WEBVTT

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Section sixteen, Experience and the special
theory of relativity. To what extent is

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the special theory of relativity supported by
experience? This question is not easily answered,

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for the reason already mentioned in connection
with the fundamental experiment of the zoe.

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The special theory of relativity has crystallized
out from the Maxwell Lorentz theory of

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electromagnetic phenomena. Thus, all facts
of experience which support the electromagnetic theory also

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support the theory of relativity, as
being of particular importance. I mention here

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the fact that the theory of relativity
enables us to predict the effects produced on

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the light reaching us from the fixed
stars. These results are obtained in an

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exceedingly simple manner, and the effects
indicated, which are due to the relative

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motion of the Earth with reference to
those fixed stars, are found to be

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in accord with experience. We refer
to the yearly movement of the apparent position

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of the fixed stars resulting from the
motion of the Earth round the Sun aberration,

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and to the influence of the radial
components of the relative motions of the

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fixed stars with respect to the Earth
on the color of the light reaching us

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from them. The latter effect manifests
itself in a slight displacement of the spectral

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lines of the light transmitted to us
from a fixed star as compared with a

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position of the same spectral lines when
they are produced by a terrestrial source of

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light Doppler principle. The experimental arguments
in favor of the Maxwell Lorenz theory,

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which are at the same time arguments
in favor of the theory of relativity,

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are too numerous to be set forth
here. In reality, they limit the

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theoretical possibilities to such an extent that
no other theory than that of Maxwell and

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Lorenz has been able to hold its
own when tested by experience. But there

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are two classes of experimental facts hitherto
obtain which can be represented in the Maxwell

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Lorenz theory only by the introduction of
an auxiliary hypothesis, which in itself,

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ie without making use of the theory
of relativity, appears extraneous. It is

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known that cathode rays, in the
so called beta rays emitted by radioactive substances,

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consists of negatively electrified particles electrons of
very small inertia and large velocity.

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By examining the deflection of these rays
under the influence of electric and magnetic fields,

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we can study the law of motion
of these particles very exactly. In

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the theoretical treatment of these electrons,
we are faced with a difficulty that electrodynamic

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theory of itself is unable to give
an account of their nature. For since

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electrical masses of one sign repel each
other, the negative electrical masses constituting the

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electron would necessarily be scattered under the
influence of their mutual repulsions, unless there

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are forces of another kind operating between
them, the nature of which has hitherto

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remained obscure to us. Footnote.
The general theory of relativity renders it likely

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that the electrical masses of an electron
are held together by gravitational forces. End

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footnote. If we now assume that
the relative distances between the electrical masses constituting

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the electron remain unchanged during the motion
of the electron rigid connection in the sense

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of classical mechanics, we arrive at
a law of motion of the electron which

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does not agree with experience guided by
purely formal points of view. H.

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A. Lorentz was the first to
introduce the hypothesis that the particles constituting the

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electron experience a contraction in the direction
of motion, and consequence of that motion,

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the amount of this contraction being proportional
to the expression the square root of

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the difference I minus the fraction V
squared OVERC squared. This hypothesis, which

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is not justifiable by any electrodynamical facts, supplies us with that particular law of

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motion which has been confirmed with great
precision in recent years. The theory of

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relativity leads to the same law of
motion without requiring any special hypothesis whatsoever as

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to the structure and the behavior of
the electron. We arrived at a similar

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conclusion in section thirteen in connection with
the experiment of the Zoo, result of

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which is foretold by the theory of
relativity, without the necessity of drawing on

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hypotheses as to the physical nature of
the liquid. The second class effects to

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which we have alluded has referenced the
question whether or not the motion of the

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Earth in space can be made perceptible
in terrestrial experiments. We have already remarked

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in section five that all attempts of
this nature led to a negative result.

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Before the theory of relativity was put
forward, it was difficult to become reconciled

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to this negative result. For reasons
now to be discussed, the inherited prejudices

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about time and space did not allow
any doubts to arise as to the prime

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importance of the Galiley transformation for changing
over from one body of reference to another.

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Now, assuming that maxweller Runs equations
hold for a reference body K,

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we then find that they do not
hold for a reference body K prime moving

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uniformly with respect to K. If
we assume that the relations of the Galilean

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transformation exist between the coordinates of K
and k prime, it thus appears that

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of all Galilean coordinate systems, one
K corresponding to our particular state of motion,

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is physically unique. This result was
interpreted physically by regarding K as at

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rest with respect a hypothetical ether of
space. On the other hand, all

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coordinate systems k prime moving relatively to
k or to be regarded as in motion

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with respect to the ether. To
this motion of K prime against the ether,

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ether drift relative to k prime were
assigned the more complicated laws which were

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supposed to hold relative to kay prime. Strictly speaking, such an ether drift

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thought also to be assumed relative to
the Earth and for a long time the

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efforts of physicists were devote to attempts
to detect the existence of an ether drift

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at the Earth's surface. In one
of the most notable of these attempts,

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Michaelson divides a method which appears as
though it must be decisive. Imagine two

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mirrors so arranged on a rigid body
that the reflecting surfaces face each other.

