Since the light is a wave motion, the two beams start out
in phase,
or in step. If the beams return in step, then the time
required for
the two trips is the same, but if they return out of step the
time
difference between the two trips is the time for hal

time interval between flashes of 2t:.to which the rest
clock reads,
28 AN INTRODUCTION TO THE SPECIAL
THEORY OF RELATIVln
the moving clock is observed by the rest observer to take
a time
2y.:lto between flashes. The speed of a moving object is
always
less

an observer in the laboratory frame, for whom the rod
clock is
moving with speed v transverse to the orientation of the
rod,
measures the time interval between flashes to be 2M, and
that
these two are related by the equation
(1-5.1)
In the literature of r

Now consider the meaning of the rod clock experiment.
The
laboratory observer and the moving observer both have
rod
clocks. Each observer knows his clock to keep correct
time. Yet
whichever is considered to be the observer at rest notes
that the
moving cl

As a first step in the application of the general principle,
let
us suppose that the constancy of the speed of light is a
physical
law; that is, the speed of light is the same in every inertial
frame.
Then the speed of light is the same in every direction

1-5.1 Find 'Y for fJ = S/5, 4/5, 5/U, 12/IS. Use the right
triangle
relationships. If sin (J =fJ, then sec (J ='Y. [1.2, 1.67,
1.08, 2.6]
1-5.2 If fJ =I - x, then 'Y-2 =2x - x2 E! 2x for small x or
for fJ close
to I. Prove that the error in 'Y in making t

measurement of time. Moving clocks do not run at the
same rate
as clocks fixed in the laboratory. We shall see that moving
clocks
Tun slow.
Let us design a clock, taking account of our new view of
the
speed of light. If the speed of light is constant, and

to the source. Such a system is an oscillator, or a rod
clock. If the
light source is initially flashed, the light will travel down
the
stick and back with speed c and be detected; the detector
will
then trigger the light source to flash again. The time
i

on the ground who measures the velocity of an airplane
with respect to his frame will arrive at an entirely
different result
than the pilot reading his airspeed indicator and his
compass
heading. If these observers communicate by radio, their
results
are

Lorentz transformation yields the results of ordinary
experience
in the limit of low velocities.
By direct substitution of Eqs. 2-1.3 into the equation of
the
light sphere in the unprimed frame, Eq. 2-1.1, we find the
equation
of the light sphere in the p

it is true at speeds comparable to the speed of light. We
will
require of statements of relativity that they agree with
ordinary
experience in the limit of low velocities. Whenever an
experimental
test has been made of deductions from the special theory
o

the ether has a length L o, and that a rod moving through
the
ether with its length aligned perpendicular to its direction
of
motion is unaltered, that is LJ. = L o Then Eq. 1-3.lc
gives the
time t:.t J. correctly, provided that we replace L by L J. or
by

shrinks, with its new length given by
(1-3.4)
we would find that
2Lo 1
Atn = t:.tJ. = ~ (1 _ V2/C2)1/2 (1-3.5)
This result would account for the failure to detect a time
difference
in the two paths in the Michelson-Morley experiment.
These explanations ha

away a negative experiment, they had no other detectable
consequence than to explain the effect for which they
were created.
But a train of thought was set into motion by the failure of
the
Michelson-Morley experiment to find an ether,
culminating in
the

same. This procedure takes account of the finite speed of
light,
and insures that there are no alterations in the clock's
behavior,
as there might be if they were carried to a central timing
station.
In anyone frame there is no doubt as to when two events

the rod will note that the light flashes in intervals of 2L/c,
assuming
the detector to be infinitely fast. An observer in the
laboratory notes that the path of the beam from the
moving
source to the moving mirror is not L long, but is from the
source
at

frame depends on where the events took place. From the
Lorentz
transformation, Eqs. 3-1.3, we note that
32 AN INTRODUCTION TO THE SPECIAL
THEORY OF RELATIVITY
t\ = 'Y(t1 - VxI/c2) and t'2 = 'Y(t2 - VX2/C2).
The difference between the time of the two event

