Gravity and Strings

8.2 The Einstein–Maxwell system 223
and, using the antisymmetry of Fµν or d 2 = 0, it is trivial to see that, since the l.h.s. of the
equation is divergence-free, the r.h.s. of the equation is also, for consistency, divergencefree,
as we knew it had to be in order to preserve the gauge invariance of the action. This is
no coincidence: the fact that the r.h.s. of the Maxwell equation is divergence-free is in fact
the gauge identity associated with the invariance under δ Aµ = ∂µ,asweare going to see.
Finally, using the Maxwell equation (8.47), we can rewrite the definition of the total
electric charge Eq. (8.44) in terms of the field strength and again use Stokes’ theorem. If
the boundary of a constant-time slice has the topology of a S 2 at infinity, we obtain
q =
S2 ⋆F. (8.48)
∞
which is a useful definition of the total electric charge of a spacetime in terms of the field
strength (the electric flux) and which we will generalize further in Part III.
This is the kind of formula that we will use because in the Einstein–Maxwell system
there are no fields explicitly written that act as sources for Aµ. Just as in the case of the
Maxwell equations in vacuum, we can obtain solutions describing the field of charges.
These solutions are singular near the place where the charge ought to be and the solution is
not a solution there (there are no charges explicitly included in the system). However, the
above expression allows us to calculate the charge that ought to be placed there to produce
the flux of electromagnetic field that we observe. 10 We have introduced sources as a device
for understanding the definition.
We could also have used the invariance of the Einstein–Maxwell action to find the conserved
Noether current and define the electric charge through it.
We studied the invariance of the Maxwell action and found the corresponding Noether
current in Minkowski spacetime in Section 3.2.1. The coupling to gravity introduces only
minor changes and the conclusion is, again, that the electric charge can be defined by
Eq. (8.48).
It is useful to consider a simple example of a source: the current associated with a particle
of electric charge q and worldline γ parametrized by X µ (ξ).Inamanifestly covariant form
it is given by
j µ
µ 1
(x) = qc dX √ δ
γ |g| (4) [x − X (ξ)], (8.49)
where dX µ = dξdX µ /dξ. Onmaking the choice ξ = X 0 and integrating over X 0 ,we
obtain
j µ (x 0
, x) = qc
=−qc dXµ
dx 0
0 dXµ
dX
dX0 1
√ δ
|g| (3) (x − X)δ(x 0 − X 0 )
δ (3) [x − X(x 0 )]
√ . (8.50)
|g|
10 Of course, this is just a covariant generalization of the Gauss theorem that relates the flux of electric field
through a closed surface to the charge enclosed by it.