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confirmed by the observation oŸ many kinds of particles, moving at speeds ranging
up to practically the speed of light. However, because the efect is ordinarily
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so small, it seems remarkable that it was discovered theoretically before it was
discovered experimentally. Empirically, at a sufficiently high velocity, the efect
is very large, but it was not discovered that way. Therefore it is interesting to see
how a law that involved so delicate a modification (at the time when it was firsb
discovered) was brought to light by a combination of experiments and physical
reasoning. Contributions to the discovery were made by a number of people, the
ñnal result of whose work was Einstein's discovery.
There are really two Hinstein theories of relativity. 'This chapter is concerned
with the Special Theory of Relativity, which dates from 1905. In 1915 Einstein
published an additional theory, called the General 'Pheory of Relativity. This
latter theory deals with the extension of the Special Theory to the case of the
law of gravitation; we shall not discuss the General 'Pheory here.
The principle of relativity was first stated by Newton, in one of his corollaries
to the laws of motion: ““The motions of bodies included in a given space are
the same among themselves, whether that space 1s at rest or moves uniformly
forward in a straight line.” 'Phis means, for example, that 1Ý a space ship is drifting
along at a uniform speed, all experiments performed in the space ship and all the
phenomena in the space ship will appear the same as if the ship were not moving,
provided, of course, that one does not look outside. 'That is the meaning of the
principle of relativity. This is a simple enough idea, and the only question 1s
whether it is £rue that in all experiments performed inside a moving system the
laws of physics will appear the same as they would if the system were standing
still. Let us frst investigate whether Newton's laws appear the same in the
1noving system.
3uppose that Moe is moving in the z-direction with a uniform velocity , and
he measures the position of a certain point, shown in Fig. 15-1. He designates
the “z-distance” of the point in his coordinate system as #”. Joe is at rest, and
JOE MOE (x,y',z")
ụ e«.P or
(x,y.Z)
Fig. 15-1. TWo coordinate systems In uniform relative motion along
thelr x-axes.
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measures the position of the same point, designating its #ø-coordinate in his
system as ø. The relationship of the coordinates in the two systems is clear from
the diagram. After time ý Moe's origin has moved a distance œ#, and if the two
systems originally coincided,
zh—=#— tt,
Ă (15.2)
zZ —=#,
TÝ we substitute this transformation of coordinates into NÑewton's laws we fnd
that these laws transform to the same laws in the primed system; that is, the laws
of Newton are of the same form in a moving system as in a stationary system,
and therefore it is impossible to tell, by making mechanical experiments, whether
the system is moving or not.
The principle of relativity has been used in mechanies for a long time. lt
was employed by various people, in particular Huygens, to obtain the rules for
the collision of billiard balls, in much the same way as we used it in Chapter 10
to discuss the conservation of momentum. In the 190h century interest in iE
was heightened as the result of investigations into the phenomena. of electricity,
magnetism, and light. A long series of careful studies of these phenomena by
many people culminated in Maxwells equations of the electromagnetic field,
which describe electricity, magnetism, and light in one uniform system. However,
the Maxwell equations did øœø# seem to obey the principle of relativity. That
is, IÝ we transform Maxwells equations by the substitution of equations (15.2),
theñr ƒorm does no‡ remain the same; therefore, in a moving space ship the
electrical and optical phenomena should be diferent from those in a stationary
ship. Thus one could use these optical phenomena to determine the speed of
the ship; in particular, one could determine the absolute speed of the ship by
making suitable optical or electrical measurements. One of the consequences of
Maxwells equations is that if there is a disturbance in the fñeld such that light is
generated, these electromagnetic waves go out in all directions equally and at
the same speed c, or 186,000 mi/sec. Another consequence oŸ the equations is
that 1f the source of the disturbance 1s moving, the light emitted goes through
space at the same speed c. 'This is analogous to the case of sound, the speed of
sound waves being likewise independent of the motion of the source.
'This independenece of the motion of the source, in the case of light, brings up
an interesting problem:
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Suppose we are riding in a car that is going at a speed , and light trom the
rear is going past the car with speed e. Diferentiating the frst equation in (15.2)
da /dt = d+/dt — tu,
which means that according to the Galilean transformation the apparent speed
of the passing light, as we measure it in the car, should not be é but should
be ce—u. For instance, if the car is going 100,000 mi/sec, and the light is
going 186,000 mi/sec, then apparently the light going past the car should go
86,000 mi/sec. In any case, by measuring the speed of the light going past the car
(ïf the Galilean transformation is correct for light), one could determine the speed
of the car. A number of experiments based on this general idea were performed to
determine the velocity of the earth, but they all failed—they gave no uelocitU
dÏl. We shall discuss one of these experiments in detail, to show exactly what was
done and what was the matter; something +0øs the matter, of course, something
was wrong with the equations of physics. What could it be?
15-2 The Lorentz transformation
'When the failure of the equations of physics in the above case came to light,
the fñrst thought that occurred was that the trouble must lie in the new Maxwell
equations of electrodynamics, which were only 20 years old at the time. It seemed
almost obvious that these equations must be wrong, so the thing to do was to
change them in such a way that under the Galilean transformation the principle
of relativity would be satisfied. When this was tried, the new terms that had to
be put into the equations led to predictions of new electrical phenomena that did
not exist at all when tested experimentally, so this attempt had to be abandoned.
'Then it gradually became apparent that Maxwell's laws of electrodynamics were
correct, and the trouble must be sought elsewhere.
In the meantime, H. A. Lorentz noticed a remarkable and curious thing when
he made the following substitutions in the Maxwell equations:
„h= % — Uuử