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, may describe the motion of the
body in respect to the measure always in the same way--i.e., as one
uniformly accelerated, as we ascribe now and again a fixed value to
the acceleration of the sphere of gravitation, in a particular case
the value of zero.
Of course, in the case here under consideration the use of a measure
fixed immovably upon the earth should merit all recommendation. But
in the spaces of the solar system we have, now that we have abandoned
the ether, no such support. We can no longer establish a system of
co-ordinates, like the one just mentioned, in a universal intermediate
matter, and if we were to arrive in one way or another at a definite
system of lines crossing each other in three directions, then we should
be able to use just as well another similar system that in respect to
the first moves this or that way. We should also be able to remodel the
system of co-ordinates in all kinds of ways, for example by extension
or compression. That in all these cases for fixed bodies that do not
participate in the movement or the remodelling of the system other
co-ordinates will be read off again and again is clear.