is always perpendicular to its direction of motion. Both
Eqs. 3-2.8
have subsequently been confirmed many times, and their
ac56
AN INTRODUCTION TO THE SPECIAL THEORY
OF RELATIVITY
curacy is now implicit in the design of charged partide
accelerators.
3-2.1
this frame. As measured by a laboratory observer for
whom the proton
is moving with speed corresponding to fJ =0.98 in the +x
direction,
what is (c) the force on the proton, (d) the acceleration of
the proton,
and (e) the angle between the force and the a
If a constant force F is moved through a displacement s,
we
say that physical work W (a scalar) has been done,
according to
the equation
W = Fs = Fscos9. (4-1.1)
The work is the product of the magnitude of the force
times the
magnitude of the displacement
advent of radioactivity made it possible to perform
experiments
at speeds close to c. Then it was found that the relativistic
form
of the expression for momentum was in good agreement
with
experiment, while the nonrelativistic form of Eq. 3-1.2
was not.
B
to describe the momentum in the relativistic case, with
Eq. 3-1.2
as its nonrelativistic limit.
32 Newton's Second Law Newton's second law states:
If-a
net force acts on a body, the momentum of the body will
be
changed; the rate of change of the momentum
momentum p of a particle of mass m moving with velocity
v is
given by relativity theory as
p = m"cfw_v. (3-1.1)
Prior to the special theory of relativity the momentum of a
particle had been given by
p = my. (3-1.2)
Deductions from Eq. 3-1.2 had always bee
impossible to distinguish between an accelerated
coordinate
frame and a gravitational field. To quote Einstein. "This
assumption
of exact physical equivalence makes it impossible for us
to speak of the absolute acceleration of the system of
reference,
jus
input information, for the device itself cannot tell whether
it is
being accelerated, or is in a gravitational field, or both.
One interpretation of the word weight is the upward force
exerted on an object by its supports. Thus we are
"heavier" when
we st
rest) if it is acted on by a force of F' =: 9.1 X 10-4 l
.dynes. (b) What
force does an observer in the laboratory frame believe to
be acting on
the electron, if he observes it to be moving at v = 0.9998c
1. with
60 AN INTRODUCTION TO THE SPECIAL
THEORY O
Eq. 3-2.8b the inertial observers know that the floor must
exert
an upward force on the puck equal to myg, and they, as
referees,
declare that this is the weight of the moving puck. Thus
we
must conclude that a body moving horizontally in a
uniform ver58
because it causes a supporting spring to be deflected),
then we
find that we can make two possible statements about it.
Either
the object is in a gravitational field of field intensity g
such that
F =mg. or it is being accelerated (along with its
supporti
in the appropriate limit. And this is as it must be, for any
physical
theory must provide an accurate and concise description
of
experience, or it never finds acceptance. When new
theories are
born that seem to upset older ideas, their predictions must
al
Iz dynes (c) 4 X 1020 Iz cmjsec2 (d) 4 X 1020 cm/sec2]
3-4.2 The mass of a proton is 1.67 X 10-27 kgm. The
proton experiences
a force of (1.67 X 10-9 N) X (I",. + Iz.) in its proper
frame, in
which it is instantaneously at rest. (a) Find the
acceleration
in computing ~:. This will involve the computation of
-d~. Now v2 = v 'v, the scalar product of the vector v
W'Ith'ItseIf.
dt
The scalar product of two vectors A and B is written as
ABand
is equal to the product of the magnitude of the two vectors
and
the
the primed frame but is accelerating with acceleration a'
with
respect to that frame, that the primed frame is moving
with
speed v in the +x direction with respect to the unprimed
or
laboratory frame, and that observers in the laboratory
frame
measure the
observer compute the electron to have, by use of the
relativistic
relations between mass. and acceleration? (d) What
acceleration does
the laboratory observer find for the electron if he
translates the acceleration
noted by the proper observer through use
electric and magnetic forces as observed in different
frames.
Once again we note that in the nonrelativistic limit C~ 00,
we find y ~ I, and the relativistic force transformation
Eqs.
3-4.4 approach the nonrelativistic Eqs. 3-4.1.
3-4.1 The mass of the el
one of taste. In this book no distinction will be made
between
rest mass and moving mass. The symbol m will always
signify the
mass detennined when the particle is at rest in its inertial
frame,
such as by comparison with a standard mass through use
of an
at the same acceleration as we would experience in free
fall.
Let us consider the weight of a moving object. What is the
weight of a puck which slides horizontally on a smooth
floor at
speed v? We interpret the word weight as meaning the
force the
floor e
The discrepancy between Newtonian mechanics and
nature
is remedied by a theory called Quantum Mechanics in the
realm
of small objects, and by the Special Theory of Relativity
in the
realm of high speeds. Both of these theories are so
constructed
that thei
viewpoint of a set of observers on an inertial frame who
see the
equivalent problem, that of a puck sliding on the floor of
an
elevator which is accelerating upward in field free space.
For
this situation must be identical with the real one where the
puck
that the two expressions agree, as required by the
correspondence
principle.
FORCE AND MOTION 53
Some authors interpret Eqs. 3-U and 3-1.2 to imply that
m~
mentum is always expressed as the product of mass by
velocity,
but that the mass m of a moving part
speed with respect to a planet or the sun.
In the same way the general theory of relativity and the
equivalence principle advise us that it is impossible to
build a
wholly self-contained accelerometer. When
accelerometers are
used in inertial navigation,
dt dt dt
(3-2.4)
The first term of Eq. 3-2.4 is in the direction of the
velocity vector.
The second term contains the derivative of the velocity
vector
with respect to time. Since the change in the velocity
vector
may be in any direction, we may resolve t
a such that F =ma (in the low velocity limit). It is
even possible that the object is experiencing some
combination
of gra'\litational field and accelerated motion, as in the
case of
a mass hanging from the ceiling of an elevator which is
accelerating
upw
perpendicular to the velocity and a component parallel
to the velocity. Writing the perpendicular component as
the perpendicular
component of the acceleration; that is, writing
(~t = a~,
we find F~ = m-ya~. (3-2.5)
Before analyzing the component equation
absolute velocity of a system." The equivalence principle
lies at
the foundation of the general theory of relativity.
FORCE AND MOTION 57
The special theory of relativity advises us that it is
impossible
for us to build an entirely self-contained speedome
Applying this result to Eq. 3-2.4, we have
F II = mv(-i)(l - v2/c2)-S/2(-2vall/c2) + m'Ya".
When we collect terms and clear fractions, the equation
becomes
F II = m'Y3a ll'
For comparison we repeat the equation for F.1'
(3-2.8a)
FJ. = m'YaJ.' (3-2.8b)
In
respect to some coordinate frame. We cannot have a
velocimeter
which is wholly built in, which makes no contact with an
external
frame of reference. In an automobile the speedometer
measures motion with respect to the road; in an airplane
the airspeed
ind
a = a'.
If we multiply both sides of the equation by m, we find
rna = rna'.
Making use of the nonrelativistic form of Newton's
second law
in Eq. 3-2.3, we write for each of the two inertial frames
that
F = rna and F' = rna'.
Thus we infer that the force a