526 M. Farsam et al. / Nuclear Physics B 938 (2019) 523–542
S =
d
4
x
√
−g
R
μ
μ
−2
16πG
+4b
2
1 −
1 +
2F
b
2
, (2.1)
where R
μ
μ
is Ricci scalar, F =
1
4
F
μν
F
μν
is the Maxwell electromagnetic field Lagrangian density
and b is the Born–Infeld parameter. G is Newton’s coupling constant and is the cosmological
constant which relates to radius of the 4D-AdS space time L as =−
3
L
2
[41]. One can infer for
b →∞the last term of the above action leads to 4b
2
[1 −
1 +2F/b
2
] ∼−4F + O(b
−2
) and
so the total action reduces to the well known Einstein–Maxwell model. Author of the ref. [41]
obtained a spherically symmetric static charged black hole metric of the model (2.1)as follows.
ds
2
=−f(r)dt
2
+
dr
2
f(r)
+r
2
d
2
2
, (2.2)
where d
2
2
=(dθ
2
+sin
2
θdφ
2
) denotes to the unit 2-sphere metric S
2
, and f(r) =1 +g(r) for
which
g(r) =−
2M
r
+
r
2
L
2
+
2b
2
r
2
3
1 −
Q
2
b
2
r
4
+1
+
4Q
2
3r
2
2
F
1
1
4
,
1
2
,
5
4
;−
Q
2
b
2
r
4
. (2.3)
2
F
1
(r) given in the above metric potential is hypergeometric function, M and Q are constants
of integral. They have related to the black hole mass and the electric charge respectively. It
is simple to calculate asymptotically behavior of the metric potential f(r) at infinity r →∞as
f(r) ∼1 −
2M
r
+
Q
2
r
2
+
r
2
L
2
−
Q
4
20b
2
r
6
which leads to a Reissner–Nordström AdS black hole in limits
b →∞[41]. One can obtain nonzero component of the electromagnetic field of the system as
F
rt
=
b
1+b
2
r
4
/Q
2
. It is equivalent with the vector potential gauge field A
μ
(r) = (A
t
(r), 0, 0, 0)
as F
rt
(r) =−
dA
t
(r)
dr
. We can integrate it to obtain A
t
(r) =
Q
r
2
F
1
(
1
4
,
1
2
,
4
4
; −
Q
2
b
2
r
4
) − in which
is a constant of integral. It is really electrostatic potential difference between the horizon and
infinity of the charged Born–Infeld black hole. A
t
(r) is a gauge field and so we can set A
t
(r
h
) =0
on the black hole horizon r
h
obtained from f(r
h
) =0(see [38,41]for more details). By applying
the latter boundary condition one can infer
(r
h
) =
Q
r
h
2
F
1
1
4
,
1
2
,
5
4
;−
Q
2
b
2
r
4
h
. (2.4)
In the context of the AdS/CFT correspondence there may be exist some black holes with varied
topologies for instance planer solutions [43] where their topology at the boundary of a 4D-AdS
space–time are R
3
instead of R × S
2
. Such a black hole exist only due to the presence of a
negative cosmological constant. To obtain such a black hole we should usually use finite-volume
re-scaling of the solutions as done in [42]by introducing a dimensionless parameter λ and then
obtain its limits at λ →∞. The latter proposal is well kno
wn as “infinite volume limit” which we
will consider it to obtain topologically deformed form of the metric solution (2.3). It should be
reminded the Born–Infeld parameter b and the AdS radius L is topological invariant quantities
for the metric solution (2.3)but its other quantities are changed under the transformations λ as
follo
ws: r →λ
1/3
r, t →−λ
−1/3
t, M →λM, Q →λ
2/3
Q, L
2
d
2
2
→λ
−2/3
(d x ·d x) for which
f(r) → λ
2
3
λ
−
2
3
+ g(r)
. After substituting the latter transformations into the metric solution
(2.2) and taking its limits for λ →∞we obtain
ds
2
=−g(r)dt
2
+
dr
2
g(r)
+
r
2
L
2
d x ·d x, (2.5)