4 H. Nadi et al. / Nuclear Physics B 949 (2019) 114822
geodesic equation for the minimal surface is obtained in this unperturbed spacetime. To get the
geodesic equation, the area functional in Eq. (5) needs to be varied:
δA =
d
d−1
ζδ
√
h. (7)
With some simple calculations we get to the Gauss equation (see appendix A):
∂
a
(
√
hh
ab
∂
b
x
μ
) +
√
hh
ab
μ
νρ
∂
a
x
ν
∂
b
x
ρ
= 0. (8)
This is, indeed, the geodesic equation for the initial d − 1 dimensional minimal surface. We
now turn to a small perturbation around AdS
d+1
. We want to derive the difference between
the minimal surface areas of the pure AdS
d+1
and the perturbation over it. Again, we need to
calculate the variation in the area functional (5). In this case, δg
(0)
μν
can be written as follows
δg
(0)
μν
=
∂g
(0)
μν
∂x
ρ
δx
ρ
+g
(1)
μν
, (9)
where, g
(1)
μν
is a perturbation over the metric g
(0)
μν
. It is clear that in this case the variation in the
area functional includes the geodesic equation and a new term in Eq. (10) that arises from the
perturbation in the metric. The variation in the area then becomes
δA =
d
d−1
ζ
1
2
√
hh
ab
[g
(1)
μν
∂
a
x
μ
∂
b
x
ν
]. (10)
Thus, the variation in the minimal surface area in the first order of the metric perturbation is
derived, which is proportional to the change in the HEE of the pure AdS
d+1
. This method is
equivalent to covariant method [6] (see appendix B to clarify this issue). Thus, one needs to
derive the induced metric, h
ab
, and the local basis, ∂
a
x
μ
, for the pure AdS
d+1
to compute the
modification of HEE in the presence of black holes in the bulk. Also, the metric perturbation
g
(1)
μν
is computed using the Fefferman-Graham expansion close to the boundary in the following
calculations for the d +1 dimensional planar AdS black holes and the rotating cylindrical black
holes in d + 1 dimensions.
3. Parametrized minimal surface and solution of equation of motion
Let us consider the Poincaré coordinate where the boundary of AdS
d+1
is located at z = 0;
we have
ds
2
=
1
z
2
(−dt
2
+dz
2
+
d−1
i=1
dx
2
i
), (11)
where, g
(0)
μν
=
1
z
2
(−1, 1, 1, ...). We consider that the subsystem at the boundary of AdS
d+1
is a
strip with a length of l in a time constant slice and also −l/2 <x
1
<l/2, −L/2 <x
i
<L/2for
i = 2, ..., d − 1. Considering the black holes as perturbations over AdS
d+1
, it will be necessary
to find ∂
a
x
μ
, the local basis of the pure AdS, for use in expression (14). For this purpose, the
(d −1) dimensional surface is parametrized with respect to parameters τ and σ
i
as [ζ
a
] =[τ, σ
i
],
while −∞ <τ <∞ and −L/2 <σ
i
<L/2. Therefore, the area functional in Eq. (5)is given
by:
A =
d
d−2
σdτ
√
h, (12)
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