非线性AdS黑洞的热力学与相变研究

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"这篇论文探讨了在非线性电动力学和爱因斯坦质量引力理论框架下，4维（4D）带电阿德萨（AdS）黑洞的解决方案及其热力学与相结构。作者Cao H. Nama在Yonsei大学物理系进行了这项研究。文章发表于Eur. Phys. J. C (2018) 78:1016，并提供了DOI链接：10.1140/epjc/s10052-018-6498-1。" 在非线性电动力学中，电荷的行为不再遵循线性规律，而是呈现出复杂的非线性特征。这种理论在处理高电荷密度或强电磁场时更为准确。在爱因斯坦质量引力的背景下，质量与引力场之间存在直接关联，导致引力效应会受到物质质量分布的影响。论文中，研究人员成功地构建了一个4D非线性带电AdS黑洞的解。阿德萨空间是具有负宇宙常数的时空中的一种特殊类型，它在量子场论和弦理论中具有重要意义。黑洞的层位结构分析揭示了其内部结构的特性。接着，作者对这个黑洞解决方案的热力学性质进行了深入研究。他们计算了关键的热力学量，如温度、熵和恒压下的热容，以及吉布斯自由能。这些量是理解系统稳定性和相变的关键。黑洞的相变行为是热力学中的重要现象，它与物质的固态、液态和气态之间的转变类似。特别地，论文指出，黑洞可以经历一阶和二阶相变，这些相变不仅依赖于黑洞自身的特性，如有效水平曲率，还取决于耦合参数$$ c_1 $$的符号，以及非线性电动力学的特征参数。这些因素共同决定了相变的可能性和方式。此外，压力也是影响相变的重要因素。最后，作者推导出了状态方程，并针对有效水平曲率的情况研究了$$ P-V $$（压力-体积）临界行为。在热力学中，$$ P-V $$图常常用于识别相变，特别是在批判现象的研究中。临界点的存在表明系统可能发生类似液体-气体相变的现象。这篇论文为理解非线性电动力学和质量引力理论如何影响AdS黑洞的热力学性质和相结构提供了新的见解。通过这些研究，我们可以更深入地认识黑洞的动态行为，这在理论物理、特别是量子引力和高能物理领域具有深远的意义。

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.

We would like to consider the spherically-symmetric and

static metric of the spacetime, given by the following ansatz

=−f (r)dt

+ f (r )

−1

d

, f (r ) = 1−

2m(r )

(6)

Following Ref . [97], we take the reference metric as

μν

= diag(0, 0, c

, c

sin

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

= U

= 0. (8)

As a result, the equations of motion derived from the action

(2)are

−



+ m





= 2



∂L(F)

∂ F

μρ

νρ

− δ

L(F)



, (9)

∇



∂L(F)

∂ F

νμ



= 0. (10)

We would like to thank Reviewer for indicating these points.

∇

∗ 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

μν



− δ



B(r,θ). (12)

It is notable that the magnetic charge Q is deﬁned by

Q =

4π



∞

F, (13)

with F =

μν

∧dx

and S

∞

to be a two-sphere at the

inﬁnity. From Eqs. (10), (11) and (13), it can derive

B(r,θ) = Q sin θ, −→ F =

. (14)

Using this result, the (t, t ) component of Eq. (9) reads

dm(r )



r + c



−

. (15)

Integrating this equation with the integral constant M



m(r) +



+ c



r→∞

= M, (16)

then substituting m(r) into f (r), we ﬁnally get

f (r ) = 1 −

−

+ m



+ c



. (17)

Here, 1+m

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 +

+ m



+ c



. (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 −

+ m



+ c



, (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

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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