HKB
ds2 = dx12 + dx22 + dx32 + dx42.
In this case relations hold in the four-dimensional continuum which are analogous to those holding in our three-dimensional measurements.
However, the Gauss treatment for ds2 which we have given above is not always possible. It is only possible when sufficiently small regions of the continuum under consideration may be regarded as Euclidean continua. For example, this obviously holds in the case of the marble slab of the table and local variation of temperature. The temperature is practically constant for a small part of the slab, and thus the geometrical behaviour of the rods is almost as it ought to be according to the rules of Euclidean geometry. Hence the imperfections of the construction of squares in the previous section do not show themselves clearly until this construction is extended over a considerable portion of the surface of the table.
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 coordinates) 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 coordinates) which differ by an indefinitely small amount are assigned to adjacent points. The Gaussian coordinate 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.
XXVI.
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.
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 fulfill the condition.[21]
[21] Cf. Appendixes 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).
dx2 + dy2 + dz2 – c2dt2 = *dx′*2 + *dy′*2 + *dz′2 – c2dt′*2.
The validity of the Lorentz transformation follows from this condition. We can express this as follows: The magnitude
ds2 = dx2 + dy2 + dz2 – c2 dt2,
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,
image034
by x1, x2, x3, x4, we also obtain the result that
ds2 = dx12 + dx22 + dx32 + dx42.