D维Bardeen-AdS黑洞：爱因斯坦-高斯-引擎盖理论新解
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"这篇研究论文探讨了D维Bardeen-AdS黑洞在Einstein-Gauss-Bonnet（EGB）重力理论中的性质。作者Arun Kumar、Dharm Veer Singh和Sushant G. Ghosh展示了Bardeen-EGB-AdS黑洞解，这是一种在爱因斯坦-高斯-引擎盖引力作用下的D维Bardeen类似Anti-de Sitter（AdS）黑洞模型。这个黑洞模型有三个关键参数：质量（M）、高斯-引擎盖参数（α）以及磁荷（e）。"
在EGB重力框架下，Bardeen-EGB-AdS黑洞的一个显著特征是其磁荷参数e的存在。对于不同的α值，存在一个临界磁荷值e_E，这导致了一个极端的规则黑洞，它的事件视界退化。当磁荷e小于e_E时，黑洞是非极端的，拥有两个独立的视界。这个模型允许研究人员精确计算出与黑洞热力学密切相关的量，如温度（T）、比热（C）和熵（S）。
有趣的是，黑洞的热容量在特定的半径r_C处发散，这个半径对应于温度的最大值。这种现象暗示了在 Hawking-Page 跃迁过程中，黑洞可能会演化成一种热力学上稳定的残余状态。Hawking-Page跃迁是指黑洞从热力学不稳定状态转变为稳定状态的过程，通常与黑洞蒸发和宇宙背景的相变有关。
该研究的发表地点是《欧洲物理杂志C》（Eur.Phys.J.C），2019年，第79卷，第275号。文章链接为：https://doi.org/10.1140/epjc/s10052-019-6773-9，表明这是一项开放获取的研究。作者们来自印度的中央理论物理中心、多学科高级研究和研究学院以及南非夸祖鲁-纳塔尔大学的天体物理学和宇宙学研究单元。
这项工作深化了我们对D维黑洞热力学行为的理解，特别是它们如何在EGB重力和AdS背景中响应磁荷变化，以及这些变化如何影响黑洞的稳定性。通过这种方式，研究可能为理解更广泛的量子引力理论，如弦理论或环量子引力提供有价值的洞察。
275 Page 4 of 16 Eur. Phys. J. C (2019) 79 :275
solution, f (r) must be real valued. So, we must consider
the value of radial co-ordinate (r) such that the term under
square root in (16) must be non-negative. For the case, 1 −
4 ˜α/l
2
≥ 0, that will always be non-negative. But for 1 −
4 ˜α/l
2
< 0, there will be some values of r for which that term
can be negative, so, for those values of r our solution is no
more real valued. Hence, we can say that the Gauss–Bonnet
coupling constant ˜α must lie in the interval [0, l
2
/4]. Besides,
the causality of dual theory demands another constraint on
Gauss–Bonnet coupling constant [67,68]
−
(3D − 1)(D − 3)
4(D + 1)
2
l
2
≤˜α ≤
(D − 3)(D − 4)(D
2
− 3D + 8)
4(D
2
− 5D + 10)
2
l
2
(18)
Thesolution(14) with (16) is a general spherically symmetric
D-dimensional solution of EGB theory coupled to nonlinear
electrodynamics in an AdS spacetime thereby generalizing
the Bardeen solution. The special case in which charge e = 0
and = 0, one get the Boulware–Deser solution [43]. It see
that solution (14) with (16) gets for other field equation. For
definiteness, henceforth, we shall call solution (16) Bardeen–
EGB–AdS black holes. In the case of no charge e = 0, Eq.
(16) reduces to D-dimensional EGB–AdS black holes [59,
60,63,69–74] and in the limit, α → 0, the negative branch
of (16)toD- dimensional Bardeen–AdS black holes [39]
ds
2
=−
1 −
μ
r
2
(r
D−2
+ e
D−2
)
D−1
D−2
+
r
2
l
2
dt
2
+
1
1 −
μ
r
2
(r
D−2
+e
D−2
)
D−1
D−2
+
r
2
l
2
dr
2
+r
2
˜γ
ij
dx
i
dx
j
.
