Description
Problems
1. Lorentz transformations: Let Λ = (Λµ
ν
) be the 4×4 matrix of a Lorentz boost in the
x direction and g = (gµν) = diag(1, −1, −1, −1) the 4×4 matrix of the metric tensor.
(a) (5 pts) Show that the Lorentz transformation fulfills ΛT
gΛ = g and consequently
leaves the scalar product of two four-vectors invariant.
(b) (5 pts) If x
µ
is a contravariant vector, what kind of object is ∂
∂xµ ? Show by
studying its transformation properties.
(c) (10 pts) Show that two Lorentz boosts both in the x direction with rapidities ζ1
and ζ2 are equivalent to a single boost with rapidity ζ3. Derive the value of ζ3 in
terms of ζ1 and ζ2.
2. Doppler effect and aberration of light:
A light source emits light of frequency ωS with a wave
vector k in the xy plane, where |k| = ωS/c. The light
is reflected from a plane mirror parallel to the yz plane.
The angle of incidence θi and the angle of reflection θr are
defined with respect to the normal to the mirror, as shown
in the figure. Now consider the entire device (both the
source and the mirror) in motion with relativistic velocity
β = v/c in the positive x-direction, with respect to the
laboratory. Predict the results of the measurements made
in the laboratory for:
x
y
k
θi
θ
r
(a) (20 pts) the frequencies of the incident and reflected waves (expressed in terms of
ωS, β, and the angle of incidence and reflection θS in the device frame),
(b) (20 pts) the cosine of the angle of incidence (expressed in terms of cos θS and β),1
(c) (10 pts) the relation between angle of incidence and angle of reflection (both in
the lab frame).
3. Lorentz transformations for acceleration, Jackson, 11.5 (30 pts): A coordinate
system K0 moves with a velocity v relative to another system K. In K0 a particle
has a velocity u
0 and an acceleration a
0
. Find the Lorentz transformation law for
accelerations, and show that in the system K the components of acceleration parallel
and perpendicular to v are
ak =
1 −
v
2
c
2
3/2
1 + v·u0
c
2
3
a
0
k, a⊥ =
1 −
v
2
c
2
1 + v·u0
c
2
3
a
0
⊥ +
v
c
2
× (a
0 × u
0
)
. (1)
1The change in direction of light between two different inertial frames is known as the aberration of light.
It also occurs classically, but the relativistic formula gives more pronounced effects at large v/c.
