Born-Infeld AdS黑洞的全息纠缠熵与热化研究
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"这篇论文是关于全息纠缠熵在小子区域和Born-Infeld AdS黑洞热化过程中的研究。作者使用Born-Infeld Anti de Sitter带电黑洞的度量，结合Ryu和Takanayagi的提议，探讨了全息纠缠熵的计算方法。在研究中，他们假设黑洞的质量和电荷随时间呈阶跃函数变化。工作主要集中在不会让空壳穿透黑洞视界的小子区域。通过微扰方法处理时间相关的HEE计算，对于边界上的纠缠区域，选取条形和球形两种形状函数。文章指出存在一个饱和时间，此时HEE达到最大值，空壳经过转折点。该研究与Camelio等人和Zeng等人的工作相呼应，但他们的研究方法不同，采用的是等时的两点相关函数和威尔逊循环，而非纠缠熵，且使用了数值方法。"
本文详细分析了全息纠缠熵（HEE）在黑洞物理中的角色，特别是在Born-Infeld Anti de Sitter (BI-AdS)黑洞背景下。HEE是一种衡量量子系统纠缠度的工具，它在量子引力理论，特别是弦理论和AdS/CFT对应中具有重要意义。Born-Infeld理论是一种非线性电磁理论，可以用来描述强磁场下的物理现象。BI-AdS黑洞则是这个理论在带有负宇宙常数的反德西特空间中的应用。
作者选择阶跃函数来模拟黑洞质量与电荷的时间演化，这是为了模拟可能的瞬时过程，如黑洞吸收物质或辐射。在小子区域的限制下，可以避免空壳穿越黑洞视界导致的复杂问题。微扰方法则允许他们处理这些小区域内的HEE，其中转折点远小于黑洞的局部平衡点。
论文中，纠缠区域的边界形状被设定为条形和球形，这两种形状代表了不同的物理情况。条形可能对应于一维量子链，而球形则更接近于高维系统的纠缠特性。通过这两种形状，可以研究不同维度和拓扑下的HEE行为。
此外，作者发现HEE有一个饱和时间，这表明系统会经历一个热化过程，直到HEE达到最大值，这可能反映了黑洞系统的热力学性质。这个结果与之前的研究相符，但使用的方法和可观察量不同，前人的工作更多依赖于数值计算，而这里则聚焦于HEE的解析分析。
该研究深入探讨了全息纠缠熵在特定黑洞模型中的动态行为，提供了对黑洞热化过程的新见解，进一步深化了我们对量子引力和黑洞信息理论的理解。
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)
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