(19)
Further, the solution also goes over to Schwarzschild–
Tangherlini black hole [75] in the absence of charge. To study
the structure of solution, we take limit r → 0 to obtain
f (r) = 1 +
r
2
l
2
ef f
, (20)
where (1/l
2
ef f
) is effective AdS length, it reads
1
l
2
ef f
=
1
2 ˜α
1 −
1 +
4μ
˜α
e
D−1
−
4 ˜α
l
2
, (21)
which is describing a de Sitter solution for ˜α>0inthe
Bardeen–EGB–AdS black hole. When, we take r e,we
get
f (r) ≈ 1 +
r
2
2 ˜α
⎡
⎣
1 −
1 +
4 ˜αμ
r
D−1
+
−
D − 1
D − 2
4 ˜αμ
e
D−2
r
2D−3
+
−
4 ˜α
l
2
+ O
e
2D−4
r
3D−5
+
⎤
⎦
. (22)
From Eq. (22), one can easily notice that the charge density
for our Bardeen–EGB–AdS solution is falling by 1/r
2D−3
+
,
but when we look for the charged EGB black hole [76], it
falls by 1/r
2D−4
+
.
The regularity of the black hole solution (16) can be seen
by behaviour of the scalar invariants, which are given by
lim
r→0
R =
D(D − 1)
2 ˜α
−1 +
1 +
4μ
˜α
e
D−1
−
4 ˜α
l
2
1/2
,
lim
r→0
R =
D(D − 1)
2
2 ˜α
2
1 +
2μ
˜α
e
D−1
−
2 ˜α
l
2
−
1 +
4μ
˜α
e
D−1
−
4 ˜α
l
2
1/2
,
lim
r→0
K =
D(D − 1)
˜α
2
1 +
2μ
˜α
e
D−1
−
2 ˜α
l
2
−
1 +
4μ
˜α
e
D−1
−
4 ˜α
l
2
1/2
. (23)
Thus, the spacetime is regular everywhere as seen from the
behavior of the invariants if e = 0 and l = 0. The D-
dimensional Bardeen–EGB–AdS black hole solution is well
defined everywhere by its curvature invariants.
The weak energy condition states that T
ab
t
a
t
b
≥ 0for
all time like vectors t
a
, i.e., the local energy density cannot
be negative for any observer. The dominant energy condition
states that T
ab
t
a
t
b
≥ 0 and T
ab
t
b
must be space like, for
any time like vector t
a
. Hence, the energy conditions require
ρ ≥ 0 and ρ + P
i
≥ 0,
ρ +P
2
=ρ +P
3
=ρ +P
4
= β
(D − 2)μ
e
D−2
(r
D−3
+ e
D−3
)
2D−3
D−2
. (24)
Where β = (D − 1) and (D − 4) are, respectively, for even
and odd dimensions. It is worthwhile to note that ρ>P
3
and
P
1
=−ρ. Thus the Bardeen–EGB–AdS black holes obey the
weak energy condition.
Next, we proceed to discuss the horizon structure of our
Bardeen–EGB–AdS black holes. The horizons radius, if
exists, are zeros of g
rr
= f (r) = 0. The numerical analysis
of f (r) = 0 reveals that it is possible to find non-vanishing
value of α and e for which metric function f (r) is minimum,
i.e, f (r) = 0 admits two roots r
±
. The smaller and larger
roots, respectively, corresponds to the Cauchy and event hori-
zon of the black holes. We have shown that for a given value
of α and fixed μ
, there exists a critical charge parameter
e
E
, and critical horizon radius r
E
, such that f (r
E
) = 0 has
a double root, i.e, r
E
= r
±
. This case corresponds to the
123
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