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 implic
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
th
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 magnitu
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,
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 bo
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
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 th
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 o
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 = 0.9
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 concl
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 =
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 th
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
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 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,
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 th
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 t
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
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 int
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 spee
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 situa
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
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
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 ve
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 fr
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~ =
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
impossi
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 th
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 wi
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
tha