Since there was no independent way of determining that
the
length of the two paths was identical, the instrument was
first
adjusted to yield a bright view to the observer and then
was rotated
by 90. By this means any difference between the parallel
and pe

can detect the absolute motion of an inertial reference
frame.
Thus there is no purpose in speaking of a velocity of light
except
with respect to the observer who measures it. We reason
further
that no one observer has a special place, superior to all
oth

length of each section is about 15 m yields an elapsed
time of
10-7 sec for the return trip. Thus a time measurement to
half
the period of oscillation of a light wave could measure the
transit
time to 10-15/10-7, or to I part in 108, as required. In the
a

define any two of the three quantities, the unit of length,
the
unit of time, or the speed of light, but not all three. At
present
it is simpler to define a fundamental unit of length and a
fundamental
unit of time, and to measure the speed of light, but

bright, while if out of step the two waves cancel, and the
view is
dark.
It is easy to see that this apparatus might have the
necessary
precision. Yellow light, of wavelength 600 millimicrons
(nanometers),
has a period of 2 X 10-15 sec, so that the time f

speed V in the -x' direction with respect to the primed
frame.
As indicated in Eqs. 1-5.2, we use the symbol y = (1 V2/C2)-1/2.
The Lorentz transformation fulfills the first two
requirements
we have stated. It is linear, and in the limit V ~ 0, y ~ 1,
the

in the limit of low speeds. Here low speeds mean low
compared to c, so that we wish to examine the
transformation in
the limit f3 ~ O.
(3) The speed of light must have the same value, c, in
every
inertial frame.
Just as the disturbance in a pond resulting

light sphere spreads out from the origin of the primed
coordinates
with speed c.
2-I.l By direct application of the Lorentz tranSiformation
equations,
show that Eq. 2-1.1 is transformed into Eq. 2-1.2.
2-1.2 By taking differentials of the Lorentz transfor

same time, the same reading of the clocks located at the
places
where the events occurred assures us that the events were
simultaneous.
Let us consider two events, I and 2, which are
simultaneous in
the unprimed frame. Then t1 =t2, wherever the events
too

pebble into the water is a system of circular ripples, so a
flash of
light spreads out as a growing sphere. We may describe
this
sphere, whose radius grows at speed c, by the equation
x2 + y2 + %2 = C2[2.
29
(2-1.1)
30 AN INTRODUCTION TO THE SPECIAL
THEOR

at position 3. For him t:.t is the time from I to 2, so that
the
interval between flashes is 2t:.t, according to
c2 lit2 = L2 + v2 lit2,
lit = !:. 1
c (1 - v2Ic2)1/2
Thus if 2t:.to is the interval between flashes as
determined by a
proper observer (for wh

Each measured event is described at least by a set of
four coordinates, giving the position and time at which the
event took place.
In any inertial frame the clocks of all observers are
synchronized.
They have been checked by stationing a third observer
m

Just as the spreading light flash forms a spreading sphere
in the
unprimed frame, so it must also form a spreading sphere
in the
primed frame. Suppose the flash of light takes place at the
instant
when both t and t' are zero, and when the origins of the
t

An ingenious postulate proposed by H. A. Lorentz (18531928),
and known as the Lorentz contraction, was that moving
objects
shrink in their direction of motion through the ether. If the
amount of shrinkage was properly chosen, the times of
flight in
the tw

capable of detecting the time differences predicted by the
Galilean
transformation, no shift in the appearance of the field of
view was detected which could be attributed to motion
through
the ether. The experiment was a failure, but this failure
was one

one which beats off time intervals, and the other which
counts
them. The counter in the laboratory which tallies the
flashes of
the moving clock will show fewer counts than the counter
which
tallies the flashes of the rest clock. The laboratory
observer m

value for observers in any inertial frame, then in principle
we
have no need for independent measures of length and
time. A
clock may be made from a rod, as in Fig. 1-3.2: a rod of
length L
provided with a mirror at one end and provided at the
other
end w

dz2 - c2dt2, and ds'2 =dx'2 +d.,'2 + dz'2 - c2dt'2.
2-2 Simultaneity and Time Sequence Suppose we now
examine
some implications of the Lorentz transformations. Let
there be as many observers in each inertial frame as are
required.
Observers in each frame