非线性电动力学中的AdS黑洞：旋转与非旋转解

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"这篇论文探讨了在非线性电动力学框架下的$$ f(\mathcal{T}) $$ 重力理论中的旋转和非旋转AdS（Anti-de Sitter）黑洞解。作者Salvatore Capozziello和Gamal G.L. Nashed在Eur. Phys. J. C (2019) 79:911中发表的研究成果，通过使用二次远平等效重力模型，他们推导出了新的带电d维黑洞解，其中$$ f(\mathcal{T}) = a_0 + a_1 \mathcal{T} + a_2 \mathcal{T}^2 $$，$$ \mathcal{T} $$是非线性电动力学的扭曲标量。" 这篇研究详细阐述了如何构建电磁函数，并确定了在电磁场存在下，与黑洞解相关的未知函数的形式。作者指出，这些黑洞解表现出AdS空间的特性，并且包含除了单极和四极项之外的更高级项，这些额外项来源于非线性电动力学效应。他们进一步分析了电磁麦克斯韦场，证明所得到的d维黑洞解与之前的工作（Awad等人在J High Energy Phys 13:1706.01773, 2017）保持一致。研究中强调的一个关键点是，与广义相对论相比，这些黑洞解的中心奇异性较弱。此外，作者还对解决方案的热力学性质进行了深入探讨，如计算了熵、霍金温度和热容量等物理量。一个重要的发现是，熵并不与黑洞的表面积成正比，这暗示了可能需要对四极项施加限制以确保正熵，否则熵可能会取负值。论文结论部分，作者讨论了这些结果对黑洞物理学的意义以及它们可能带来的新见解，特别是在非线性电动力学与AdS黑洞的相互作用背景下。这个研究为理解重力与非线性电动力学耦合的复杂系统提供了新的洞察，并且可能对黑洞信息悖论和其他相关领域的研究产生影响。该工作为理论物理学领域，特别是引力理论与量子场论的交叉部分，贡献了重要的新知识，它深化了我们对AdS空间中非线性电动力学黑洞的理解，并可能对未来的黑洞研究和宇宙学模型产生重要影响。

Eur. Phys. J. C (2019) 79 :911 Page 3 of 12 911

Using Eq. (4), the torsion and contorsion tensors are

μν

:= 

νμ

− 

μν

:= −

(

μν

− T

νμ

− T

μν

)

. (5)

From Eq. (5), one can deﬁne the superpotential tensor

λμ



μν

+ δ

βν

− δ

βμ



. (6)

From Eqs. (5) and (6) the torsion scalar is provided as

T := T

μν

. (7)

The Lagrangian of TEGR theory is constructed from the tor-

sion scalar given by Eq. (7).

Let us now consider f (T ) gravity minimally coupled with

non-linear electrodynamics. Thence, the action of this theory

is given by

L :=

2κ



|h| f (T ) d

x +



|h|L(F) d

x, (8)

with |h|=

√

−g = det





being the determinant of

the metric and κ a dimensional constant with the form

κ = 2(d − 3)

d−1

, with G

being the Newtonian grav-

itational constant in d-dimensions and 

d−1

a (d − 1)-

dimensional unitary volume whit the form



d−1

2π

(d−1)/2

((d − 1)/2)

, (9)

where  is the -function (when d = 4, it is 2(d −

3)

= 8π). The electromagnetic Lagrangian L(F) is

gauge-invariant and depends on the invariant F deﬁned as

F =

αβ

[69]. The antisymmetric Faraday tensor is

deﬁned as

αβ

= E

α,β

− E

β,α

, (10)

where E

is its gauge potential 1-form. In the Maxwell theory,

the Lagrangian L(F ) is L(F) = 4F . Here, we consider

a more general choice of the electromagnetic Lagrangian.

From Action (8), the non-linear electrodynamics is described

by nonlinear terms in F

αβ

and its invariants. However, we can

provide a dual representation in terms of an auxiliary ﬁeld

αβ

. This method is proved to be highly beneﬁt to derive

exact solutions in GR, speciﬁcally for the electric case [70,

71]. The dual form can be obtained adopting the Legendre

transformation below:

ℵ=2FL

− L, where L

∂L

∂F

, (11)

where ℵ is an arbitrary function depending on the invariant

P, deﬁned as P =

αβ

.FromEq.(11), non-linear

electrodynamics can be recast in terms of P according to the

formulas

μν

= L

μν

, F

μν

=ℵ

μν

, L = 2Pℵ

−ℵ,

(12)

where the standard Maxwell theory is obtained for L

= 1.

As it is clear from the above equations, ℵ is a function of P ,

where [70,71]

ℵ

∂ℵ

∂P

The variation of Lagrangian (8) with respect to the vielbeins

leads to

= S

ρν

∂

T f



−1

∂



ρν



− T

λμ

νλ



−

κT

nl em

≡ 0, (13)

and the Maxwell equations for non-linear electrodynamics

become [70]

∂



√

−gP

μν



= 0. (14)

The stress-energy tensor of non-linear electrodynamic is

nl em

:= 2(ℵ

μα

να

− δ

[2Pℵ

−ℵ]). (15)

It is worth saying that Eq. (15) has a non-vanishing trace

unlike the stress-energy tensor coincides with the Maxwell

one. It is worth noticing that the electric ﬁeld of linear elec-

trodynamics is obtained as

E = F

=ℵ

. (16)

In this context, black hole solutions can be found.

3 Anti-de-Sitter black hole solutions in non-linear

electrodynamics

Let us search now for charged AdS black hole solutions

in non-linear electrodynamics assuming, in general, d-

dimensions in the framework of f (T ) gravity. Using the

following vierbein diagonal ansatz in d-dimensions (t, r , η

, ···, η

, ξ

···ξ

), with l = 1, 2 ··· d − n − 2, in

which 0 ≤ r < ∞, −∞ < t < ∞,0 ≤ η

< 2π and

−∞ <ξ

< ∞, we assume the vielbein [49,62]:









A(r),

√

A(r)g(r )

, r, r, r ···



, (17)

123

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非线性电动力学中的AdS黑洞：旋转与非旋转解

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