12
1 A Brief Stroll in Special Relativity
S
moves with respect to system S
at β 2 . To see how fast system S
moves
with respect to us multiply the individual Lorentz transformations using the
matrix representation (1.18). The product will be a Lorentz transformation, that
is A(β 1 )A(β 2 ) = A(β) with β = (β 1 + β 2 )/(1 + β 1 β 2 ). This is the addition
law for velocities. Notice that for small velocities it agrees with the classical
law, while for both velocities approaching c the total velocity remains less than
c, approaching c from below.
1.4 Derive the general expression for the Doppler shift,
λ ob = (1 − β cos θ )γ λ p ,
where θ is the angle between the velocity of the source and a line between
the source and the observer. Notice that even for θ = 90° there is a shift; this
is called the transverse Doppler shift and is not present in the classical theory
(Taylor 1963).
1.5 We defined rapidity as a convenient alternative measure of velocity in the text.
It can also be motivated if we consider rotations in 2 dimensions as an analog.
The usual matrix representation of a rotation is
R(θ ) =
cos θ − sin θ
sin θ cos θ
.
Show that angles are additive, that is R(θ 1 )R(θ 2 ) = R(θ 1 + θ 2 ).
However we could also measure the rotation by the tangent of θ , call it α. In
this case the rotation matrix would be
R(α) =
1
√
1 + α 2
1 −α
α 1
.
Show that with the α measure the rotations are not additive, that is
R(α 1 )R(α 2 ) = R(α 1 + α 2 ). We may conclude that α is not a very convenient rotation measure. Notice the similarity between the matrix above and the
Lorentz transformation matrix in (1.18); that is what leads to the hyperbolic
definition of rapidity.
1.6 One could have a solution to (1.16) with a 11 (v) = 1/(1 − v/c). Investigate how
this would affect the rate of a moving clock according to (1.20) and show that
it is not compatible with the isotropy of space.
1 A Brief Stroll in Special Relativity
S
moves with respect to system S
at β 2 . To see how fast system S
moves
with respect to us multiply the individual Lorentz transformations using the
matrix representation (1.18). The product will be a Lorentz transformation, that
is A(β 1 )A(β 2 ) = A(β) with β = (β 1 + β 2 )/(1 + β 1 β 2 ). This is the addition
law for velocities. Notice that for small velocities it agrees with the classical
law, while for both velocities approaching c the total velocity remains less than
c, approaching c from below.
1.4 Derive the general expression for the Doppler shift,
λ ob = (1 − β cos θ )γ λ p ,
where θ is the angle between the velocity of the source and a line between
the source and the observer. Notice that even for θ = 90° there is a shift; this
is called the transverse Doppler shift and is not present in the classical theory
(Taylor 1963).
1.5 We defined rapidity as a convenient alternative measure of velocity in the text.
It can also be motivated if we consider rotations in 2 dimensions as an analog.
The usual matrix representation of a rotation is
R(θ ) =
cos θ − sin θ
sin θ cos θ
.
Show that angles are additive, that is R(θ 1 )R(θ 2 ) = R(θ 1 + θ 2 ).
However we could also measure the rotation by the tangent of θ , call it α. In
this case the rotation matrix would be
R(α) =
1
√
1 + α 2
1 −α
α 1
.
Show that with the α measure the rotations are not additive, that is
R(α 1 )R(α 2 ) = R(α 1 + α 2 ). We may conclude that α is not a very convenient rotation measure. Notice the similarity between the matrix above and the
Lorentz transformation matrix in (1.18); that is what leads to the hyperbolic
definition of rapidity.
1.6 One could have a solution to (1.16) with a 11 (v) = 1/(1 − v/c). Investigate how
this would affect the rate of a moving clock according to (1.20) and show that
it is not compatible with the isotropy of space.