16. Expereince and the Special Theory of Relativity17. Minkowski's Four-dimensial Space

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Episode Transcript

Speaker 1 (00:00):
Correction for this chapter in mathematical formulae instead of I
here one. This is a LibriVox recording. All LibriVox recordings
are in the public domain. For more information or to volunteer,
please visit LibriVox dot org. Recording by Kelly Buscher. Relativity,

(00:26):
The Special and General Theory by Albert Einstein, continuing part one,
section sixteen and seventeen 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,

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

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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

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the apparent position of the fixed stars resulting from the
motion of the Earth around 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

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fixed star as compared with the position of the dam
the same spectral lines when they are produced by a
terrestrial source of light Doppler principle. The experimental arguments in
favor of the Maxwell Lorentz 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,

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they limit the theoretical possibilities to such an extent that
no other theory than that of Maxwell and Lorentz has
been able to hold its own when tested by experience.
But there are two classes of experimental facts hitherto obtained
which can be represented in the Maxwell Lurentz theory. Only
by the introduction of an auxiliary hypothesis, which in itself

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i e. 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

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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 the 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

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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 electrical masses of an electron are held together
by gravitational forces en footnote. If we now assume that

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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.
Lorenz was the first to introduce the hypothesis that the

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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 over
C squared. This hypothesis, which is not justifiable by any
electrodynamical facts, supplies us with that particular law of motion

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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

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hypotheses as to the physical nature of the lud 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

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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 Galilei transformation for changing over from one
body of reference to another. Now, assuming that maxweller Run's

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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

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particular state of motion, is physically unique. This result was
interpreted physically by regarding K as at rest with respectable
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

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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 ought also to be assumed relative
to the Earth, and for a long time the efforts
of physicists were devoted 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 devised a

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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 see
be at rest with respect to the ether. It is
found by calculation, however, that a slight different time t

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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 t 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

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estimated difference between these two times is exceedingly small, Michaelson
and Morley performed in 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

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that the motion of the body relative to the ether
produces a contraction of the body in the direction of motion,
the amount of contract 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
the theory of relativity, this solution of the difficulty was
the right one, but on the basis of the theory

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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 two fundamental

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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

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system of Michaelson and Morley is not shortened, and 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

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mysterious shuddering when he years 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,

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where 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 some one z some one,
which may be as near as we choose the respective
values of the coordinate x y z of the first point.
In virtue of the latter property, we speak of a continuum,

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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

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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 realize or
at least thinkable as we care to choose the coordinates
x of one and why sub one zs and onae
tis of one, of which differ by an indefinitely small
amount from those of the events x y z T.

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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 played a different
and 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

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a matter of fact, according to classical mechanics, time is absolute,
i e. It 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 in the theory of relativity,

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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 enumerator
is t minus the fraction vx over C squared and
the denominator is the square root of the difference ius

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the fraction v squared over C squared. Moreover, according to
this equation, the time difference deals to 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 k vanishes. Pure
space distance of two events with respect to k results

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in time distance of the same events with respect to
k 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 relationship to the

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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 te why in
imaginary magnitude square root of negative eye c t proportional

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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. Formerly these four coordinates correspond
exactly to the three space coordinates in Euclidean geometry. It
must be clear even to the non mathematician that as

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

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farther than its long clothes. Minkowski's work is doubtless difficult
of access to anyone inexperienced in mathematics. But since it
is not necessary to have a very exact grasp of
this work in order to understand the fundamental life ideas
of either the special or the general theory of relativity,
I shall at present leave it here, and shall revert

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to it only towards the end of Part two, end
of section seventeen, end of Part one,