- Consider two Cartesian reference frames,
and
, in the
standard configuration. Suppose that
moves with constant
velocity
with respect to
along their common
-axis.
Demonstrate that the Lorentz transformation between coordinates in the
two frames can be written

where . Show that the previous transformation is equivalent to a rotation through an angle , in the - plane, in ( , , , ) space. - Show that, in the standard configuration, two successive Lorentz transformations with velocities
and
are equivalent to
a single Lorentz transformation with velocity
- Let
and
be the displacement vectors of some particle in the Cartesian reference frames
and
, respectively. Suppose that frame
moves with
velocity
with respect to frame
. Demonstrate that a
general Lorentz transformation takes the form
(1982)

where . If and are the particle's velocities in the two reference frames, respectively, demonstrate that a general velocity transformation is written - Let
be the Earth's approximately constant orbital speed.
Demonstrate that the direction of starlight incident at right-angles to the Earth's
instantaneous direction of motion appears slightly shifted in the Earth's
instantaneous rest frame by an angle
. This
effect is known as the
*abberation*of starlight. Estimate the magnitude of (in arc seconds). - Let
and
be the electric and magnetic field, respectively,
in some Cartesian reference frame
. Likewise, let
and
be the electric and magnetic field, respectively, in some other Cartesian
frame
, which moves with velocity
with respect to
.
Demonstrate that the general transformation of fields takes the form

where . - A particle of rest mass
and charge
moves relativistically
in a uniform magnetic field of strength
. Show that the particle's
trajectory is a helix aligned along the direction of the field, and that
the particle drifts parallel to the field at a uniform
velocity, and gyrates in the plane perpendicular to the field with constant
angular velocity
- Let
and
. Prove the
following statements, assuming that
and
are not both zero.
- At any given event,
is perpendicular to
either in all
frames of reference, or in none. Moreover, each of the three relations
,
, and
holds in all frames or in none.
- If then the field is said to be null. For a null field, is perpendicular to , and , in all frames.
- If
and
then there are infinitely many frames (with
a common relative direction of motion) in which
or
, according
as
or
, and none other. Precisely one of these frames
moves in the direction
, its velocity being
or
, respectively.
- If then there are infinitely many frames (with a common direction of motion) in which is parallel to , and none other. Precisely one of these moves in the direction , its velocity being given by the smaller root of the quadratic equation , where , and . In order for to be real we require . Demonstrate that this is always the case.

- At any given event,
is perpendicular to
either in all
frames of reference, or in none. Moreover, each of the three relations
,
, and
holds in all frames or in none.
- In the rest frame of a conducting medium, the current density satisfies
Ohm's law
, where
is the conductivity,
and primes denote quantities in the rest frame.
- Taking into account the possibility of convection currents, as
well as conduction currents, show that the covariant generalization
of Ohm's law is
- Show that if the medium has a velocity

- Taking into account the possibility of convection currents, as
well as conduction currents, show that the covariant generalization
of Ohm's law is
- Consider the relativistically covariant form of Maxwell's equations
in the presence of magnetic monopoles. Demonstrate that it
is possible to define a proper-4-current

respectively. Finally, if is the electromagnetic field tensor, and its dual, show that Maxwell's equations are equivalent to

- Prove that the electromagnetic energy tensor satisfies the following
two identities:
- A charge
moves in simple harmonic motion along the
axis,
such that its retarded position is
.
- Show that the instantaneous power radiated per unit solid angle is
- By time averaging, show that the average power radiated per unit solid angle is
- Sketch the angular distribution of the radiation for non-relativistic and ultra-relativistic motion.

- Show that the instantaneous power radiated per unit solid angle is
- The trajectory of a relativistic particle of charge
and rest mass
in a uniform magnetic field
is a helix aligned with the field. Let the pitch angle of the
helix be
(so,
corresponds to circular motion).
By arguments similar to those used for synchrotron radiation, show
that an observer far from the charge would detect radiation with a
fundamental frequency