*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.*

.

## 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 *x _{1}, x_{2}, x_{3}, x_{4},* 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.*

.