WEBVTT

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Section thirteen theorem of the addition of
velocities the experiment of Faso. Now,

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in practice, we can move clocks
in measuring rods only with velocities that are

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small compared with the velocity of light. Hence, we shall hardly be able

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to compare the results of the previous
section directly with the reality. But on

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the other hand, these results must
strike you as being very singular, and

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for that reason I shall now draw
another conclusion from the theory, one which

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can easily be derived from the foregoing
considerations, and which has been most elegantly

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confirmed by experiment. In section six, we derive the theorem of the addition

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of velocities in one direction in the
form which also results from the hypotheses of

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classical mechanics. This theorem can also
be deduced readily from the Galiley transformation section

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eleven. In place of the man
walking inside the carriage, we introduce a

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point moving relatively to the coordinate system
k prime in accordance with the equation x

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prime equals w t prime. By
means of the first and fourth equations of

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the Galiley transformation, we can express
x prime and t prime in terms of

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x and t and We then obtain
x equals parentheses V plus W unparentheses T.

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This equation expresses nothing else than the
law of motion of the point with

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reference to the system K of the
man with reference to the embankment. We

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denote this velocity by the symbol capital
W, and we then obtain, as

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in section six, capital W equals
V plus W equation A. But we

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can carry out this consideration just as
well on the basis of the theory of

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relativity. In the equation x prime
equals w t prime. We must then

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express x prime and T prime in
terms of x and t, making use

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of the first and fourth equations of
the Lorentz transformation instead of the equation A.

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We then obtain the equation capital w
equals the sum V plus W over

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the sum I plus v W over
c squared equation B, which corresponds to

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the theorem of addition for velocities in
one direction according to the theory of relativity.

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The question now arises as to which
of these two theorems is the better

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in accord with experience. On this
point, where enlightened by a most important

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experiment which the brilliant physicist faces,
O perform more than half a century ago,

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and which has been repeated since then
by some of the best experimental physicists,

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so that there can be no doubt
about its result. Experiment is concerned

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with a following question. Light travels
in a motionless liquid with a particular velocity

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W. How quickly does it travel
in the direction of the arrow in the

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tube T see the accompanying diagram Fig. Three. When the liquid above mentioned

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is flowing through the tube with a
velocity V in accordance with the principle of

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relativity, we shall certainly have to
take for granted that the propagation of light

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always takes place with the same velocity
W with respect to the liquid, whether

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the latter is in motion with reference
to other bodies or not. The velocity

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of light relative to the liquid and
the velocity of the latter relative to the

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tube are thus known, and we
require the velocity of light relative to the

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tube. It is clear that we
had the problem the section six again before

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us. The two plays a part
of the railway embankment or of the coordinate

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system K. The liquid plays a
part of the carriage or of the coordinate

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system K prime, and finally,
the light plays a part of the man

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walking along the carriage or of the
moving point in the present section. If

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we denote the velocity of the light
relative to the tube by capital W,

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then this is a given by the
equation A or B according as a Galiley

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transformation or the Lorentz transformation corresponds to
the facts. Experiment decides in favor of

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equation B derived from the theory of
relativity, and the agreement is indeed very

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exact. Footnote I say so foul
capital W equals W plus v open parentheses,

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I minus I over n squared close
parentheses, where n equals c OVERW

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is the index of refraction of the
liquid. On the other hand, owing

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to the smallness of v w oversea
squared as compared with I, we can

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replace B in the first place by
capital W equals open parentheses W plus v

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close parentheses, open parentheses, I
minus the fraction v w oversea squared close

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parentheses, or to the same order
of approximation by W plus v open parentheses,

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I minus I over in squared close
parentheses, which agrees with Vasso's resolved

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n. Footnote. According to recent
and most excellent measurements by Zemen, the

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influence of the velocity of flow v
on the propagation of light is represented by

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formula B to within one percent.
Nevertheless, we must now draw attention to

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the fact that a theory of this
phenomenon was given by H. A.

