Section sixteen, Experience and the special theory of relativity. To what extent is
the special theory of relativity supported by experience? This question is not easily answered,
for the reason already mentioned in connection with the fundamental experiment of the zoe.
The special theory of relativity has crystallized out from the Maxwell Lorentz theory of
electromagnetic phenomena. Thus, all facts of experience which support the electromagnetic theory also
support the theory of relativity, as being of particular importance. I mention here
the fact that the theory of relativity enables us to predict the effects produced on
the light reaching us from the fixed stars. These results are obtained in an
exceedingly simple manner, and the effects indicated, which are due to the relative
motion of the Earth with reference to those fixed stars, are found to be
in accord with experience. We refer to the yearly movement of the apparent position
of the fixed stars resulting from the motion of the Earth round the Sun aberration,
and to the influence of the radial components of the relative motions of the
fixed stars with respect to the Earth on the color of the light reaching us
from them. The latter effect manifests itself in a slight displacement of the spectral
lines of the light transmitted to us from a fixed star as compared with a
position of the same spectral lines when they are produced by a terrestrial source of
light Doppler principle. The experimental arguments in favor of the Maxwell Lorenz theory,
which are at the same time arguments in favor of the theory of relativity,
are too numerous to be set forth here. In reality, they limit the
theoretical possibilities to such an extent that no other theory than that of Maxwell and
Lorenz has been able to hold its own when tested by experience. But there
are two classes of experimental facts hitherto obtain which can be represented in the Maxwell
Lorenz theory only by the introduction of an auxiliary hypothesis, which in itself,
ie without making use of the theory of relativity, appears extraneous. It is
known that cathode rays, in the so called beta rays emitted by radioactive substances,
consists of negatively electrified particles electrons of very small inertia and large velocity.
By examining the deflection of these rays under the influence of electric and magnetic fields,
we can study the law of motion of these particles very exactly. In
the theoretical treatment of these electrons, we are faced with a difficulty that electrodynamic
theory of itself is unable to give an account of their nature. For since
electrical masses of one sign repel each other, the negative electrical masses constituting the
electron would necessarily be scattered under the influence of their mutual repulsions, unless there
are forces of another kind operating between them, the nature of which has hitherto
remained obscure to us. Footnote. The general theory of relativity renders it likely
that the electrical masses of an electron are held together by gravitational forces. End
footnote. If we now assume that the relative distances between the electrical masses constituting
the electron remain unchanged during the motion of the electron rigid connection in the sense
of classical mechanics, we arrive at a law of motion of the electron which
does not agree with experience guided by purely formal points of view. H.
A. Lorentz was the first to introduce the hypothesis that the particles constituting the
electron experience a contraction in the direction of motion, and consequence of that motion,
the amount of this contraction being proportional to the expression the square root of
the difference I minus the fraction V squared OVERC squared. This hypothesis, which
is not justifiable by any electrodynamical facts, supplies us with that particular law of
motion which has been confirmed with great precision in recent years. The theory of
relativity leads to the same law of motion without requiring any special hypothesis whatsoever as
to the structure and the behavior of the electron. We arrived at a similar
conclusion in section thirteen in connection with the experiment of the Zoo, result of
which is foretold by the theory of relativity, without the necessity of drawing on
hypotheses as to the physical nature of the liquid. The second class effects to
which we have alluded has referenced the question whether or not the motion of the
Earth in space can be made perceptible in terrestrial experiments. We have already remarked
in section five that all attempts of this nature led to a negative result.
Before the theory of relativity was put forward, it was difficult to become reconciled
to this negative result. For reasons now to be discussed, the inherited prejudices
about time and space did not allow any doubts to arise as to the prime
importance of the Galiley transformation for changing over from one body of reference to another.
Now, assuming that maxweller Runs equations hold for a reference body K,
we then find that they do not hold for a reference body K prime moving
uniformly with respect to K. If we assume that the relations of the Galilean
transformation exist between the coordinates of K and k prime, it thus appears that
of all Galilean coordinate systems, one K corresponding to our particular state of motion,
is physically unique. This result was interpreted physically by regarding K as at
rest with respect a hypothetical ether of space. On the other hand, all
coordinate systems k prime moving relatively to k or to be regarded as in motion
with respect to the ether. To this motion of K prime against the ether,
ether drift relative to k prime were assigned the more complicated laws which were
supposed to hold relative to kay prime. Strictly speaking, such an ether drift
thought also to be assumed relative to the Earth and for a long time the
efforts of physicists were devote to attempts to detect the existence of an ether drift
at the Earth's surface. In one of the most notable of these attempts,
Michaelson divides a method which appears as though it must be decisive. Imagine two
mirrors so arranged on a rigid body that the reflecting surfaces face each other.
