130_notes.dvi

∇·~ E~ = 4πρ

∇×~ B~−^1

c

∂E

∂t

=

4 π

c

~j.

TheLorentz Forceis

F~=−e(E~+^1

c

~v×B~).

When we change toRationalized Heaviside-Lorentz units, the equations become

∇·~ B~= 0

∇×~ E~+^1

c

∂B

∂t

= 0

∇·~ E~=ρ

∇×~ B~−^1

c

∂E

∂t

=

1

c

~j

F~=−e(E~+^1

c

~v×B~)

That is, the equations remain the same except the factors of 4πin front of the source terms disappear.
Of course, it would still be convenient to setc= 1 since this has been confusing us about 4D geometry
andcis the last unnecessary constant in Maxwell’s equations. For our calculations, we can setc= 1
any time we want unless we need answers in centimeters.

32.2 The Electromagnetic Field Tensor

The transformation of electric and magnetic fields under a Lorentzboost we established even before
Einstein developed the theory of relativity. We know that E-fields can transform into B-fields and
vice versa. For example, a point charge at rest gives an Electric field. If we boost to a frame in which
the charge is moving, there is an Electric and a Magnetic field. This means that the E-field cannot
be a Lorentz vector. We need to put the Electric and Magnetic fieldstogether into one (tensor)
object to properly handle Lorentz transformations and to write our equations in a covariant way.

The simplest way and the correct way to do this is to make the Electricand Magnetic fields com-
ponents of arank 2 (antisymmetric) tensor.

Fμν=







0 Bz −By −iEx

−Bz 0 Bx −iEy

By −Bx 0 −iEz

iEx iEy iEz 0







The fields can simply be written in terms of thevector potential, (which is a Lorentz vector)

Aμ= (A,iφ~ ).

Fμν=

∂Aν

∂xμ

−

∂Aμ

∂xν