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A ray of light requires a perfectly
definite time tea to pass from one mirror

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to the other and back again,
if the whole system be at rest with

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respect to the ether. It is
found by calculation, however, that a

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slight different time tea prime is required
for this process if the body, together

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with the mirrors, be moving relatively
to the ether. And yet another point,

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it is shown by calculation that,
for a given velocity v with reference

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to the ether, this time tea
prime is different when the body is moving

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perpendicularly to the planes of the mirrors, from that resulting when the motion is

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parallel to these planes. Although the
estimated difference between these two times is exceedingly

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small, Michaelson and Morally performed an
experiment involving interference in which this result should

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have been clearly detectable, but the
experiment gave a negative result, a fact

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very perplexing to physicists. Lorentz and
Fitzgerald rescued this theory from this difficulty by

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assuming that the motion of the body
relative to the ether produces a contraction of

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the body in the direction of motion, the amount of contraction being just sufficient

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to compensate for the difference in time
mentioned above. Comparison with the discussion in

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section twelve shows us that from the
standpoint also of the theory of relativity,

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the solution of the difficulty was the
right one. But on the basis of

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the theory of relativity, the method
of interpretation is incomparably more satisfactory. According

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to this theory, there is no
such thing as especially favored, unique coordinate

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system to occasion the introduction of the
ether idea, and hence there can be

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no either drift nor any experiment with
which to demonstrate it. Here, the

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contraction of moving bodies follows from the
fundamental principles of the theory, without the

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introduction of particular hypotheses, and as
the prime factor involved in this contraction,

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we find not the motion in itself, to which we cannot attach any meaning

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but the motion with respect to the
body of reference chosen in the particular case

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in point. Thus, for a
coordinate system moving with the Earth, the

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mirror system of michaelsonnemorphy is not shortened, but it is shortened for a coordinate

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system which is at rest relatively to
the Sun. End of section sixteen,

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Section seventeen Minkowski's four dimensional space.
The non mathematician is seized by a mysterious

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shuddering when a year is of four
dimensional things, by feeling not unlike that

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awakened by thoughts of the occult.
And yet there is no more commonplace statement

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than that the world in which we
live is a four dimensional space time continuum

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space is a three dimensional continuum.
By this we mean that it is possible

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to describe the position of a point
at rest by means of three numbers or

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coordinates x, y z, and
that there is an indefinite number of points

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in the neighborhood of this one,
the position of which can be described by

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coordinates such as x of one y
so of one z s one, which

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may be as near as we choose
the respective values of the x, y

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z of the first point in virtue
of the latter property, we speak of

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a continuum, and owing to the
fact that there are three coordinates, we

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speak of it as being three dimensional. Similarly, the world of physical phenomena,

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which was briefly called world by Minkowski, is naturally four dimensional in the

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space time sense, for it is
composed of individual events, each of which

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is described by four numbers, namely
three space coordinates x, y z and

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a time coordinate the time value T. The world is in this sense also

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a continuum, for to every event, there are as many neighboring events realized

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or at least thinkable as we care
to choose the coordinates x of one in

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whits of one ze tsi, one
of which differ by an indefinitely small amount

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from those of the events x y
ZT. Originally considered that we have not

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been accustomed to regard the world in
this sense as a four dimensional continuum is

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due to the fact that in physics, before the advent of the theory of

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relativity, time plated different in more
independent role as compared with the space coordinates.

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It is for this reason that we
have been in the habit of treating

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time as an independent continuum. As
a matter of fact, according to classical

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mechanics, time is absolute, ie
eve is independent of the position and the

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condition of the motion of the system
of coordinates. We see this expressed in

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the last equation of the Galilean transformation
t prime equals t. The four dimensional

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mode of consideration of the world is
natural on the theory of relativity, since

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according to this theory, time is
robbed of its independence. This is shown

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by the fourth equation of the Lorentz
transformation t prime equals the fraction, in

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which the numerator is t minus the
fraction v x over c squared and the

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denominator is the square root of the
difference i minus the fraction v squared over

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c squared. Moreover, according to
this equation, the time difference delta T

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prime of two events with respect to
k prime does not in general vanish,

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even when the time difference of delta
t of the same events with reference to

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Kay vanishes. Pure space distance of
two events with respect de kay results in

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time distance of the same events with
respect to Kay prime. But the discovery

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of Minkowski, which was of importance
in the formal development of the theory of

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relativity, does not lie here.
It is to be found rather in the

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fact of his recognition that the four
dimensional space time continuum of the theory of

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relativity, in its most essential formal
properties, shows a pronounced for relationship to

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the three dimensional continuum of Euclidean geometrical
space. Begin footnote. Compare the somewhat

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more detailed discussion in Appendix two end
footnote. In order to give due prominence

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to this relationship, however, we
must replace the usual time coordinate T by

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an imaginary magnitude square root of negative
eye set proportional to it. Under these

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conditions, the natural laws, satisfying
the demands of the special theory of relativity

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assume mathematical forms in which the time
coordinate plays exactly the same role as the

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three space coordinates. Formally, these
four coordinates correspond exactly to the three space

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coordinates in Euclidean geometry. It must
be clear even to the non mathematician that,

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as a consequence of this purely formal
addition to our knowledge, the theory

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purports gained clearless. In no mean
measure, these inadequate remarks can give the

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reader only a vague notion of the
important idea contributed by Minkowski. Without it

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the general theory of relativity, of
which the fundamental ideas are developed in the

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following pages, would perhaps have got
no farther than its long clothes. Minkowski's

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work is doubtless difficult of access to
anyone in inexperienced in mathematics. But since

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it is not necessary to have a
very exact grasp of this work in order

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to understand the fundamental ideas of either
the special or the general theory of relativity,

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I shall at present leave it here, and shall revert to it only

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towards the end of Part two end
of section seventeen.