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Lorentz long before the statement of the
theory of relativity. This theory was of

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a purely electrodynamical nature and was obtained
by the use of particular hypotheses as to

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the electromagnetic structure of matter. This
circumstance, however, does not in the

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least diminish the conclusiveness of the experiment
as a crucial test in favor of the

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theory of relativity. For the electrodynamics
of Maxims Well Orents, in which the

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original theory was based, in no
way opposes the theory of relativity. Rather,

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has the latter been developed from electrodynamics
as an astoundingly simple combination and generalization

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of the hypotheses formally independent of each
other on which electrodynamics was built. End

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of Section thirteen. Section fourteen.
The heuristic value of the theory of relativity

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our train of thought in the foregoing
pages can be epiimized in the following manner.

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Experiences led to the conviction that on
the one hand, the principle of

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relativity holds true, and that,
on the other hand, the velocity of

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transmission of light invokwell has to be
considered equal to a constant C. By

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uniting these two postulates, we obtain
the law of transformation for the rectangular coordinates

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x, y, c and the
time T of the events which constitute the

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processes of nature. In this connection, we did not obtain the Galiley transformation,

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but, differing from classical mechanics,
the Lorentz transformation, the law of

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transmission of light, the acceptance of
which is justified by our actual knowledge,

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played an important part in this process
of thought. Once in possession of the

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Lorentz transformation, however, we can
combine this with a principle of relativity and

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sum up the theory. Thus,
every general law of nature must be so

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constituted that it is transformed into a
law of exactly the same form when instead

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of the space time variables xy c
t of the original coordinate system k,

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we introduce new space time variables x
prime y prime c prime t prime fe

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coordinate system k prime. In this
connection, the relation between the ordinary and

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the accent of magnitudes is given by
the Lorentz transformation, or in brief,

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general laws of nature are co variant
with respect to Lorentz transformation. This is

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a definite mathematical condition that the theory
of relativity demands of a natural law,

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and in virtue of this the theory
becomes valuable heuristic aid in the search for

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general laws of nature. If a
general law of nature were to be found

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which did not satisfy this condition,
then at least one of the two fundamental

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assumptions of the theory would have been
disproved. Let Us now examine what general

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results the latter theory has hitherto evinced. End of Section fourteen. Section fifteen

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General results of the theory. It
is clear from our previous considerations that the

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special theory relativity has grown out of
electrodynamics and optics. In these fields,

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it has not appreciably altered the predictions
of theory, but it has considerably simplified

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the theoretical structure, ie the derivation
of laws, and what is incomparably more

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important, it has considerably reduced a
number of independent hypotheses forming the basis of

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theory. The special theory of relativity
is rendered the Maxwell Orens theory so plausible

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that the latter would have been generally
accepted by physicists even if experiment had decided

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less unequivocally in its favor. Classical
mechanics require to be modified before it could

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come into line with the demands of
the special theory of relativity. For the

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main part, however, this modification
affects only the laws for rapid motions in

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which the velocities of matter fee are
not very small as compared with the velocity

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of light. We have experience of
such rapid motions only in the case of

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electrons and ions. For other motions, the variations from the laws of classical

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mechanics are too small to make themselves
evident in practice. We shall not consider

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the motion of stars until we come
to speak of the general theory of relativity.

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In accordance with the theory of relativity, the kinetic energy of a material

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point of mass M is no longer
given by the well known expression m v

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squared over two, but by the
expression mc squared over the square root of

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the difference I minus the fraction v
squared over C squared. This expression approaches

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infinity as the velocity V approaches the
velocity of light c. The velocity must

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therefore always remain less than C.
However great may be the energies used to

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produce acceleration. If we develop the
expression for the kinetic energy in the form

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of a series, we obtain mc
squared plus m v squared over two plus

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three ace m fee to the fourth
over C squared plus c. When v

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squared over c squared is small compared
with unity, the third of these terms

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is always small in comparison with a
second, which last is alone considered in

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classical mechanics. The first term mc
squared does not contain the velocity and requires

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no consideration. If we are only
dealing with a question as to how the

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energy of a point mass depends on
the velocity, we shall speak of its

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essential significance later. The most important
result of a general character to which the

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special theory of relativity has led is
concerned learned with the conception of mass.

