Eur. Phys. J. C (2018) 78:641 Page 3 of 11 641
3 Black hole solution for Einstein–Maxwell model of
arbitrary power
The general metric ansatz assuming circular symmetry is
given by
ds
2
=−f (r)dt
2
+ g(r)dr
2
+r
2
dφ
2
. (6)
Note that, in the classical solution, it is possible to deduce the
Schwarzschild relation, namely g(r) = f (r)
−1
. The classi-
cal (2 + 1)-dimensional Einstein–Maxwell black hole solu-
tion (for an arbitrary index β) is obtained after solving f
0
(r)
and E
0
(r) and was previously found in Ref. [21]. As we
will compare these results with the scale-dependent solution
provided in Sect. 6, here we will briefly comment the main
features of the classical case. Then, solving the Einstein field
equations we obtain:
f
0
(r) = Br
1−α
+
C
α − 1
, (7)
E
0
(r) = A
e
α+1
0
r
α
. (8)
where the set {A, B, C} are constants of integration which
must be fixed. According to Ref. [21], the parameter C is
related with the mass of the black hole M
0
while B takes into
account the classical charge Q
0
(the same for the parameter
A). In addition, note that the auxiliary parameter α is defined
as follow:
α =
1
2β − 1
. (9)
The next step consists in computing the horizon of this black
hole, which is
r
0
=
C
B(1 −α)
1
1−α
. (10)
By writing the lapse function in terms of the classical horizon
we have
f
0
(r) =
C
α − 1
1 −
r
0
r
α−1
. (11)
Another important point is the thermodynamics of the sys-
tem. We can then define three quantities, i. e., the Hawking
temperature, T
H
, the Bekenstein–Hawking entropy, S, and
the specific heat, C
Q
. Their corresponding expressions are
given by
T
0
(r
0
) =
1
4π
C
r
0
, (12)
S
0
(r
0
) =
A
0
4G
0
, (13)
C
0
(r
0
) = T
∂ S
∂T
r
0
, (14)
being A
0
the horizon area defined as
A
0
=
dx
√
h = 2πr
0
. (15)
where h
ij
is the induced metric at the horizon r
0
.
4 Scale dependent coupling and scale setting
This section summarizes the equations of motion for the
scale-dependent Einstein-power-Maxwell theory with arbi-
trary index. The idea and notation follows [49–58]. The scale-
dependent couplings of the theory are (i) the Newton’s cou-
pling G
k
(which can be related with the gravitational cou-
pling by κ
k
≡ 8π G
k
), and (ii) the electromagnetic coupling
1/e
k
. Furthermore, there are three independent fields, which
are the metric g
μν
(x), the electromagnetic four-potential
A
μ
(x), and the scale field k(x). The effective action for this
theory reads
Γ [g
μν
, A
μ
, k]=
d
3
x
√
−g
1
2κ
k
R −
1
e
2β
k
L(F)
. (16)
The equations of motion obtained from a variation of (16)
with respect to g
μν
(x) are
G
μν
=
κ
k
e
2β
k
T
eff
μν
, (17)
where
T
eff
μν
= T
EM
μν
−
e
2β
k
κ
k
Δt
μν
. (18)
Note that T
EM
μν
is given by (3) and the additional contribution
Δt
μν
is
Δt
μν
= G
k
g
μν
−∇
μ
∇
ν
G
−1
k
. (19)
The equations of motion for the four-potential A
μ
(x) taking
into account the running of e
k
are
D
μ
L
F
F
μν
e
2β
k
= 0. (20)
It is important to note that, in any quantum field theory the
renormalization scale k has to be set to a quantity characteriz-
ing the physical system under consideration. Thus, for back-
ground solutions of the gap equations, it is not constant any-
more. However, having an arbitrarily chosen non-constant
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