Problems Week 11
1. Photons can scatter against each other even though it happens rarely.
An observer sees two photons going towards each other, in his orthogonal
space, with energies E1 and E2. One of them is scattered an angle
θ. The orthogonal space picture is
E '¡
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d
d
d
d‚
E1 E2
E
θ
Calculate its energy as measured by the observer.
2. Consider two parallel null lines and a vector ¯r connecting them which
is orthogonal to them. Show that all vectors connecting the lines are
orthogonal to them. Show also that all such vectors have the same
length.
3. Protons (p+
) and anti-protons (p−
) have the same mass, m. Protons
are accelerated to energy E and then collide with a proton at rest, all
with respect to the laboratory. What energy E is needed to form an
anti-proton through the process
p+
+ p+
→ p+
+ p+
+ p+
+ p−
?
4. A particle travels along a worldline given by ¯R = ¯R(τ) where τ is the
proper time. Along the whole worldline we have
ˆu · ¯A = 0 ,
with ˆu a constant four-velocity and ¯A the four-acceleration of the particle.
Express this in a simple way in terms of the particle’s velocity
relative to ˆu.
5. Four spaceships with travel times τ1, τ1, τ2, τ2 part and meet according
to the planar spacetime diagram
1
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d
d
d
ds
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d
d
d
ds
¯A
¯C ¯B
¯D
τ1
τ1
τ2
τ2
Show that opposite worldlines are parallel.
[Hint: Calculate ( ¯A + ¯B) · ( ¯A − ¯B) and ( ¯C + ¯D) · ( ¯A − ¯B).]
6. A rocket is propelled by some of its mass being ejected backwards with
velocity v relative to the rocket. It moves in a straight line, i.e. the
motion is in a 2-plane in spacetime. It is driven until its velocity with
respect to the initial state is u. Calculate the ratio of final mass to
initial mass of the rocket.
7. Let ¯n be a null vector. Show that if ¯k · ¯n = 0 then ¯k is either space-like
or proportional to ¯n.
8. A light source is moving away from an observer U with velocity v. Another
observer V is also moving away with velocity v but perpendicular
to the direction of motion of the light source. When the source emits
a light signal it is as far from the origin as V is when he receives it.
Calculate the Doppler shift ωV /ω0 where ω0 is the frequency measured
by an observer traveling with the light source.
9. Two observers currently at the same location observe a small distant
object in their direction of travel. One observes the object to be twice
as big as the other. What is their relative velocity.
2