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Before the advent of relativity, physics
recognized two conservation laws of fundamental importance,

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namely, the law of conservation of
energy and the law of the conservation of

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mass. These two fundamental laws appeared
to be quite independent of each other.

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By means of the theory of relativity, they have been united into one law.

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We shall now briefly consider how this
unification came about and what meaning is

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to be attached to it. The
principle of relativity requires that the law of

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the conservation of energy should hold not
only with reference to a coordinate system K,

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but also with respect to every coordinate
system K prime which is in a

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state of uniform motion of translation relative
to K, or briefly, relative to

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every Galilean system of coordinates. In
contrast to classical mechanics, the Lorentz transformation

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is the deciding factor, and the
transition from one such system to another.

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By means of comparatively simple considerations,
we are led to draw the following conclusions

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from these premisses, in conjunction with
the fundamental equations of the electrodynamics of Maxwell.

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A body moving with a velocity V
which absorbs footnote one, e sum

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zero is the energy taken up as
judged from a coordinate system moving with a

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body end footnote. An amount of
energy e sum zero in the form of

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radiation without suffering an alteration in velocity
in the process, has as a consequence

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its energy increased by an amount e
sum zero over the square root of the

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difference I minus V squared over C
squared. In consideration of the expression given

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above for the connection energy of the
body, the required energy of the body

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comes out to be the sum M
plus e sub zero over c squared times

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c squared over the square root of
the difference i v squared over c squared.

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Thus, the body has the same
energy as a body of mass M

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plus e sub zero over c squared
moving with a velocity v. Hence,

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we can say if a body takes
up an amount of energy e sub zero,

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then its inertial mass increases by an
amount e sum zero over c squared.

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The inertial mass of a body is
not a constant, but varies according

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to the change in the energy of
the body. The inertial mass of a

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system of bodies can even be regarded
as a measure of its energy. The

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law of the conservation of the mass
of a system becomes identical with the law

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of the conservation of energy, and
is only valid provided that the system neither

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takes up nor sends out energy.
Writing the expression for the energy in the

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form the sum mc squared plus e
sub zero over the square root of the

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difference i v squared over a C
squared. We see that the term mc

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squared, which has hitherto attracted our
attention, is nothing else than the energy

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possessed by the body footnote two,
as judged from a coordinate system moving with

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a body and footnote before it absorbs
the energy he sum zero. A direct

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comparison of this relation with experiment is
not possible at the present time. Note

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The equation e equals mc squared has
been thoroughly proved time and again since this

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time n note Owing to the fact
that the changes in energy e zero to

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which we can subject a system are
not large enough to make themselves perceptible as

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a change in the inertial mass of
the system E zero overseas squared is too

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small in comparison with a mass M, which was present before the alteration of

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the energy. It is owing to
this circumstance that classical mechanics was able to

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establish successfully the conservation of mass as
a law of independent validity. Let me

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add a final remark of a fundamental
nature. The success of the Faraday Maxwell

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interpretation of electromagnetic action at a distance
resulted in physicists becoming convinced that there are

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no such things as instantaneous actions at
a distance not involving an intermediary medium of

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the type of Newton's law of gravitation. According to the theory of relativity,

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action at a distance with the velocity
of light always takes a place of instantaneous

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action at a distance, or of
action at a distance with an infinite velocity

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of transmission. This is connected with
the fact that the velocity c plays a

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fundamental role in this theory. In
Part two we shall see in what way

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this result becomes modified in the general
theory of relativity. End of section fifteen.