E
Small Magnetic Black Hole
E.1
The Electromagnetic Field
For an exact tetrad given via the 1-forms (7.27)–(7.30) and an exact potential 1-form
given by (7.40) the tetrad components of the perturbed Maxwell tensor are given by
F (1)(2) =
ˆ
p 2
r 2
∂ ˆ
M
∂x
−
∂ ˆ
L
∂y
,
(E.1)
F (1)(3) =
ˆ
p
r
−e
− ˆ
α cosh ˆ
β
∂ ˆ
L
∂r
+ e
ˆ
α sinh ˆ
β
∂ ˆ
M
∂r
,
(E.2)
F (2)(3) =
ˆ
p
r
e
− ˆ
α sinh ˆ
β
∂ ˆ
L
∂r
− e
ˆ
α cosh ˆ
β
∂ ˆ
M
∂r
,
(E.3)
F (1)(4) = −
ˆ
p ˆ
b
r
∂ ˆ
M
∂x
−
∂ ˆ
L
∂y
+
ˆ
p
r
e
− ˆ
α cosh ˆ
β
∂ ˆ
K
∂x
−
∂ ˆ
L
∂u
−
ˆ
p
r
e
ˆ
α sinh ˆ
β
∂ ˆ
K
∂y
−
∂ ˆ
M
∂u
+
ˆ
p ˆ
c
2 r
e
− ˆ
α cosh ˆ
β
∂ ˆ
L
∂r
−
ˆ
p ˆ
c
2 r
e
ˆ
α sinh ˆ
β
∂ ˆ
M
∂r
,
(E.4)
F (2)(4) =
ˆ
p ˆ
a
r
∂ ˆ
M
∂x
−
∂ ˆ
L
∂y
−
ˆ
p
r
e
− ˆ
α sinh ˆ
β
∂ ˆ
K
∂x
−
∂ ˆ
L
∂u
+
ˆ
p
r
e
ˆ
α cosh ˆ
β
∂ ˆ
K
∂y
−
∂ ˆ
M
∂u
−
ˆ
p ˆ
c
2 r
e
− ˆ
α sinh ˆ
β
∂ ˆ
L
∂r
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. A. Hogan, D. Puetzfeld, Frontiers in General Relativity, Lecture Notes
in Physics 984, https://doi.org/10.1007/978-3-030-69370-1
241