Gravity and Strings

8.2 The Einstein–Maxwell system 219
where E = (E1, E2, E3) and B = (B1, B2, B3) are the electric and magnetic 3-vector fields
in that coordinate system, and, thus, with ∇ =(∂1,∂2,∂3)
Ei = F0i,
Bi =− 1
2ɛijkFkl, ⇔
⎧
⎪⎨ E =−∇φ −
⎪⎩
1 ∂
c ∂t A,
(8.18)
B = ∇ × A.
The field strength (and the action) is invariant under the Abelian gauge transformations
A ′ µ = Aµ + ∂µ (8.19)
with smooth, gauge parameter . Depending on which gauge group we consider (R or
U(1)), must be a single-valued or multivalued function. 6 In differential-forms language
A = Aµdx µ , A ′ = A + d, F = 1
2 Fµνdx µ ∧ dx ν = dA, (8.20)
and the gauge invariance of F is a consequence of d 2 = 0. Using these differential forms,
the Maxwell action can be rewritten as follows:
SM[A] = 1
F ∧
8c
⋆ F. (8.21)
Observe that there is no matter charged with respect to Aµ in this system. This is analogous
to the presence of no matter fields in the Einstein–Hilbert action. However, the
Einstein–Hilbert action contains the self-coupling of gravity and therefore the presence
of a coupling constant in it makes sense, whereas in the Maxwell theory there are no direct
interactions between photons and, in principle, there is neither an electromagnetic coupling
constant nor a unit of electric charge. We will see that things are a bit more complicated in
the presence of gravity, through which photons do interact.
The equations of motion of gµν and Aµ are
Gµν − 8πG(4)
N
c4 Tµν = 0, (8.22)
∇µF µν = 0 (Maxwell’s equation), (8.23)
where
Tµν = −2c δSM[A]
√
|g| δg µν = Fµρ Fν ρ − 1
4 gµν F 2
(8.24)
is the energy–momentum tensor of the vector field, which is traceless7 in d = 4. The
tracelessness of the electromagnetic energy–momentum tensor implies that R = 0 and the
6 If the gauge group is R, the elements of the group will be e/L , whereas, if it is U(1), they will be ei/L ,
where L is a constant introduced to make the exponent dimensionless because is dimensionful. In the
second case will have to be identified with + 2π L. When there is a unit of charge, L is related to it.
7 This property is associated with the invariance of the Maxwell Lagrangian in curved spacetime under Weyl
rescalings of the metric,
g ′ µν = 2 (x)gµν. (8.25)
In fact, if = e σ , then for infinitesimal transformations δσ gµν = 2σ(x)gµν we have
δσ SM = δSM
δσ gµν ∼ σ T
δgµν
µν gµν = 0. (8.26)