1.3 Some Elementary Properties and Applications
11
Fig. 1.4 The light source at two different times. It moves directly toward the observer
We can relate this to the period T p in the proper frame with the time dilation
equation (1.20) to find
λ ob = (1 − β)cT = (1 − β)γ cT p = (1 − β)γ λ p ,
(1.27)
where λ p = cT p is the wavelength in the proper frame; the contribution of relativity to
this expression is the factor of γ . If the source moves toward us we therefore observe
a shorter wavelength, that is a blue shift. You should think through the argument for
a source that moves away from the observer and verify that the sign of the velocity
in (1.27) changes and one observes a red shift. The general case in which the source
moves at an arbitrary angle is not difficult; see Exercise 1.4.
There is another velocity measure, called rapidity, that is often more useful than β.
Notice that under successive Lorentz transformations the velocities are not additive;
that is A(β 1 )A(β 2 ) = A(β 1 + β 2 ) (see Exercise 1.3). Rapidity is defined so that
the rapidities of successive Lorentz transformations do add; specifically, we define
rapidity θ by β = tanh θ , so the Lorentz transformation (1.18) may be written as
A(θ ) =
cosh θ − sinh θ
− sinh θ cosh θ
, β = tanh θ, γ = cosh θ,
(1.28)
(see Exercise 1.5). It is easy to verify that A(θ 1 )A(θ 2 ) = A(θ 1 + θ 2 ). That is, rapidity
is an additive measure, as desired (see Exercise 1.3 for further motivation for the
definition). We will find this property useful when we discuss accelerated motion in
Chap. 3.
Exercises
1.1 At the SLAC National Accelerator Center electrons were accelerated to have
γ = 4 × 10
4 so they were moving at nearly c. To see how near set β = 1 − ε
then calculate ε approximately.
1.2 Suppose that you will live for another 100 years or so. The universe is about 10
billion light years across—as we will later discuss. About how fast must you
move to cross it in your lifetime?
1.3 Velocities behave rather differently in relativity than in classical mechanics.
Study the addition of velocities by considering 3 inertial systems as follows.
We are at rest in S, system S
moves with respect to us at β 1 , while system
11
Fig. 1.4 The light source at two different times. It moves directly toward the observer
We can relate this to the period T p in the proper frame with the time dilation
equation (1.20) to find
λ ob = (1 − β)cT = (1 − β)γ cT p = (1 − β)γ λ p ,
(1.27)
where λ p = cT p is the wavelength in the proper frame; the contribution of relativity to
this expression is the factor of γ . If the source moves toward us we therefore observe
a shorter wavelength, that is a blue shift. You should think through the argument for
a source that moves away from the observer and verify that the sign of the velocity
in (1.27) changes and one observes a red shift. The general case in which the source
moves at an arbitrary angle is not difficult; see Exercise 1.4.
There is another velocity measure, called rapidity, that is often more useful than β.
Notice that under successive Lorentz transformations the velocities are not additive;
that is A(β 1 )A(β 2 ) = A(β 1 + β 2 ) (see Exercise 1.3). Rapidity is defined so that
the rapidities of successive Lorentz transformations do add; specifically, we define
rapidity θ by β = tanh θ , so the Lorentz transformation (1.18) may be written as
A(θ ) =
cosh θ − sinh θ
− sinh θ cosh θ
, β = tanh θ, γ = cosh θ,
(1.28)
(see Exercise 1.5). It is easy to verify that A(θ 1 )A(θ 2 ) = A(θ 1 + θ 2 ). That is, rapidity
is an additive measure, as desired (see Exercise 1.3 for further motivation for the
definition). We will find this property useful when we discuss accelerated motion in
Chap. 3.
Exercises
1.1 At the SLAC National Accelerator Center electrons were accelerated to have
γ = 4 × 10
4 so they were moving at nearly c. To see how near set β = 1 − ε
then calculate ε approximately.
1.2 Suppose that you will live for another 100 years or so. The universe is about 10
billion light years across—as we will later discuss. About how fast must you
move to cross it in your lifetime?
1.3 Velocities behave rather differently in relativity than in classical mechanics.
Study the addition of velocities by considering 3 inertial systems as follows.
We are at rest in S, system S
moves with respect to us at β 1 , while system