What way Einstein had to follow is now apparent. He must--this
hardly needs to be said--in calculating definite, particular cases
make use of a chosen system of co-ordinates, but as he had no means
of limiting his choice beforehand and in general, he had to reserve
full liberty of action in this respect. Therefore he made it his aim
so to arrange the theory that, no matter how the choice was made, the
phenomena of gravitation, so far as its effects and its stimulation
by the attracting bodies are concerned, may always be described in
the same way--i.e., through comparisons of the same general form,
as we again and again give certain values to the numbers that mark
the sphere of gravitation. (For the sake of simplification I here
disregard the fact that Einstein desires that also the way in which
time is measured and represented by figures shall have no influence
upon the central value of the comparisons.)
Whether this aim could be attained was a question of mathematical
inquiry. It really was attained, remarkably enough, and, we may say, to
the surprise of Einstein himself, although at the cost of considerable
simplicity in the mathematical form; it appeared necessary for the
fixation of the field of gravitation in one or the other point in
space to introduce no fewer than ten quantities in the place of the
one that occurred in the example mentioned above.
In this connection it is of importance to note that when we exclude
certain possibilities that would give rise to still greater intricacy,
the form of comparison used by Einstein to present the theory is
the only possible one; the principle of the freedom of choice in
co-ordinates was the only one by which he needed to allow himself to
be guided. Although thus there was no special effort made to reach a
connection with the theory of Newton, it was evident, fortunately,
at the end of the experiment that the connection existed. If we
avail ourselves of the simplifying circumstance that the velocities
of the heavenly bodies are slight in comparison with that of light,
then we can deduce the theory of Newton from the new theory, the
"universal" relativity theory, as it is called by Einstein. Thus
all the conclusions based upon the Newtonian theory hold good, as
must naturally be required. But now we have got further along. The
Newtonian theory can no longer be regarded as absolutely correct in all
cases; there are slight deviations from it, which, although as a rule
unnoticeable, once in a while fall within the range of observation.
Now, there was a difficulty in the movement of the planet Mercury
which could not be solved. Even after all the disturbances caused by
the attraction of other planets had been taken into account, there
remained an inexplicable phenomenon--i.e., an extremely slow turning
of the ellipsis described by Mercury on its own plane; Leverrier had
found that it amounted to forty-three seconds a century. Einstein
found that, according to his formulas, this movement must really
amount to just that much. Thus with a single blow he solved one of
the greatest puzzles of astronomy.
Still more remarkable, because it has a bearing upon a phenomenon which
formerly could not be imagined, is the confirmation of Einstein's
prediction regarding the influence of gravitation upon the course
of the rays of light. That such an influence must exist is taught
by a simple examination; we have only to turn back for a moment to
the following comparison in which we were just imagining ourselves
to make our observations. It was noted that when the compartment is
falling with the acceleration of 981 the phenomena therein will occur
just as if there were no attraction of gravitation. We can then see
an object, A, stand still somewhere in open space. A projectile,
B, can travel with constant speed along a horizontal line, without
varying from it in the slightest.
A ray of light can do the same; everybody will admit that in each case,
if there is no gravitation, light will certainly extend itself in a
rectilinear way. If we limit the light to a flicker of the slightest
duration, so that only a little bit, C, of a ray of light arises,
or if we fix our attention upon a single vibration of light, C, while
we on the other hand give to the projectile, B, a speed equal to that
of light, then we can conclude that B and C in their continued motion
can always remain next to each other. Now if we watch all this, not
from the movable compartment, but from a place on the earth, then we
shall note the usual falling movement of object A, which shows us that
we have to deal with a sphere of gravitation. The projectile B will,
in a bent path, vary more and more from a horizontal straight line,
and the light will do the same, because if we observe the movements
from another standpoint this can have no effect upon the remaining
next to each other of B and C.