Lecture 7
Handouts:
5. Solutions, Homework 1
6. Homework 2
We further discussed the transformations of observables and operators under Lorentz boosts, en route to writing Maxwell's equations in a Lorentz covariant way.
Reviewed how the four-vector 
transforms as a
vector under Lorentz transformations: that is, the components ct and
mix with each other in such a way that the four-vector 
is just matrix-multiplied by a Lorentz tranformation matrix 
.
(Later in lecture we clarified why our index summation conventions mean
that 
is just matrix multiplication of on
the vector 
.)
We then explored the consequences of Lorentz transformations for other observables. Redefining space and time (as Lorentz transformations do) induces redefinitions of other observables: for example, the velocity depends explicitly on space and time, and changes when those change due to a Lorentz transformation. Similarly, densities depend on the definition of a volume element, and so change under Lorentz transformations, as do currents which depend both on densities and velocities.
We noted that we can group observables together to find objects which
change under a Lorentz transformation in exactly the same way as
-- that is, they form four-vectors which get matrix-multiplied by
under a Lorentz transformation. Such four-vectors include
: energy mixes with momentum in such a way that
the rest mass 
is invariant. That 
and 
are also four-vectors
is shown in Appendix B of Rolnick.
We also noted that determines a four-dimensional
gradient operator, 
, whose components are
and 
. We get tired of writing so many
's and label this operator 
. Calculus tells you
that under a change of variables (and a Lorentz transformation is just
a specific change of variables) independent variables remain
independent. That is, initially we know that the partial derivative
of with respect to is 1 if 
and 
are the same index,
and 0 otherwise. If the same thing is to be true in our
Lorentz-tranformed frame, multiplication of by must be
accompanied by multiplication of by 
. Thus
really transforms like a covector, and its lower index
makes sense.
Now that we have grouped the relevant observables and operators into objects that transform in a well-defined way under Lorentz tranformations (as vectors or covectors), we seek to write the laws of electromagnetism in a Lorentz-covariant way. We focus on Maxwell's equations; you will see a covariant formulation of the Lorentz force law in your homework.
I will not reproduce the details of the derivation
here. Schematically, we started with Maxwell's equations in terms of
the potentials 
and 
. These looked like
We worked as follows:
1)
Given that and 
, we identified their components in object 1
to show that object 1 
. This object is a scalar: since it has 1 upper and 1 lower
index, it is multiplied by one factor of and one factor of
under Lorentz transformations, which cancel to leave it
invariant. Thus the Lorentz condition, 
, is
Lorentz covariant.
2)
We showed that 
. To show this, we
digressed on how raising to determine 
introduces signs on the spatial components, due to matrix
multiplication by 
. The dot product
then gives 
. We also calculated the sum 
explicitly, showing how the 4
nonzero diagonal entries fix the signs so that we get 
.
3)
Having simplified those parts, we came back to the original equations,
and used the identities , and 
-- the sign
on comes from raising . We read off their
components in the Maxwell equations to get a simple, covariant
equation, in which a four-vector on the left-hand side equals another
four-vector on the right-hand-side.
-- KB