Gravity and Strings

224 The Reissner–Nordström black hole
If the particle is at rest at the origin in the chosen coordinate system, the current is
j µ (x 0 , x) =−qcδ µ0 δ(3) (x)
√ |g| , (8.51)
and it is easy to see that q is indeed the electric charge according to the above definitions.
j µ is a conserved current:
∇µ j µ ∼ ∂
∂x µ
|g(x)|
dξ ˙X µ 1
√ δ
|g(X)| (4)
[x − X (ξ)]
= dξ ˙X
µ ∂
∂x µ δ(4)
[x − X (ξ)] =− dξ ˙X
µ ∂
∂ X µ δ(4) [x − X (ξ)]
=− dξ d
dξ δ(4) [x − X (ξ)] =−δ (4) [x − X (ξ)] ξ2 = 0, (8.52)
ξ1
generically, except for the initial and final positions of the particle X µ (ξ1) and X µ (ξ2),
which look like a 1-particle source and a sink and can be taken to infinity.
Observe that, for the current (8.49), the interaction term (8.40) becomes the integral of
the 1-form A over the worldline γ:
− q
c
γ(ξ)
Aµ ˙x µ dξ =− q
A. (8.53)
c γ
This term has to be added to the action of the particle, Eq. (3.255), (3.257) or (3.258), in
order to obtain the worldline action of a massive electrically charged particle,
S[X µ
(ξ)] =−Mc
S 2 ∞
dξ
gµν(X) ˙X µ ˙X ν − q
c
dξ Aµ ˙X µ , (8.54)
or that of a massless one. That kind of term is known as a Wess–Zumino (WZ) term. In
this form it is easy to see that, under a gauge transformation, the action changes by a total
derivative. The integral of the total derivative vanishes exactly only for special boundary
conditions, though.
This action can be used as a source, but it also describes the motion of a charged particle
in a gravitational/electromagnetic background. In the special-relativistic limit, taking ξ =
X 0 = ct, the action takes the standard form
S ∼ dt −Mc 2 + 1
2 Mv2 − qφ + q
c A
· v . (8.55)
If there is a point-like charge q at rest at the origin the only non-vanishing components
of F are F0r and they should depend only on r because of the spherical symmetry of
the problem. Using the above definition of charge and working in general static spherical
coordinates Eq. (7.22), we find
ɛµνρσ
q =
4 √ |g| F ρσ dx µ ∧ dx ν
= d 2 r 2 2
F0r = ω(2) lim r F0r , (8.56)
r→∞
S 2 ∞