which the special theory of relativity has led is concerned with the conception of mass. Before the advent of relativity, physics recognised two conservation laws of fundamental importance, namely, the law of the conservation of energy and the law of the conservation of mass; these two fundamental laws appeared to be quite independent of each other. By means of the theory of relativity they have been united into one law.

The success of the Faraday-Maxwell interpretation of electromagnetic action at a distance resulted in physicists becoming convinced that there are no such things as instantaneous actions at a distance (not involving an intermediary medium) of the type of Newton's law of gravitation. According to the theory of relativity, action at a distance with the velocity of light always takes the place of instantaneous action at a distance or of action at a distance with an infinite velocity of transmission. This is connected with the fact that the velocity c plays a fundamental rôle in this theory.

Our principle rather asserts what follows: If we formulate the general laws of nature as they are obtained from experience, by making use of the embankment as reference-body, the railway carriage as reference-body, then these general laws of nature (e.g. the laws of mechanics or the law of the propagation of light in vacuo) have exactly the same form in both cases.

In contrast to this we wish to understand by the “general principle of relativity” the following statement: All bodies of reference K, K' , etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion.

We can sum this up as follows: Gauss invented a method for the mathematical treatment of continua in general, in which “size relations” (“distances” between neighbouring points) are defined. To every point of a continuum are assigned as many numbers (Gaussian co-ordinates) as the continuum has dimensions. This is done in such a way, that only one meaning can be attached to the assignment, and that numbers (Gaussian co-ordinates) which differ by an indefinitely small amount are assigned to adjacent points. The Gaussian co-ordinate system is a logical generalisation of the Cartesian co-ordinate system. It is also applicable to non-Euclidean continua, but only when, with respect to the defined “size” or “distance,” small parts of the continuum under consideration behave more nearly like a Euclidean system, the smaller the part of the continuum under our notice.

“All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.”

According to the general theory of relativity, on the other hand, by application of arbitrary substitutions of the Gauss variables x1, x2, x3, x4, the equations must pass over into equations of the same form; for every transformation (not only the Lorentz transformation) corresponds to the transition of one Gauss coordinate system into another.