PHY4222, Section 3801, Spring 2016, Homework 12
Due at the start of class on Friday, April 15. Half credit will be available for late
homework submitted no later than the start of class on Monday, April 18.
Answer all questions. Please write neatly and include your name on the front page of your
answers. You must also clearly identify all your collaborators on this assignment. To gain
maximum credit you should explain your reasoning and show all working.
1. This question involves working with space-time tensors. These obey similar transformation
laws to co- and contra-variant vectors, but may have more (or fewer) indices.
a) Let:


x
y
xµ =  
z
ct
be the 4-position of a particle and:

dx/dτ
dxµ  dy/dτ 
,
=
Vµ =
dz/dτ 
dτ
cdt/dτ

its 4-velocity, where τ is proper time along its world-line. Prove that V µ transforms
correctly as a contra-variant (equivalent to a column-like) vector, i.e., that:
′
′
′
Vµ
∂xµ dxµ
dxµ
=
.
=
dτ
∂xµ dτ
b) Let:
∇µ f (x, y, z, ct) ≡
∂f
∂x
∂f
∂y
∂f
∂z
∂f
c∂t
Prove that this transforms as a space-time covariant (that is, row-like) vector, i.e.,
that:
∇µ′ f (x′ , y ′ , z ′ , ct′ ) =
∂xµ
′ ∇µ f (x, y, z, ct)
∂xµ
c) For a fixed world-line, prove that V µ ∇µ f is a space-time scalar, i.e., show that it
is invariant under a coordinate transformation. Note: you will need to use:
′
∂xµ ∂xν
ν
′ = δµ ,
µ
µ
∂x ∂x
′
or
∂xν ∂xµ
ν′
′ = δµ′
µ
µ
∂x ∂x
′
where δµν (and δµν ′ ) is the unit matrix.
d) Let V µ and U ν be two different space-time vectors. Prove that T µν = V µ U ν
transforms correctly as a contra-variant space-time tensor of rank 2.
e) The entity:
ds2 = gµν dxµ dxν
is a space-time scalar, because the metric gµν is a covariant space-time tensor of
rank 2. How does gµν transform under a coordinate transformation?
f) What sort of a space-time quantity (tensor, vector, scalar) is V µ U ν gµν ? How does
it transform under a coordinate transformation? How does U ν gµν (often denoted
Uµ for convenience) transform?
2. One way to define the electromagnetic field tensor is by:
Fµν = ∇µ Aν − ∇ν Aµ ,
where Aµ = (A, −φ/c) and A = Ai is a covariant 3-vector.
a) If E = −∇φ − dA/dt and B = ∇ × A, show that

0
B3
−B2
 −B3
0
B1
Fµν = 
B2
−B1
0
−E1 /c −E2 /c −E3 /c
b) Consider the Lorentz transformation:


γ
−γβ


1
µ′
Λµ = 
 and its inverse
1
−γβ
γ

E1 /c
E2 /c 
.
E3 /c 
0


µ
Λµ′ = 
γ
γβ
1
1
γβ
γ


,
and construct
µ
Fµ′ ν ′ = Fµν Λµ′ Λνν ′
c) Show that your result in b) is equivalent to:
E1′ = E1 ,
B1′ = B1 ,
E2′ = γ(E2 − βcB3 ),
B2′ = γ(B2 + βE3 /c),
E3′ = γ(E3 + βcB2 ),
and
B3′ = γ(B3 − βE2 /c).
d) Hence prove that E · B and E2 − c2 B2 are both invariant under Lorentz transformations.
e) Thus prove that if E and B are perpendicular in one inertial frame S, they are
perpendicular in all other inertial frames S ′ .
f) Similarly, show that if E > cB in frame S, there cannot exist a frame in which
E = 0.
2
3.
a) Show that if we make the change of variables ξ = x − ct and η = x + ct then:
2
∂ 2 u(x, t)
2 ∂ u(x, t) = −4c2 ∂ ∂u(x(ξ, η), t(ξ, η)) ,
−
c
∂ξ
∂η
∂t2
∂x2
and hence write down a complete solution for a wave on an infinite string.
b) Consider a string of length L and clamped at both ends. At t = 0, suppose that
the displacement of the string is given by u(x, 0) = u0 sin(nπx/L) where n is an
integer and its instantaneous velocity satisfies u̇(x, 0) = 0. By the separation of
variables (or some other approach) find the full solution describing the motion of
the string.
c) Given that:
1 ∂
∇ f=√
g ∂xi
2
√
∂f
gg
∂xj
ij
show that, in spherical polar coordinates (with gij = diag(1, r 2, r 2 sin2 θ)), this is
equivalent to:
1 ∂ 2 rf
∂ 2f
∂
1
∂f
1
2
∇ f=
+
.
sin
θ
+
r ∂r 2
∂θ
r 2 sin θ ∂θ
r 2 sin2 θ ∂φ2
d) Since waves in three dimensions obey:
∂ 2 f (xi, t)
− c2 ∇2 f (xi , t) = 0,
∂t2
show that a spherical light wave (away from the origin r = 0) can be given by:
F0 exp i(kr ± ωt)
i
,
f (x , t) = ℜ
r
where f is any component of A, and find the relation between k and ω.
e) There can be an infinite number of such solutions. What quantities can be varied in
creating sums of such solutions? Argue that, if the integral exists and is sufficiently
differentiable, then:
f (xi , t)
Z ∞
F (ω) exp i(k(ω)r − ωt)
=ℜ
dω,
r
−∞
is also a solution (away from r = 0).
3