Lecture 5
Today we reviewed electromagnetism and began discussing Lorentz invariance.
We reviewed the Lorentz force: a charged particle experiences the
force 
along the electric field lines. A moving charged
particle experiences the force 
, perpendicular
to both the particle's velocity and the magnetic field lines, which
makes it spiral around the field lines.
We then reviewed Maxwell's equations: Gauss' law, that charge is the
source for the electric field; Ampere's law, that current or a
changing 
field induces net circulation in the 
field; Gauss' law
for magnetism, that there are no isolated magnetic sources; and
Faraday's law, that a changing field induces net circulation in the
field (causing current loops). We wrote Maxwell's equations in
differential form, noting that applying the Divergence and Stokes'
theorems produces the integral form.
We then introduced the electromagnetic potentials 
and
. These came from finding the most general forms possible for
the and fields, consistent with the last two Maxwell
equations. Thus we wrote as the curl of some vector potential
, and as 
times the partial time derivative of
minus the gradient of some scalar potential .
We pointed out that only the electric and magnetic fields -- not the
potentials themselves -- had physically observable
consequences. These physical fields don't uniquely determine the
potentials and . Instead, and are unchanged by "gauge
transformations". Gauge transformations are determined by some
spacetime function f: if we both subtract the gradient of f from
, and add 
times the partial time derivative of f to
, neither nor changes. The original pair 
, and the new pair 
are redundant
descriptions of the same physical state. Each choice for f gives us
yet another redundant description.
This is not necessarily bad; however, we must be vigilant. Each
redundant description had better give us the same prediction about
observations made of the physical state 
. Otherwise,
our formalism is nonsense. That is to say, if we write Maxwell's laws
in terms of the electromagnetic potentials and , those
laws must be "gauge invariant". For now, we'll assume that they are;
we will explicitly show it after an interlude on Lorentz invariance.
We then derived Maxwell's equations for and , by
substituting for and in the remaining (first) two
Maxwell equations. Their form looked a bit complicated in general, so
we chose a specific gauge to work in which they simplified. In this
gauge (the Lorentz gauge), we obtained the simple equations 
, and 
(where 
is the
d'Alembertian, or wave-equation, operator). This showed us 1) that
and play similar roles in Maxwell's equations; and 2)
that electromagnetic waves in free space are solutions to Maxwell's
equations.
The general form for Maxwell's equations for and (with no
gauge choice) will turn out to be just right to make Lorentz
invariance (that is, causality and locality) clear. We will see this
next time.
We then introduced Lorentz transformations, discussing how
measurements of space and time intervals vary for frames related by a
Lorentz boost in the x-direction (that is, one frame moves with
velocity v with respect to the other, in the x-direction). We found
that the quantities x and ct mix among themselves, due to the
boost. The coefficients of the mixing, 
and 
(where 
is the dimensionless velocity, and 
is
are such that the quantity 
, called the
proper distance, is unchanged by the transformation.
-- KB