Problems Week 12
1. Two light signals follow parallel worldlines, with a vector ¯r connecting
them being orthogonal to them. Show that an observer who measures
them finds that they travel in the same direction. Also show that the
spatial distance between them is constant independent of the observer.
2. A plane light wave with wave four-vector ¯K1 is incident on a mirror.
The reflected wave has wave four-vector ¯K2. Calculate the normal
vector to the mirror.
Consider the case where incidence/reflection is along the normal direction.
Calculate the four-velocity of the mirror in this case.
3. A particle follows the worldline ¯R = ¯R(τ) with τ the proper time. At a
certain point ¯R0 its four-velocity is ˆu. An unaccelerated observer with
this four-velocity measures the particle’s velocity ¯v, acceleration ¯a and
its derivative d¯a
dt
. Give ˆv, dˆv
dτ
and d2ˆv
dτ2 at ¯R0 in terms of ˆu, ¯a0 and (d¯a
dt
)0.
4. An observer has four-velocity ˆt. A rod has slope m relative to the x-axis
in his orthogonal space, i.e. y/x = m (z = 0). The rod is moving with
velocity u in the ˆx-direction. Another observer moves with velocity v in
the ˆx-direction. Her x-axis can be taken to lie in the (ˆt, ˆx)-plane. What
is the slope of the rod relative to this axis in her orthogonal space?
5. An observer shoots out two particles with velocity v in perpendicular
directions in her orthogonal space. Calculate the two particle’s relative
velocity.
6. Consider a central elastic collision of a ball of mass m and an object
of mass M. Central means all four-velocities lie in a 2-plane and elastic
means that the masses are not changed in the collision.
Let v be the velocity of the ball after the collision as seen by an observer
who sees the object at rest before the collision. Show that
γ =
1
√
1 − v2
≤
m2
+ M2
2mM
.
7. Consider the reaction p+
+ γ → n0
+ π+
where mp+ = mn0 = M and
mπ+ = m. Calculate the lower limit on the photon’s energy for this to
happen, as measured by an observer who sees the proton at rest.
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8. Protons are bombarded with pions. What energy do the pions need to
have in the rest frame of the protons for the reaction
π−
+ p+
→ π+
+ π−
+ n0
to take place? (Mπ± = m and Mp+ = Mn0 = M)
9. Two particles with masses m1 and m2 move on a line in an observer’s
orthogonal space. She measures their velocities to be u1 and u2. The
particles collide and form a new particle. Calculate its mass and veloc-
ity.
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