63 Page 4 of 17 Eur. Phys. J. C (2019) 79 :63
where φ
X
= dφ/dXand Q
∗
is an integration constant which
can be regarded as the charge of the black hole. Note that the
last equality in Eq. (3.3) can be obtained from the conser-
vation equation of NED at the background level [45]. From
Eq. (3.3), we get
F
(0)(2)
=
Q
∗
r
2
φ
X
. (background level) (3.4)
In the general case where the perturbations are taken into
account, the metric functions and the field strength could
depend on t, r, and θ . In this case, the Bianchi identity of the
field strength F
[(a)(b)|(c)]
= 0 leads to
e
ψ+μ
2
F
(1)(2)
,3
+
e
ψ+μ
3
F
(3)(1)
,2
= 0, (3.5)
e
ψ+ν
F
(0)(1)
,2
+
e
ψ+μ
2
F
(1)(2)
,0
= 0, (3.6)
e
ψ+ν
F
(0)(1)
,3
+
e
ψ+μ
3
F
(1)(3)
,0
= 0, (3.7)
e
μ
2
+ν
F
(0)(2)
,3
+
e
μ
3
+ν
F
(3)(0)
,2
+
e
μ
2
+μ
3
F
(2)(3)
,0
=−e
ψ+ν
q
3,2
− q
2,3
F
(0)(1)
+e
ψ+μ
2
σ
,3
−q
3,0
F
(1)(2)
+ e
ψ+μ
3
σ
,2
− q
2,0
F
(3)(1)
. (3.8)
Note that the comma here denotes the partial derivative with
respect to the tensor indices. This derivative is related to the
directional derivative shown in Eq. (2.7)byH
(a)(b),(c)
=
e
μ
(c)
H
(a)(b),μ
[59]. In addition, Eq. (3.5) is a redundant equa-
tion because it is just an integrability condition for Eqs. (3.6)
and (3.7).
On the other hand, the conservation equation for NED
η
(b)(c)
(F
(a)(b)
φ
X
)
|(c)
= 0 can be written explicitly as follows
e
ψ+μ
3
F
(0)(2)
φ
X
,2
+
e
ψ+μ
2
F
(0)(3)
φ
X
,3
= 0, (3.9)
e
ψ+μ
2
F
(0)(3)
φ
X
,0
+
e
ψ+ν
F
(3)(2)
φ
X
,2
= 0, (3.10)
e
ψ+ν
F
(2)(3)
φ
X
,3
+
e
ψ+μ
3
F
(0)(2)
φ
X
,0
= 0, (3.11)
e
ν+μ
3
F
(1)(2)
φ
X
,2
+
e
ν+μ
2
F
(1)(3)
φ
X
,3
+
e
μ
2
+μ
3
F
(0)(1)
φ
X
,0
=
e
ψ+μ
3
σ
,2
−q
2,0
F
(0)(2)
+e
ψ+μ
2
σ
,3
−q
3,0
F
(0)(3)
+ e
ψ+ν
q
3,2
− q
2,3
F
(2)(3)
φ
X
. (3.12)
Again, it can be shown that Eq. (3.9) is a redundant equation
since it is an integrability condition for Eqs. (3.10) and (3.11).
To linearize the equations above, the scalar field φ
X
and
the gauge field invariant X should be decomposed into the
background and the 1st order parts:
φ
X
→ φ
X
+ δφ
X
, X → X + δ X, (3.13)
where
δφ
X
= φ
XX
δ X = 2φ
XX
F
(0)(2)
δF
(0)(2)
. (3.14)
In the above expressions, we have decomposed F
(0)(2)
as
F
(0)(2)
→ F
(0)(2)
+δ F
(0)(2)
in which the background F
(0)(2)
is given in
1
Eq. (3.4). Note that the quantity X at the back-
ground level is given by X = F
2
(0)(2)
.
After the decomposition, the linearized Eqs. (3.6), (3.7),
and (3.8)are
re
ν
sin θ F
(0)(1)
,r
+re
μ
2
sin θ F
(1)(2),0
= 0, (3.15)
re
ν
F
(0)(1)
sin θ
,θ
+r
2
sin θ F
(1)(3),0
= 0, (3.16)
e
μ
2
+ν
δF
(0)(2),θ
+ F
(0)(2)
(
δμ
2
+ δν
)
,θ
+
re
ν
F
(3)(0)
,r
+re
μ
2
F
(2)(3),0
= 0. (3.17)
On the other hand, the linearized Eqs. (3.10), (3.11) and
(3.12) can be written as
φ
X
re
μ
2
F
(0)(3),0
+
φ
X
re
ν
F
(3)(2)
,r
= 0, (3.18)
φ
X
δF
(0)(2),0
+ F
(0)(2)
(
δψ + δμ
3
)
,0
+
φ
X
e
ν
r sin θ
F
(2)(3)
sin θ
,θ
+ δφ
X,0
F
(0)(2)
= 0, (3.19)
re
ν
φ
X
F
(1)(2)
,r
+ e
ν+μ
2
φ
X
F
(1)(3),θ
+ re
μ
2
φ
X
F
(0)(1),0
= r
2
sin θ F
(0)(2)
φ
X
σ
,2
− q
2,0
,
(3.20)
where δν, δμ
2
, δμ
3
, and δψ are the 1st order parts of the
metric functions ν, μ
2
, μ
3
, and ψ, respectively.
For the sake of abbreviation, we define the field perturba-
tion
B ≡ F
(0)(1)
sin θ. (3.21)
After differentiating Eq. (3.20) with respect to x
0
= t and
using Eqs. (3.15) and (3.16), we have
φ
X
e
ν−μ
2
re
ν
B
,r
,r
+ φ
X
e
2ν+μ
2
r
B
,θ
sin θ
,θ
sin θ
− φ
X
re
μ
2
B
,00
=−r
2
F
(0)(2)
φ
X
σ
,20
− q
2,00
sin
2
θ.
(3.22)
Recall that in the derivation of Eq. (3.22), we have only used
the Maxwell equation of the NED source. Eq. (3.22) would
become one of the coupled master equations in the sense
that the linear order metric functions on the right hand side
of Eq. (3.22) are coupled with the field perturbation B (or
the 1st order field strength F
(0)(1)
) on the left hand side. In
1
We only use a delta into the linear order perturbations of quantities
whose values at the background level do not vanish. The quantities that
vanish at the background level, such as the metric functions σ , q
2
,and
q
3
, and the Maxwell tensor components F
(i)( j)
(ij = 02 or 20), shall
be regarded as linear order perturbation quantities directly.
123
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