Section thirteen theorem of the addition of velocities the experiment of Faso. Now,
in practice, we can move clocks in measuring rods only with velocities that 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 strike you as 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, 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 Galiley transformation section
eleven. In 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 Galiley transformation, we can express x prime and t prime 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 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 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 equation A.
We 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 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, where enlightened by a most important
experiment which the brilliant physicist faces, O perform more than half a century ago,
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 a 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. 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 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 had the problem the section six again before
us. The two plays a part of the railway embankment or of the coordinate
system K. The liquid plays a part of the carriage or of the coordinate
system K prime, and finally, the light plays a 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 tube by capital W,
then this is a given by the equation A or B according as a Galiley
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 I say so foul capital W equals W plus v open parentheses,
I minus I over n squared close parentheses, where n equals c OVERW
is the index of refraction of the liquid. On the other hand, owing
to the smallness of v w oversea 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 w oversea squared close
parentheses, or to the same order of approximation by W plus v open parentheses,
I minus I over in squared close parentheses, which agrees with Vasso's resolved
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 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 of relativity. For the electrodynamics of Maxims Well Orents, 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 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 epiimized in the following manner.
Experiences led to the conviction that on the one hand, the principle of
relativity holds true, and that, on the other hand, the velocity of
transmission of light invokwell has to be considered equal to a constant C. By
uniting these two postulates, we obtain the law of transformation for the rectangular coordinates
x, y, c and the time T of the events which constitute the
processes of nature. In this connection, we did not obtain the Galiley 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, we can combine this with a 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 xy c t of the original coordinate system k,
we introduce new space time variables x prime y prime c prime t prime fe
coordinate system k prime. In this connection, the relation between the ordinary and
the accent of magnitudes is given by the Lorentz transformation, or in brief,
general laws of nature are co variant 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 valuable heuristic aid in the search for
general laws of nature. If 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 the latter theory has hitherto evinced. End of Section fourteen. Section fifteen
General results of the theory. It is clear from our previous considerations that the
special theory relativity has grown out of electrodynamics and optics. In these fields,
it has not appreciably altered the predictions of theory, but it has considerably simplified
the theoretical structure, ie the derivation of laws, and what is incomparably more
important, it has considerably reduced a 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
less unequivocally in its favor. Classical mechanics require 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 fee are not very small 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 until we come to speak of the general theory of relativity.
In accordance with the theory of relativity, the kinetic energy of a material
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 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 acceleration. If we develop the expression for the kinetic energy in the form
of a series, we obtain mc squared plus m v squared over two plus
three ace m fee to the fourth over C squared plus c. When v
squared over c squared is small compared with unity, the 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
essential significance later. The most important result of a general character to which the
special theory of relativity has led is concerned learned 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 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.
We 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 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 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, e sum
zero is the energy taken up as judged from a coordinate system moving with a
body end footnote. An amount of energy e sum zero in the form of
radiation without suffering an alteration in velocity in the process, has as a consequence
its energy increased by an 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 connection energy of the body, the required energy of the body
comes out to be the sum M plus e sub zero over c squared times
c squared over the square root of the difference i v squared over c squared.
Thus, the body has the same energy as a body of mass M
plus e sub zero over c 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 sum zero over c squared.
The inertial mass of a body is not a constant, but varies according
to 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 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
difference i v squared over a C squared. We see that the term mc
squared, which has hitherto attracted our attention, is nothing else than the energy
possessed by the body footnote two, as judged from a coordinate system moving with
a body and footnote before it absorbs the energy he sum zero. A direct
comparison of this relation with experiment is not possible at the present time. Note
The equation e equals mc squared has been thoroughly proved time and again since this
time n 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 overseas squared is too
small in comparison with a mass M, which was present before the alteration of
the energy. It is owing to this circumstance that classical mechanics was able to
establish successfully the 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 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 the velocity of light always takes a place of instantaneous
action at a distance, or of action at a distance with an infinite velocity
of transmission. This is connected with the fact that the velocity c 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. End of section fifteen.