Eur. Phys. J. C (2018) 78 :1016 Page 3 of 13 1016
Before proceeding, we stop here to more clarify the sys-
tem described by the action (2) and indicate its suggestions.
In this action, apart from considering the UV modification of
the matter, we consider the modification of the gravity at the
large distances or IR regime. Interestingly, at the short dis-
tances or UV regime, the gravity may obtain the corrections
arising from a quantum theory of the gravity such as string
theory. One of such UV modified gravity theories is the f (R)
gravity where f (R) is a function of the scalar curvature R
(see Ref. [110] for a review). Considering the modification of
the gravity in both UV and IR regimes, corresponding to the
f (R) nonlinear massive gravity, and its cosmological impli-
cations are studied in Refs. [111,112]. In addition, in this
work, the reference metric f is treated as a non-dynamical
object. However, in general the reference metric f may be
a dynamical object, and consequently we have a non-linear
bimetric theory which consists of a massless spin-2 field cou-
pling to a massive spin-2 field [113]. Therefore, it is signifi-
cant to find the non-linear charged black hole solutions in the
massive gravity with including the f (R) correction as well as
considering the reference metric f to be a dynamical object.
Also, an extension of the non-linear electrodynamics into the
cosmology of the massive gravity [114–119] is expected to
lead to interesting implications. All of these points will be
studied in our future works.
1
We would like to consider the spherically-symmetric and
static metric of the spacetime, given by the following ansatz
ds
2
=−f (r)dt
2
+ f (r )
−1
dr
2
+r
2
d
2
2
, f (r ) = 1−
2m(r )
r
.
(6)
Following Ref . [97], we take the reference metric as
f
μν
= diag(0, 0, c
2
, c
2
sin
2
θ), (7)
where c is a positive constant. With given spacetime and
reference metrics, one can easily write explicitly the massive
gravity term as
U
1
=
2c
r
, U
2
=
2c
2
r
2
, U
3
= U
4
= 0. (8)
As a result, the equations of motion derived from the action
(2)are
G
ν
μ
−
3
l
2
+ m
2
cc
1
r
+
c
2
c
2
r
2
δ
ν
μ
= 2
∂L(F)
∂ F
F
μρ
F
νρ
− δ
ν
μ
L(F)
, (9)
∇
μ
∂L(F)
∂ F
F
νμ
= 0. (10)
1
We would like to thank Reviewer for indicating these points.
∇
μ
∗ F
νμ
= 0. (11)
Now let us look for a black hole solution of the mass M and
magnetic charge Q, with the ansatz of the magnetic field as
F
μν
=
δ
θ
μ
δ
ϕ
ν
− δ
θ
ν
δ
ϕ
μ
B(r,θ). (12)
It is notable that the magnetic charge Q is defined by
Q =
1
4π
S
∞
2
F, (13)
with F =
1
2
F
μν
dx
μ
∧dx
ν
and S
∞
2
to be a two-sphere at the
infinity. From Eqs. (10), (11) and (13), it can derive
B(r,θ) = Q sin θ, −→ F =
Q
2
2r
4
. (14)
Using this result, the (t, t ) component of Eq. (9) reads
dm(r )
dr
+
3r
2
2l
2
+
m
2
2
cc
1
r + c
2
c
2
=
Q
2
2r
2
e
−
k
2r
. (15)
Integrating this equation with the integral constant M
m(r) +
r
3
2l
2
+
m
2
2
cc
1
r
2
2
+ c
2
c
2
r
r→∞
= M, (16)
then substituting m(r) into f (r), we finally get
f (r ) = 1 −
2M
r
e
−
k
2r
+
r
2
l
2
+ m
2
cc
1
r
2
+ c
2
c
2
. (17)
Here, 1+m
2
c
2
c
2
can be understood as effective horizon cur-
vature which can be positive, zero, or negative corresponding
to the sphere, flat, or hyperbolic effective horizon. In the case
of the massless graviton (m = 0), it leads to the non-linear
charged AdS black hole sourced by the non-linear electro-
dynamics (4). For M = Q = 0, we can derive the vacuum
solution as
f (r ) = 1 +
r
2
l
2
+ m
2
cc
1
r
2
+ c
2
c
2
. (18)
Note that, the vacuum solution itself has a singularity at the
origin r = 0 because the mass of the graviton is non-zero. In
the limit k → 0, we have
f (r ) = 1 −
2M
r
+
r
2
l
2
+ m
2
cc
1
r
2
+ c
2
c
2
, (19)
which implies that the black hole becomes the 4D
Schwarzschild-AdS black hole in the massive gravity. This
is because, for k → 0, the squared charge Q
2
of the black
hole is extremely small compared to its mass M and thus
the black hole could be considered to be electrically neutral.
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