13. Theorem of the Addition of Velocities. The Experiment of Fizeau14. The Hueristic Value of the Theory of Relativity15. General Results of

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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 Linda lew Relativity

(00:25):
the Special and General Theory by Albert Einstein, continuing part one,
section thirteen through fifteen. Section thirteen theorem of the addition
of velocities the experiment a faeso. Now, in practice we
can move clocks and measuring rods only with velocities that

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are small compared with the velocity of light. Hence, we
shall hardly be able to compare the results of the
previous section directly with the reality. But on the other hand,
these results must strikes being very singular, and for that
reason I shall now draw another conclusion from the theory,
one which can easily be derived from the foregoing considerations,

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and which has been most elegantly confirmed by experiment. In
section six, we derive the theorem of the addition of
velocities in one direction in the form which also results
from the hypotheses of classical mechanics. This theorem can also
be deduced readily from the Galilei transformation section eleven. In

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place of the man walking inside the carriage, we introduce
a point moving relatively to the coordinate system K prime
in accordance with the equation x prime equals W t prime.
By means of the first and fourth equations of the
Galilei transformation, we can express x prime and t prime

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in terms of x and t, and we then obtain
x equals parentheses V plus W unparentheses T. This equation
expresses nothing else than the law of motion of the
point with reference to the system K of the man

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with reference to the embankment. We denote this velocity by
the symbol capital W, and we then obtain, as in
section six, capital w equals V plus w equation A.
But we can carry out this consideration just as well
on the basis of the theory of relativity. In the

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equation x prime equals w t prime, we must then
express x prime and t prime in terms of x
and t, making use of the first and fourth equations
of the Lorentz transformation instead of the equip A. We

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then obtain the equation capital w equals the sum V
plus W over the sum I plus v W over
C squared equation B, which corresponds to the theorem of addition

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for velocities in one direction according to the theory of relativity.
The question now arises as to which of these two
theorems is the better in accord with experience. On this
point we are enlightened by a most important experiment, which
the brilliant physicist Faseo performed more than half a century ago,

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and which has been repeated since then by some of
the best experimental physicists, so that there can be no
doubt about its result. Experiment is concerned with the following question.
Light travels in a motionless liquid with a particular velocity W,
how quickly does it travel in the direction of the
arrow in the tube T see the accompanying diagram Fig. Three.

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When the liquid above mentioned is flowing through the tube
with a velocity V in accordance with the principle of relativity,
we shall certainly have to take for granted that the
propagation of light always takes place with the same velocity
W with respect to the liquid, whether the latter is
in motion with reference to other bodies or not. The

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velocity of light relative to the liquid and the velocity
of the latter relative to the tube are thus known,
and we require the velocity of light relative to the tube.
It is clear that we have the problem in section
six again before us the tube plays a part of
the railway embankment or of the coordinate system K, the

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liquid plays a part of the carriage or the coordinate
system K prime, and finally, the light plays the part
of the man walking along the carriage or of the
moving point. In the present section, if we denote the
velocity of the light relative to the two by capital W,
then this is the given by the equation A or

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B according as a Galilee transformation or the Lorentz transformation
corresponds to the facts. Experiment decides in favor of equation
B derived from the theory of relativity, and the agreement
is indeed very exact. Footnote phaseothu capital w equals W

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plus v open parentheses I minus I over in squared
close parentheses. Where in equals c over W is the
index of refraction of the liquid. On the other hand,

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owing to the smallness of v W over C squared
as compared with I, we can replace B in the
first place by capital W equals open parentheses W plus
V close parentheses, open parentheses I minus the fraction v

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W over C squared close parentheses, or to the same
order of approximation y W plus v open parentheses, I
minus I over n squared close parentheses, which agrees with

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Vaso's result n footnote. According to recent and most excellent
measurements by Zemen, the influence of the velocity of flow
V on the propagation of light is represented by formula
B to within one percent. Nevertheless, we must now draw
attention to the fact that a theory of this phenomenon

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was given by H. A. Lorentz long before the statement
of the theory of relativity. This theory was of a
purely electrodynamical nature and was obtained by the use of
particular hypotheses as to the electromagnetic structure of matter. This circumstance, however,
does not in the least diminish the conclusiveness of the
experiment as a crucial test in favor of the theory

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of relativity. For the electrodynamics of Maxwell or rents, in
which the original theory was based, in no way opposes
the theory of relativity. Rather, has the latter been developed
from electrodynamics as an astoundingly simple combination and generalization of
the hypotheses formally independent of each other on which electrodynamics

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was built end of Section thirteen. Section fourteen. The heuristic
value of the theory of relativity our train of thought
in the foregoing pages can be epitomized in the following manner.
Experience has led to the conviction that, on the one hand,
the principle of relativity holds true, and that, on the

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other hand, the velocity of transmission of light in Vaquoll
has to be considered equal to a constant C. By
uniting these two postulates we obtained the law of transformation
for the rectangular coordinates x, y z and the time
t of the events which constitute the processes of nature.

