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55
Relativity: The Special and General Theory
1
2
3
4
Albert Einstein: Relativity
Part II: The General Theory of Relativity
The Space?Time Continuum
of the Speical 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 17. 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 dx¹, dy¹, dz¹, dt¹. Then these magnitudes
always fulfil the condition
1)
dx² + dy² + dz² ? c²dt² = dx¹ ² + dy¹ ² + dz¹ ² ? c²dt¹ ².
The validity of the Lorentz transformation follows from this condition. We can express this as
follows: The magnitude
ds² = dx² + dy² + dz² ? c²dt²,
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,
, by x1, x2, x3, x
4
,
we
also obtaill the result that
ds² = dx
2
+ dx
2
+ dx
2
+ dx
2
.
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.
Thus, if we choose as time?variable the imaginary variable
instead of the real quantity t, we
can regard the space?time contintium — accordance with the special theory of relativity — as a ",
Euclidean " four?dimensional continuum, a result which follows from the considerations of the
preceding section.
Next: The Space?Time Continuum of the General Theory of Realtiivty is Not a Eculidean Continuum