154
7 Small Magnetic Black Hole
We begin with the Minkowskian line element in rectangular Cartesian coordinates and time X i = (T , X, Y, Z):
ds
2
= dT
2
− dX
2
− dY
2
− dZ
2
= η ij dX
i dX
j .
(7.1)
For a magnetic field the Maxwell 2-form takes the form
F =
1
2
F ij dX
i
∧ dX
j
= B 1 dZ ∧ dY + B 2 dX ∧ dZ + B 3 dY ∧ dX ,
(7.2)
with
B = (B 1 , B 2 , B 3 ) ,
(7.3)
the magnetic 3-vector. For a magnetic pole located at X = Y = Z = 0 with
monopole moment g = constant the magnetic 3-vector is given by
B =
g
r 3 (X, Y, Z) with r =
X 2 + Y 2 + Z 2 .
(7.4)
Introducing spherical polars r, θ, φ in the usual way,
X = r sin θ cos φ , Y = r sin θ sin φ , Z = r cos θ ,
(7.5)
(7.1), and (7.2) with (7.4), take the form
ds
2
= −r
2 (dθ
2
+ sin
2 θ dφ
2 ) − dr
2
+ dT
2 ,
(7.6)
and
F = −g sin θ dθ ∧ dφ = d(g cos θ dφ) ,
(7.7)
respectively. The final equality in (7.7) is an exterior derivative. Introducing
stereographic coordinates x, y and retarded time u via
x + i y = 2 e
i φ cot
θ
2
, u = T − r ,
(7.8)
we have
ds
2
= −r
2 p
−2
0 (dx
2
+ dy
2 ) + 2 du dr + du
2 with p 0 = 1 +
1
4
(x
2
+ y
2 ) , (7.9)
and
F = g p
−2
0 dx ∧ dy = dA ,
(7.10)