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Book by David Michalets
An (original) line precedes original content from the source.
A (remark) line precedes my remark from my review of the preceding original content.
My remark applies to only this section of the original.
Section XXVI of 35
CONSIDERED AS A EUCLIDEAN CONTINUUM
(original)
E 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-imensional 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 fulfil the condition.
dx^2 + dy^2 + dz^2 − c dt^2 = dx'^2 + dy'^2 + dz'^2 − c^2 dt'^2
The validity of the Lorentz transformation follows from this condition. We can express this as follows: The magnitude ds^2 = dx^2 + dy^2 + dz^2 − c^2 dt^2 ,
which belongs to two adjacent points of the fourdimensional space-time continuum, has the same value for all selected (Galileian) reference-bodies.
If we replace x, y, z, square-root(−1) ct, by x1, x2, x3, x4, we also obtain the result that
ds^2 = dx^2 + dx1^2 + dx2^2 + dx3^2 + dx4^2 *
is independent of the choice of the body of refer1 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).
ence. We call the magnitude ds the "distance" apart of the two events or four-dimensional points.
Thus, if we choose as time-variable the imaginary variable (square root of −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.
(remark)
When starting with space-time which baased on a misunderstanding of a gravitational field, a continuum of the mistake is a non-starter.
A continuum implies the universe has structure. It doesn't. See Background for Newton's absolute space and time.
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last change 05/07/2022