The Einstein Theory of Relativity | Page 5

one meter apart, points in
a level plain, like the angles or squares on a chess board extending out
into infinity, and finally, points in space as they are obtained by
repeatedly shifting that level spot a distance of a meter in the direction
perpendicular to it. If, consequently, one of the points is chosen as an
"original point" we can, proceeding from that point, reach any other

point through three steps in the common perpendicular directions in
which the points are arranged. The figures showing how many meters
are comprized in each of the steps may serve to indicate the place
reached and to distinguish it from any other; these are, as is said, the
"co-ordinates" of these places, comparable, for example, with the
numbers on a map giving the longitude and latitude. Let us imagine
that each point has noted upon it the three numbers that give its
position, then we have something comparable with a measure with
numbered subdivisions; only we now have to do, one might say, with a
good many imaginary measures in three common perpendicular
directions. In this "system of co-ordinates" the numbers that fix the
position of one or the other of the bodies may now be read off at any
moment.
This is the means which the astronomers and their mathematical
assistants have always used in dealing with the movement of the
heavenly bodies. At a determined moment the position of each body is
fixed by its three co-ordinates. If these are given, then one knows also
the common distances, as well as the angles formed by the connecting
lines, and the movement of a planet is to be known as soon as one
knows how its co-ordinates are changing from one moment to the other.
Thus the picture that one forms of the phenomena stands there as if it
were sketched on the canvas of the motionless ether.

EINSTEIN'S DEPARTURE
Since Einstein has cut loose from the ether, he lacks this canvas, and
therewith, at the first glance, also loses the possibility of fixing the
positions of the heavenly bodies and mathematically describing their
movement--i.e., by giving comparisons that define the positions at
every moment. How Einstein has overcome this difficulty may be
somewhat elucidated through a simple illustration.
On the surface of the earth the attraction of gravitation causes all bodies
to fall along vertical lines, and, indeed, when one omits the resistance
of the air, with an equally accelerated movement; the velocity increases
in equal degrees in equal consecutive divisions of time at a rate that in
this country gives the velocity attained at the end of a second as 981
centimeters (32.2 feet) per second. The number 981 defines the
"acceleration in the field of gravitation," and this field is fully

characterized by that single number; with its help we can also calculate
the movement of an object hurled out in an arbitrary direction. In order
to measure the acceleration we let the body drop alongside of a vertical
measure set solidly on the ground; on this scale we read at every
moment the figure that indicates the height, the only co-ordinate that is
of importance in this rectilinear movement. Now we ask what would
we be able to see if the measure were not bound solidly to the earth, if
it, let us suppose, moved down or up with the place where it is located
and where we are ourselves. If in this case the speed were constant,
then, and this is in accord with the special theory of relativity, there
would be no motion observed at all; we should again find an
acceleration of 981 for a falling body. It would be different if the
measure moved with changeable velocity.
If it went down with a constant acceleration of 981 itself, then an object
could remain permanently at the same point on the measure, or could
move up or down itself alongside of it, with constant speed. The
relative movement of the body with regard to the measure should be
without acceleration, and if we had to judge only by what we observed
in the spot where we were and which was falling itself, then we should
get the impression that there was no gravitation at all. If the measure
goes down with an acceleration equal to a half or a third of what it just
was, then the relative motion of the body will, of course, be accelerated,
but we should find the increase in velocity per second one-half or
two-thirds of 981. If, finally, we let the measure rise with a uniformly
accelerated movement, then we shall find a greater acceleration than
981 for the body itself.
Thus we see that we, also when the measure is not attached to the earth,
disregarding its displacement,