A ray of light requires a perfectly definite time tea to pass from one mirror
to the other and back again, if the whole system be at rest with
respect to the ether. It is found by calculation, however, that a
slight different time tea prime is required for this process if the body, together
with the mirrors, be moving relatively to the ether. And yet another point,
it is shown by calculation that, for a given velocity v with reference
to the ether, this time tea prime is different when the body is moving
perpendicularly to the planes of the mirrors, from that resulting when the motion is
parallel to these planes. Although the estimated difference between these two times is exceedingly
small, Michaelson and Morally performed an experiment involving interference in which this result should
have been clearly detectable, but the experiment gave a negative result, a fact
very perplexing to physicists. Lorentz and Fitzgerald rescued this theory from this difficulty by
assuming that the motion of the body relative to the ether produces a contraction of
the body in the direction of motion, the amount of contraction being just sufficient
to compensate for the difference in time mentioned above. Comparison with the discussion in
section twelve shows us that from the standpoint also of the theory of relativity,
the solution of the difficulty was the right one. But on the basis of
the theory of relativity, the method of interpretation is incomparably more satisfactory. According
to this theory, there is no such thing as especially favored, unique coordinate
system to occasion the introduction of the ether idea, and hence there can be
no either drift nor any experiment with which to demonstrate it. Here, the
contraction of moving bodies follows from the fundamental principles of the theory, without the
introduction of particular hypotheses, and as the prime factor involved in this contraction,
we find not the motion in itself, to which we cannot attach any meaning
but the motion with respect to the body of reference chosen in the particular case
in point. Thus, for a coordinate system moving with the Earth, the
mirror system of michaelsonnemorphy is not shortened, but it is shortened for a coordinate
system which is at rest relatively to the Sun. End of section sixteen,
Section seventeen Minkowski's four dimensional space. The non mathematician is seized by a mysterious
shuddering when a year is of four dimensional things, by feeling not unlike that
awakened by thoughts of the occult. And yet there is no more commonplace 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 or
coordinates x, y z, and that there is an indefinite number of points
in the neighborhood of this one, the position of which can be described by
coordinates such as x of one y so of one z s one, which
may be as near as we choose the respective values of the 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 coordinates, 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 coordinates x, y z and
a time coordinate the time value T. The world is in this sense also
a continuum, for to every event, there are as many neighboring events realized
or at least thinkable as we care to choose the coordinates x of one in
whits of one ze tsi, one of which differ by an indefinitely small amount
from those of the events x y ZT. 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 plated different in 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, ie eve is independent of the position and the
condition of the motion of the system of coordinates. We see this expressed in
the last equation of the Galilean transformation t prime equals 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 t prime equals the fraction, in
which the numerator is t minus the fraction v x over c squared and the
denominator is the square root of the difference i minus the fraction v squared over
c squared. Moreover, according to this equation, the time difference delta T
prime of two events with respect to k prime does not in general vanish,
even when the time difference of delta t of the same events with reference to
Kay vanishes. Pure space distance of two events with respect de kay results in
time distance of the same events with respect to Kay prime. But the discovery
of Minkowski, which was of importance in 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 for relationship to
the three dimensional continuum of Euclidean geometrical space. Begin footnote. Compare the somewhat
more detailed discussion in Appendix two end footnote. In order to give due prominence
to this relationship, however, we must replace the usual time coordinate T by
an imaginary magnitude square root of negative eye set proportional to it. Under these
conditions, the natural laws, satisfying the demands of the special theory of relativity
assume mathematical forms in which the time coordinate plays exactly the same role as the
three space coordinates. Formally, these four coordinates correspond exactly to the three space
coordinates in Euclidean geometry. It must be clear even to the non mathematician that,
as a consequence of this purely formal addition to our knowledge, the theory
purports gained clearless. In no mean measure, these inadequate remarks can give the
reader only a vague notion of the important idea contributed by Minkowski. Without it
the general theory of relativity, of which the fundamental ideas are developed in the
following pages, would perhaps have got no farther than its long clothes. Minkowski's
work is doubtless difficult of access to anyone in inexperienced in mathematics. But since
it is not necessary to have a very exact grasp of this work in order
to understand the fundamental ideas of either the special or the general theory of relativity,
I shall at present leave it here, and shall revert to it only
towards the end of Part two end of section seventeen.