1.3 Some Elementary Properties and Applications
9
Fig. 1.2 The rocket nose is at x = L and the tail at x = 0 at t = 0 in our lab frame
Example 1.2 Muons have a lifetime of about 2 μs in their rest frame. In a
universe with absolute time they could travel only about 600 m before decaying
if moving at nearly c. In fact they have been observed to travel many km. The
“little clock inside the muon” must indeed run slow.
Length contraction is one of the best-known properties of relativity. It involves
two facets of the theory—the definition of length and the relativity of simultaneity.
Suppose an object such as a rocket ship is at rest in system S
with its tail at x
= 0
and its nose at x
= L p , which of course we call its proper length. In our lab in S we
observe the ship pass by so that at t = 0 its tail is at x = 0 and its nose is at x = L,
which we call its length in the lab system. This is shown in Fig. 1.2. From this the
Lorentz transformation (1.18) gives a relation between L and L p ,
L p = x
= γ x − βγ ct = γ x = γ L , L = L p /γ ,
(1.21)
since t = 0 in the lab. That is, in the lab we observe the moving rocket to be shorter
than its length in the rest or proper frame. Note that this nonintuitive result is obtained
since the positions of nose and tail are observed simultaneously at t = 0 in the lab,
a fundamental part of the definition of length implied in the above. Observers in the
rocket’s proper frame will not consider the measurement in the lab frame to be valid
since they will see a time difference between the nose and tail measurements of
ct
= γ ct − βγ x = −βγ L = 0.
(1.22)
That is the simultaneous measurements of nose and tail positions in the lab are not
simultaneous in the proper system; simultaneity is relative to the system. This was
one of Einstein’s great insights which led to special relativity.
Example 1.3 Do objects visually appear to be contracted according to (1.21)?
They do not. The definition of length in the above example does not involve
visual appearance. Consider a rocket moving directly toward us on the x axis.
We ask where we see the nose and tail of the rocket as it moves toward us,
and take this as the definition of its visual length. Figure 1.3 shows the rocket
at two times; one photon from the tail (T ) and one from the nose (N) of the
9
Fig. 1.2 The rocket nose is at x = L and the tail at x = 0 at t = 0 in our lab frame
Example 1.2 Muons have a lifetime of about 2 μs in their rest frame. In a
universe with absolute time they could travel only about 600 m before decaying
if moving at nearly c. In fact they have been observed to travel many km. The
“little clock inside the muon” must indeed run slow.
Length contraction is one of the best-known properties of relativity. It involves
two facets of the theory—the definition of length and the relativity of simultaneity.
Suppose an object such as a rocket ship is at rest in system S
with its tail at x
= 0
and its nose at x
= L p , which of course we call its proper length. In our lab in S we
observe the ship pass by so that at t = 0 its tail is at x = 0 and its nose is at x = L,
which we call its length in the lab system. This is shown in Fig. 1.2. From this the
Lorentz transformation (1.18) gives a relation between L and L p ,
L p = x
= γ x − βγ ct = γ x = γ L , L = L p /γ ,
(1.21)
since t = 0 in the lab. That is, in the lab we observe the moving rocket to be shorter
than its length in the rest or proper frame. Note that this nonintuitive result is obtained
since the positions of nose and tail are observed simultaneously at t = 0 in the lab,
a fundamental part of the definition of length implied in the above. Observers in the
rocket’s proper frame will not consider the measurement in the lab frame to be valid
since they will see a time difference between the nose and tail measurements of
ct
= γ ct − βγ x = −βγ L = 0.
(1.22)
That is the simultaneous measurements of nose and tail positions in the lab are not
simultaneous in the proper system; simultaneity is relative to the system. This was
one of Einstein’s great insights which led to special relativity.
Example 1.3 Do objects visually appear to be contracted according to (1.21)?
They do not. The definition of length in the above example does not involve
visual appearance. Consider a rocket moving directly toward us on the x axis.
We ask where we see the nose and tail of the rocket as it moves toward us,
and take this as the definition of its visual length. Figure 1.3 shows the rocket
at two times; one photon from the tail (T ) and one from the nose (N) of the