Gravity and Strings

222 The Reissner–Nordström black hole
Let us first introduce the electric charge using sources. A source for the Maxwell field is
described by a current j µ , which naturally couples to the vector field through a term in the
action of the form
1
d 4 x |g| −Aµ j µ . (8.40)
c 2
This additional interaction term spoils the action’s gauge invariance unless the source j µ
is divergence-free,
∇µ j µ = 0 ⇔ d ⋆ j = 0 ( j ≡ jµdx µ ), (8.41)
which implies the continuity equation for the vector density j µ ≡ √ |g| j µ ,
∂µj µ = 0. (8.42)
The continuity equation can be used to establish the local conservation of the electric
charge, as explained in Section 2.3, if the electric charge contained in a three-dimensional
volume at a given time t,V3 t ,isdefined by9
q(t) =− 1
d
c
3 x j 0 , (8.43)
or, in a more covariant form,
q(t) = 1
c V3 ⋆
j.
t
(8.44)
As explained in Section 2.3, this quantity is not constant: its variation is related to the flux
of charge through the boundary of V 3 t .IfV3 t
V 3 t
is a constant-time slice of the whole spacetime
with no boundary, then the above integrals give the total charge, which will be constant in
time. If we can foliate our spacetime with constant-time hypersurfaces, then we take the
four-dimensional spacetime V 4 contained in between two constant-time slices V 3 t1 and V3 t2 ,
integrate the continuity equation over it, and use Stokes’ theorem. The boundary of the
four-dimensional region we have proposed is made up of the two constant-time slices with
opposite orientations, so
0 =
V4 d ⋆
j =
V3
⋆
j −
t1 V3 ⋆
j, (8.45)
t2 and the total electric charge is constant in time.
Thus, gauge invariance of the action implies that the source is divergence-free and from
this the local conservation of the electric charge (and the global conservation of the total
electric charge) follows.
On the other hand, in the presence of the source, the Maxwell equation is modified into
∇µF µν = 1
c j ν , (8.46)
or, equivalently,
9 The sign is conventional.
d ⋆ F = 1 ⋆
j, (8.47)
c