General Relativity and Cosmology
Problem Sheet 1
João G. Rosa
Universidade de Coimbra
1. Verify that Lorentz boosts of the form:
βi βj
Λi j = δij + (γ − 1) , Λi 0 = Λ0 i = βi γ , Λ0 0 = γ ,
β2
1
where βi = vi /c and γ = √ , are elements of the proper orthocronous Lorentz group.
1−β 2
2. Consider Maxwell’s equations describing the electromagnetic field in vacuum:
ρ ∂E
∇ · E = 0 , ∇ × B = µ0 J + 0
∂t
∂B
∇·B=0 , ∇×E+ =0.
∂t
(a) Show that these equations can be written in covariant form in terms of the antissymmetric
field strength tensor Fµν = −Fνµ , where F0i = Ei /c and Fij = −ijk Bk :
∂µ F µν = µ0 J ν , µνρσ ∂ν Fρσ = 0 ,
where the 4-current J µ = (cρ, J) and c = (0 µ0 )−1/2 is the speed of light.
(b) Show that Maxwell’s equations imply the conservation of electric charge by obtaining the
continuity equation ∂µ j µ = 0.
(c) Show that, in terms of the 4-vector potential Aµ = (Φ/c, A):
Fµν = ∂µ Aν − ∂ν Aµ .
(d) Using the above result, show that the field strength tensor is invariant under gauge
transformations of the form Aµ → Aµ − ∂µ λ, for some λ(xµ ).
(e) In the Lorentz gauge, defined by the condition ∂µ Aµ = 0, show that the 4-vector potential
satisfies a wave equation of the form:
2Aµ = µ0 J µ .
(f) Show that, in the absence of currents, the dual field strength tensor F̃µν = 12 µνρσ F ρσ
also satisfies Maxwell’s equations, and interpret this duality in terms of the electric and
magnetic fields.
1
3. Show that the solution of the Poisson equation for a point mass M located at the origin,
ρ = M δ 3 (r), is given by:
GM
Φ(r) = − ,
r
where r = |r| and hence obtain the gravitational field created by the point mass. Is this result
valid for any other mass distribution with total mass M ? Compute the value of |Φ|/c2 at the
surface of the Earth and of the Sun.
4. Show that, in spherical coordinates, the non-vanishing components of the tidal tensor generated
by a point mass M at a distance r from its position are given by:
GM GM
Err = −2 , Eθθ = Eϕϕ = .
r3 r3
Hence, estimate the relative importance of the Sun and the Moon to ocean tides on the Earth’s
surface. Consider M = 2 × 1030 kg and M$ = 7 × 1022 kg, as well as circular orbits of radius
r = 1.5 × 108 km and r$ = 4 × 105 km, respectively.
5. The Eötvös experiment was designed to test the Weak Equivalence Principle by using a torsion
balance to measure differences between the inertial and gravitational mass of test bodies. This
torsion balance consisted of a rigid rod supporting two masses on either side, as well as a wire
attached to the centre of the rod to measure the torsion, as shown in the figure below.
lA lB
A B
(a) Show that the system will rotate about the rod axis unless the inertial and gravitational
masses of the test bodies are equal. You should take into account the centrifugal acce-
leration due to the Earth’s rotation, although noting that it is much smaller than the
(vertical) gravitational acceleration.
(b) Determine the latitude on Earth that is the best for performing the experiment.
2
6. A test particle of mass m scatters of a massive body of mass M with an impact parameter b
and velocity v.
(a) Show that, in Newtonian gravity, the particle’s trajectory is deflected by an angle θ that
satisfies:
θ GM
tan = 2 ,
2 v b
which is independent of the test particle’s mass.
(b) Use the above result to determine the Newtonian deflection angle of a light ray grazing
the Sun’s surface.
7. Consider the non-relativistic Schrödinger equation for a free particle moving in one dimension:
~2 ∂ 2 ψ ∂ψ
− 2
= i~ .
2m ∂x ∂t
By performing a Galilean transformation to a uniformly accelerated frame x0 = x + vt + 21 at2
and t0 = t, obtain the Schrödinger equation in a uniform gravitational field:
~2 ∂ 2 ψ 0 0 0 ∂ψ 0
− + mgx ψ = i~ ,
2m ∂x02 ∂t0
where ψ = eiθ ψ 0 . Determine the form of the phase θ(x, t) and discuss the validity of Einstein’s
Equivalence Principle.
