The size of a radio antenna is closely related to ... - Light and Matter

system, velocities are always unitless. This sort of thing happens
frequently in physics. For instance, before James Joule discovered
conservation of energy, nobody knew that heat and mechanical energy
were different forms of the same thing, so instead of measuring
them both in units of joules as we would do now, they measured
heat in one unit (such as calories) and mechanical energy in another
(such as foot-pounds). In ordinary metric units, we just need an
extra conversion factor c, and the equation becomes
1
γ = √
1 − ( )
.
v 2
c
k / The γ factor.
Here’s why we care about γ. Figure k defines it as the ratio of two
times: the time between two events as expressed in one coordinate
system, and the time between the same two events as measured in
the other one. The interpretation is:
Time dilation
A clock runs fastest in the frame of reference of an observer
who is at rest relative to the clock. An observer in motion
relative to the clock at speed v perceives the clock as running
more slowly by a factor of γ.
l / The ruler is moving in frame
1, represented by a square, but
at rest in frame 2, shown as a
parallelogram. Each picture of
the ruler is a snapshot taken
at a certain moment as judged
according to frame 2’s notion
of simultaneity. An observer in
frame 1 judges the ruler’s length
instead according to frame 1’s
definition of simultaneity, i.e.,
using points that are lined up
vertically on the graph. The ruler
appears shorter in the frame in
which it is moving. As proved
in figure m, the length contracts
from L to L/γ.
m / This figure proves, as claimed in figure l, that the length contraction
is x = 1/γ. First we slice the parallelogram vertically like a salami
and slide the slices down, making the top and bottom edges horizontal.
Then we do the same in the horizontal direction, forming a rectangle with
sides γ and x. Since both the Lorentz transformation and the slicing
processes leave areas unchanged, the area γx of the rectangle must
equal the area of the original square, which is 1.
As proved in figures l and m, lengths are also distorted:
Length contraction
A meter-stick appears longest to an observer who is at rest
relative to it. An observer moving relative to the meter-stick
at v observes the stick to be shortened by a factor of γ.
self-check A
What is γ when v = 0? What does this mean? ⊲ Answer, p. 924
390 Chapter 7 Relativity