Environmental Engineering Reference
In-Depth Information
The factor 1
/
1
−
v
2
/c
2
appears so often in Special Relativity that it is given
its own symbol, i.e.
1
γ
≡
1
(6.4)
−
v
2
/c
2
and
t
=
γt
0
.
(6.5)
For
v/c
1 it follows that
γ>
1 and for
v/c >
1 the theory doesn't appear
to make much sense (unless we are prepared to entertain the idea of imaginary
time).
To conclude this section, let us quickly check that
t
≤
=
t
0
in classical theory.
Replacing
c
in Eq. (6.2) by
(c
2
v
2
)
1
/
2
+
gives
v
2
)
1
/
2
d
2
1
/
2
v
2
t
2
4
2
t
=
+
(6.6)
(c
2
+
which has the solution
t
2
d/c
as expected.
Eq. (6.5) is quite astonishing: it really does violate our intuition that time is
absolute. We emphasise that this effect has nothing to do with the fact that we
have considered light bouncing between two mirrors. We used light because it
allows us to make use of Einstein's 2nd postulate. If we had used a bouncing ball
then we would have become stuck when we had to figure out the speed of the
ball in
S
because we are not entitled to assume that velocities add in the classical
manner. When we have a little more knowledge and know how velocities add we
will be able to return to the bouncing ball and we shall conclude that time is dilated
exactly as for the light-clock. Clearly this must be the case for we are talking about
the time interval between actual events.
The fact that time is actually different from our intuitive perception of it is no
problem for physics, no matter how odd it may seem to us. There is a lesson to be
learnt here. Namely, we should not expect our intuition based upon everday expe-
riences to necessarily hold true in unfamiliar circumstances. In relativity theory,
the unfamiliar circumstance is when objects are travelling close to the speed of
light. The lesson also applies when tackling quantum theory. In this case common
sense breaks down when we explore systems on very small length scales.
=
Example 6.1.1
Muons are elementary particles rather like electrons but 207 times
heavier. Unlike electrons, muons are unstable and they decay to an electron and a
pair of neutrinos with a characteristic lifetime. For a muon at rest, this lifetime is
2
.
2
s.
Muons are created when cosmic rays impact upon the Earth's atmosphere at an
altitude of 20 km and are observed to reach the Earth's surface travelling at close
to the speed of light. (a) Use classical theory to estimate how far a typical muon
would travel before it decays (assume the muon is travelling at the speed of light).
(b) Now use time dilation to explain why the muons are able to travel the full 20 km
without decaying.
µ
