XXVI. The SpaceTime Continuum of the Special Theory of Relativity Considered as a Euclidean Continuum
Relativity: The Special and General TheoryAlbert Einstein
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 coordinate systems are given preference for the description of the fourdimensional, spacetime continuum. We called these “Galileian coordinate systems.” For these systems, the four coordinates $x, y, z, t$, which determine an event or—in other words, a point of the fourdimensional 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 fourdimensional continuum is given with respect to a Galileian referencebody $K$ by the space coordinate differences $dx, dy, dz$ and the timedifference $dt$. 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^{1}.
$dx^2 + dy^2 + dz^2  c^2dt^2 = dx' 2 + dy' 2 + dz' 2  c^2dt'^2$The validity of the Lorentz transformation follows from this condition. We can express this as follows: The magnitude
$ds^2 = dx^2 + dy^2 + dz^2  c^2dt^2$which belongs to two adjacent points of the fourdimensional spacetime continuum, has the same value for all selected (Galileian) referencebodies. If we replace $x, y, z$, $\sqrt{I} \cdot ct$ , by $x_1, x_2, x_3, x_4$, we also obtaill the result that
$ds^2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2$is independent of the choice of the body of reference. We call the magnitude ds the “distance” apart of the two events or fourdimensional points.
Thus, if we choose as timevariable the imaginary variable $\sqrt{I} \cdot ct$ instead of the real quantity $t$, we can regard the spacetime contintium—accordance with the special theory of relativity—as a “Euclidean” fourdimensional continuum, a result which follows from the considerations of the preceding section.
XXVII. The SpaceTime Continuum of the General Theory of Relativity is Not a Euclidean Continuum

Cf. Appendixes I and II. The relations which are derived there for the coordlnates themselves are valid also for coordinate differences, and thus also for coordinate differentials (indefinitely small differences).
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