8
1 A Brief Stroll in Special Relativity
This is the famous Lorentz transformation for motion in the x direction; γ is termed
the Lorentz contraction factor, which we will often call simply the γ factor. There is
a wealth of interesting physics in this transformation, a little of which we will discuss
next.
1.3 Some Elementary Properties and Applications
Many of the most interesting results of special relativity theory can be obtained using
only the simple Lorentz transformation above (Taylor 1963). We will give a rather
cursory discussion of some of the more important features, appropriate to a review:
time dilation of a moving clock, length contraction of a moving rod, and the Doppler
shift of light emitted by a moving object. The interested reader may consult the
references for much more material.
First note that the Lorentz transformation contains the factor γ , which is greater
than 1. If γ is not to be infinite or imaginary then the velocity parameter β must be
less than 1; thus systems and objects cannot move faster than c, a famous result of
relativity.
Example 1.1 What is “fast”? It is clear from the above, and we will soon see
further, that the γ factor is a good indicator of when relativistic effects become
important. For zero velocity it is equal to 1, and for velocity equal to c it is
infinite. We may, somewhat arbitrarily, take the velocity at which γ = 1.1 to
be fast, that is for which relativistic effects are of order of 10%. Then “fast”
means β = 1/
1 − 1/γ 2 = 0.42 or v = 1.25 × 10
7 m/s. It turns out that
this implies that classical mechanics is rather accurate for surprisingly large
velocities.
Time dilation in a moving system is an effect peculiar to relativity, which distinguishes it sharply from classical theory with its absolute time. Suppose a clock at
rest at the origin in the moving system S
ticks at t
= 0 and again at t
= t
. Then
in the system S, where we suppose our lab to be, it is seen to tick at t = 0 at x = 0
and again at t = t at x = vvt. With the Lorentz transformation in (1.18) we may
relate these time intervals,
ct
= γ ct − βγx = γ ct − βγvvt = ct/γ or t = γ γt
. (1.20)
Thus, since γ ≥ 1, the moving clock appears to run slower as seen in the lab in S. We
refer to the system in which a clock is at rest as its rest system or proper system or
rest frame. Time in the proper system is usually called proper time and often denoted
by τ .
1 A Brief Stroll in Special Relativity
This is the famous Lorentz transformation for motion in the x direction; γ is termed
the Lorentz contraction factor, which we will often call simply the γ factor. There is
a wealth of interesting physics in this transformation, a little of which we will discuss
next.
1.3 Some Elementary Properties and Applications
Many of the most interesting results of special relativity theory can be obtained using
only the simple Lorentz transformation above (Taylor 1963). We will give a rather
cursory discussion of some of the more important features, appropriate to a review:
time dilation of a moving clock, length contraction of a moving rod, and the Doppler
shift of light emitted by a moving object. The interested reader may consult the
references for much more material.
First note that the Lorentz transformation contains the factor γ , which is greater
than 1. If γ is not to be infinite or imaginary then the velocity parameter β must be
less than 1; thus systems and objects cannot move faster than c, a famous result of
relativity.
Example 1.1 What is “fast”? It is clear from the above, and we will soon see
further, that the γ factor is a good indicator of when relativistic effects become
important. For zero velocity it is equal to 1, and for velocity equal to c it is
infinite. We may, somewhat arbitrarily, take the velocity at which γ = 1.1 to
be fast, that is for which relativistic effects are of order of 10%. Then “fast”
means β = 1/
1 − 1/γ 2 = 0.42 or v = 1.25 × 10
7 m/s. It turns out that
this implies that classical mechanics is rather accurate for surprisingly large
velocities.
Time dilation in a moving system is an effect peculiar to relativity, which distinguishes it sharply from classical theory with its absolute time. Suppose a clock at
rest at the origin in the moving system S
ticks at t
= 0 and again at t
= t
. Then
in the system S, where we suppose our lab to be, it is seen to tick at t = 0 at x = 0
and again at t = t at x = vvt. With the Lorentz transformation in (1.18) we may
relate these time intervals,
ct
= γ ct − βγx = γ ct − βγvvt = ct/γ or t = γ γt
. (1.20)
Thus, since γ ≥ 1, the moving clock appears to run slower as seen in the lab in S. We
refer to the system in which a clock is at rest as its rest system or proper system or
rest frame. Time in the proper system is usually called proper time and often denoted
by τ .