# Einstein

Relativity: The Special and General Theory

Albert Einstein: Relativity

Part I: The Special Theory of Relativity

Minkowski's Four−Dimensional Space

The non−mathematician is seized by a mysterious shuddering when he hears of

"four−dimensional" things, by a feeling not unlike that awakened by thoughts of the occult. And yet

there is no more common−place statement than that the world in which we live is a

four−dimensional space−time continuum.

Space is a three−dimensional continuum. By this we mean that it is possible to describe the

position of a point (at rest) by means of three numbers (co−ordinales) x, y, z, and that there is an

indefinite number of points in the neighbourhood of this one, the position of which can be described

by co−ordinates such as x1, y1, z1, which may be as near as we choose to the respective values of

the co−ordinates x, y, z, of the first point. In virtue of the latter property we speak of a " continuum,"

and owing to the fact that there are three co−ordinates we speak of it as being "

three−dimensional."

Similarly, the world of physical phenomena which was briefly called " world " by Minkowski is

naturally four dimensional in the space−time sense. For it is composed of individual events, each of

which is described by four numbers, namely, three space co−ordinates x, y, z, and a time

co−ordinate, the time value t. The" world" is in this sense also a continuum; for to every event there

are as many "neighbouring" events (realised or at least thinkable) as we care to choose, the

co−ordinates x1, y1, z1, t1 of which differ by an indefinitely small amount from those of the event x,

y, z, t originally considered. That we have not been accustomed to regard the world in this sense as

a four−dimensional continuum is due to the fact that in physics, before the advent of the theory of

relativity, time played a different and more independent role, as compared with the space

coordinates. It is for this reason that we have been in the habit of treating time as an independent

continuum. As a matter of fact, according to classical mechanics, time is absolute, i.e. it is

independent of the position and the condition of motion of the system of co−ordinates. We see this

expressed in the last equation of the Galileian transformation (t1 = t)

The four−dimensional mode of consideration of the "world" is natural on the theory of relativity,

since according to this theory time is robbed of its independence. This is shown by the fourth

equation of the Lorentz transformation:

Moreover, according to this equation the time difference ”t1 of two events with respect to K1 does

not in general vanish, even when the time difference ”t1 of the same events with reference to

K vanishes. Pure " space−distance " of two events with respect to K results in " time−distance " of

the same events with respect to K. But the discovery of Minkowski, which was of importance for the

formal development of the theory of relativity, does not lie here. It is to be found rather in the fact of

his recognition that the four−dimensional space−time continuum of the theory of relativity, in its

most essential formal properties, shows a pronounced relationship to the three−dimensional

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