Stars as Laboratories for Fundamental Physics - MPP Theory Group

204 Chapter 6
The field-strength tensor contains six independent degrees of freedom,
the E and B fields. The redundancy imposed by the constraint of
the homogeneous equations was removed by introducing the vector potential.
There still remains one redundant degree of freedom related to
a constraint imposed by current conservation. The Maxwell equations
remain invariant under a “gauge transformation” A → A − ∂α where α
is an arbitrary scalar function. The modified A yields the same fields
E and B which are the physically measurable quantities.
The relationship to current conservation is easiest recognized if one
recalls that Maxwell’s equations can be derived from a Lagrangian
− 1 4 F 2 − J · A where F 2 = F µν F µν . A gauge transformation introduces
an additional term J ·∂α which is identical to a total divergence ∂ ·(αJ)
if ∂ · J = 0 and thus leaves the Euler-Langrange equations unchanged.
Indeed, current conservation is a necessary and sufficient condition for
the gauge invariance of the theory (Itzykson and Zuber 1983).
A judicious choice of gauge can simplify the equations enormously.
Two important possibilities are the Lorentz gauge and the Coulomb,
transverse, or radiation gauge, based on the conditions
∂ · A = 0,
Lorentz gauge,
∇ · A = 0, Coulomb gauge. (6.20)
Maxwell’s equations are then found to be (Jackson 1975)
Φ = ρ, A = J, Lorentz gauge,
−∇ 2 Φ = ρ, A = J T , Coulomb gauge, (6.21)
where J T is the transverse part of J characterized by ∇ · J T = 0.
In the absence of sources (ρ = 0 and J = 0) the potential Φ vanishes
in the Coulomb gauge while A obeys a wave equation. A Fourier transformation
leads to (−k 2 + ω 2 )A = 0 whence the propagating modes
have the dispersion relation k 2 = ω 2 corresponding to massless particles.
Because of the transversality condition k · A = 0 there are only
two polarizations, the usual transverse electromagnetic waves. They
are characterized by an electric field E transverse to k and a magnetic
field of the same magnitude transverse to both.
6.3.2 Linear Response of the Medium
Maxwell’s equations allow one to calculate the electromagnetic fields
in the presence of prescribed external currents. However, the charged