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In this connection, we do not obtain the Galilee transformation, but,
differing from classical mechanics, the Lorentz transformation, the law of
transmission of light, the acceptance of which is justified by
our actual knowledge, played an important part in this process
of thought. Once in possession of the Lorentz transformation, however,

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we can combine this with the principle of relativity and
sum up the theory. Thus every general law of nature
must be so constituted that it is transformed into a
law of exactly the same form. When instead of the
space time variables x y, z t of the original

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coordinate system k, we introduce new space time variables x
prime y prime, z prime t prime fe coordinate system
k prime. In this connection, the relation between the ordinary
and the accent in magnitudes is given by the Lorentz transformation, or,

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in brief, general laws of nature are covariant with respect
to Lorentz transformation. This is a definite mathematical condition that
the theory of relativity demands of a natural law, and
in virtue of this, the theory becomes a valuable hereistic
aid in the search for general laws of nature. If

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a general law of nature were to be found which
did not satisfy this condition, then at least one of
the two fundamental assumptions of the theory would have been disproved.
Let us now examine what general results of 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 special theory of relativity has grown
out of electro dynamics and optics. In these fields it
has not appreciably altered the predictions of theory, but it
has considerably simplified the theoretical structure i e. The derivation

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of laws, and what is incomparably more important, it has
considerably reduced the number of independent hypotheses forming the basis
of theory. The special theory of relativity is rendered the
Maxwell Orens theory so plausible 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 required to be
modified before it could come into line with the demands
of the special theory of relativity. For the main part, however,
this modification affects only the laws for rapid motions in
which the velocities of matter v are not very small

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as compared with the velocity of light. We have experience
of such rapid motions only in the case of electrons
and ions. For other motions, the variations from the laws
of classical mechanics are too small to make themselves evident
in practice. We shall not consider the motion of stars

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until we come to speak of the general theory of relativity.
In accordance with the theory of relativity, the kinetic energy
of a material burial point of mass M is no
longer given by the well known expression m v squared
over two, but by the expression mc squared over the
square root of the difference I minus the fraction V

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squared over C squared. This expression approaches infinity as the
velocity V approaches the velocity of light C. The velocity
must therefore always remain less than C, however great may
be the energies used to produce the 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 three a s m v to
the fourth over C squared plus et cetera. When v
squared over C squared is small compared with unity, the

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third of these terms is always small in comparison with
a second, which last is alone considered in classical mechanics,
the first term mc squared does not contain the velocity
and requires no consideration. If we are only dealing with
a question as to how the 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 special theory of relativity has led
is concerned with the conception of mass. Before the advent
of relativity, physics recognized two conservation laws of fundamental importance, namely,
the law of conservation of energy and the law of

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the conservation of mass. These two fundamental laws appeared to
be quite independent of each other. By means of the
theory of relativity, they have been united into one law.
Shall now briefly consider how this unification came about and
what meaning is to be attached to it. The principle
of relativity requires that the law of the conservation of

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energy should hold not only with reference to a coordinate
system K, but also with respect to every coordinate system
K prime which is in a state of uniform motion
of translation relative to K, or briefly, relative to every
Galilean system of coordinates. In contrast to classical mechanics, the

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Lorentz transformation is the deciding factor and the transition from
one such system to another. By means of comparatively simple considerations,
we are led to draw the following conclusions from these premisses.
In conjunction with the fundamental equations of the electrodynamics of Maxwell.
A body moving with a velocity V which absorbs footnote one,

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E sum zero is the energy taken up, as judged
from a coordinate system moving with a body n footnote.
An amount of energy e sum zero in the form
of radiation without suffering an alteration and velocity in the
process has as a consequence, its energy increased by an

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amount e sum zero over the square root of the
difference I minus V squared over C squared. In consideration
of the expression given above for the kinetic energy of
the body, the required energy of the body comes out
to be the sum M plus e sub zero over

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c squared times c squared over the square root of
the difference I minus v squared over c squared. Thus,
the body has the same energy v g as a
body of mass M plus e sum zero over c

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squared moving with a velocity V. Hence, we can say
if a body takes up an amount of energy e
sub zero, then its inertial mass increases by an amount
e sub zero over c squared. The inertial mass of
a body is not a constant, but varies according to

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the change in the energy of the body. The inertial
mass of a system of bodies can even be regarded
as a measure of its energy. The law of the
conservation of the mass of a system becomes identical with
the law of the conservation of energy, and is only
valid provided that the system neither takes up nor sends

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out energy. Writing the expression for the energy in the
form the sum mc squared plus e sub zero over
the square root of the diff friends i minus v
squared over c squared, we see that the term mc squared,
which has hitherto attracted our attention, is nothing else than

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the energy possessed by the body Footnote two, as judged
from a coordinate system moving with a body en footnote
before it absorbs the energy e sum zero. A direct
comparison of this relation with experiment is not possible at
the present time. Note The equation e equals mc squared

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has been thoroughly proved time and again since this time
en note. Owing to the fact that the changes in
energy e zero to which we can subject a system
are not large enough to make themselves perceptible as a
change in the inertial mass of the system e zero
over c squared is too small in comparison with the

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mass M which was present before the alteration of the
n energy. It is owing to this circumstance that classical
mechanics was able to establish successfully conservation of mass as
a law of independent validity. Let me add a final
remark of a fundamental nature. The success of the Faraday
Maxwell interpretation of electromagnetic action at a distance resulted in

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physicists becoming convinced that there are no such things as
instantaneous actions at a distance not involving an intermediary medium
of the type of Newton's law of gravitation. According to
the theory of relativity, action at a distance with a
velocity of light always takes a place of instantaneous action

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at a distance, or of action at a distance with
an infinite velocity of transmission. This is connected with the
fact that the velocity see plays a fundamental role in
this theory. In Part two we shall see in what
way this result becomes modified in the general theory of relativity.

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End of section fifteen