Reference: Einstein’s 1920 Book
This paper presents Part II, Chapter 9 from the book RELATIVITY: THE SPECIAL AND GENERAL THEORY by A. EINSTEIN. The contents are from the original publication of this book by Henry Holt and Company, New York (1920).
The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding. Feedback on these comments is appreciated.
The heading below is linked to the original materials.
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The Space-Time Continuum of the Special Theory of Relativity Considered as a Euclidean Continuum
We are now in a position to formulate more exactly the idea of Minkowski, which was only vaguely indicated in Section XVII. In accordance with the special theory of relativity, certain co-ordinate systems are given preference for the description of the four-dimensional, space-time continuum. We called these “Galileian co-ordinate systems.” For these systems, the four co-ordinates x, y, z, t, which determine an event or—in other words—a point of the four-dimensional continuum, are defined physically in a simple manner, as set forth in detail in the first part of this book. For the transition from one Galileian system to another, which is moving uniformly with reference to the first, the equations of the Lorentz transformation are valid. These last form the basis for the derivation of deductions from the special theory of relativity, and in themselves they are nothing more than the expression of the universal validity of the law of transmission of light for all Galileian systems of reference.
The space coordinates (x, y, z) and the time coordinate (t) represent two very different dimensions. The Lorentz transformation uses the speed of light as an absolute basis for all other velocities.
Minkowski found that the Lorentz transformations satisfy the following simple conditions. Let us consider two neighbouring events, the relative position of which in the four-dimensional continuum is given with respect to a Galileian reference-body K by the space co-ordinate differences dx, dy, dz and the time-difference dt. With reference to a second Galileian system we shall suppose that the corresponding differences for these two events are dx’, dy’, dz’, dt’. Then these magnitudes always fulfil the condition. [Cf. Appendices I and II. The relations which are derived there for the co-ordinates themselves are valid also for co-ordinate differences, and thus also for co-ordinate differentials (indefinitely small differences).]
The validity of the Lorentz transformation follows from this condition. We can express this as follows: The magnitude
which belongs to two adjacent points of the four-dimensional space-time continuum, has the same value for all selected (Galileian) reference-bodies. If we replace x, y, z, √(-1) ct, by x1, x2, x3, x4, we also obtain the result that
is independent of the choice of the body of reference. We call the magnitude ds the “distance” apart of the two events or four-dimensional points.
Einstein relates spatial and time coordinates through the speed of light.
Thus, if we choose as time-variable the imaginary variable √(-1) ct instead of the real quantity t, we can regard the space-time continuum—in accordance with the special theory of relativity—as a “Euclidean” four-dimensional continuum, a result which follows from the considerations of the preceding section.
From mathematical considerations, time is being treated as an imaginary variable in the Euclidean space.
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FINAL COMMENTS
The space coordinates (x, y, z) and the time coordinate (t) represent two very different dimensions, but they are related as follows:
(1) The substance stays at a location in space (x, y, z) for a certain duration of time (t). This duration of substance provides the time-density of quanta at that location. We may identify this time-density as the inertia of quanta.
(2) The greater is the duration of substance at a location the more slowly the quanta is traversing across it. We may say that the natural velocity of quanta is inversely proportional to its inertia.
(3) Therefore, the more thinly quanta is spread in time the greater velocity may be associated with it. In other words, the lesser is the inertia, the greater is the velocity; and vice versa.
(4) Therefore, in the field domain the inertia is very small but the velocity is very high. And, in the material domain the inertia is very large but the velocity is very low.
(5) The space and time densities are proportional to each other, where the proportionality constant is c.
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