Description
1. Consider a particle launched in the x, y plane at a speed V = 3/5 relative to an inertial
reference frame with coordinates (t, x, y, z), with this velocity vector making an angle of
⇡/3 with the x axis.1
a) Calculate the components of the velocity four-vector relative to this frame.
b) Show that this velocity four-vector is normalized, namely, that
˜
u ·
˜
u = 1. (1)
Now consider another inertial reference frame with coordinates (t
0
, x0
, y0
, z0
) moving at a
constant speed v = 4/5 relative to the aforementioned inertial frame.
c) Calculate the components of the velocity four-vector relative to this S0 frame.
d) Show that this velocity four-vector is normalized, namely, that
˜
u0 ·
˜
u0 = 1. (2)
2. Consider a particle moving along the x-axis whose velocity four-vector is described by
u↵(t) = ⇣p1 + f 2(t), f(t), 0, 0
⌘
, (3)
where f(t) is a generic function of coordinate time t.
a) Explicitly show that this velocity four-vector is normalized, namely, that
˜
u ·
˜
u = 1. (4)
b) Calculate the components of the acceleration four-vector, a↵(t), and write the
components in an analogous way to Eq. (3).
c) Explicitly show that the four-acceleration is orthogonal to the four-velocity, namely,
that
˜
u ·
˜
a = 0. (5)
1Here we’re working in units where c = 1
1
for any generic function f(t).
d) Construct an expression for the particles velocity, V (t) = dx/dt in terms of f(t).
e) Find expressions for ⌧ (t) and x(t), which here take the form of integrals of f(t).
f) Calculate the components of the four-force and the three-force acting on the particle.
3. The worldline of a massive particle is described parametrically by the spacetime coordinates
t(⌧ ) = 1
↵ ln (tan(↵⌧ ) + sec(↵⌧ ))
x(⌧ ) = 1
↵ ln (cos(↵⌧ )). (6)
a) Explicitly show that the parameter ⌧ represents the proper time of the particle.
b) Calculate the components of the four-velocity, u↵, and four-acceleration, a↵, of the
particle and display the components as the elements of a row.
Now consider an inertial reference frame with coordinates (t
0
, x0
) moving at a constant
speed v relative to the aforementioned inertial frame with coordinates (t, x).
c) Calculate the components of the four-velocity, u↵0
, and four-acceleration, a↵0
, of the
particle in this S0 frame and display the components as the elements of a row.
d) Explicitly show that in this S0 frame the velocity four-vector is normalized and the
four-acceleration is orthogonal to the four-velocity, namely, that
˜
u0 ·
˜
u0 = 1
˜
u0 ·
˜
a0 = 0. (7)
4. A particle of mass M is moving in the laboratory with speed (3/5)c. It decays into a
particle of mass M0, where M0 is at rest in the laboratory, and a photon of energy 12,000
MeV. Using MeV units, find
a) the mass M,
b) the mass M0, and
c) the momentum of M.
2
m
m
M (4/5)c
5. Two particles of mass m = (1/3)M collide and fuse together, as shown in the accompanying
picture. The newly formed particle of mass M moves o↵ to the right at a speed of
V = (4/5)c. Find the x and y components of the initial velocity of each particle of mass
m before the collision occurs.
