多势与规范场的电 Dilaton 黑洞：相变研究

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本文主要探讨了一个具有多（M）个Liouville势和N个规范场的Einstein-Maxwell-dilaton理论的精确解。在这个理论框架下，M和N被设定为任意整数，这使得研究者能够构建出一个广泛的引力模型，其中包括了Lifshitz黑洞和拓扑Dilaton黑洞两种特殊的引力系统。Lifshitz黑洞以其独特的空间时间尺度对称性而闻名，而拓扑Dilaton黑洞则涉及了额外的拓扑结构和Dilaton场的作用。该研究的核心内容是提出并分析了一个全新的黑洞解，它能够在两者之间进行连续变换，这在物理上意味着不同的引力环境可能通过这种理论得到统一的描述。这个解的发现揭示了黑洞在特定参数下的热力学性质，特别是当标准系综中的电荷或在grand canonical ensemble中的电势处于特定范围内时，可能会出现相变现象。相变在这里指的是系统从一种稳定状态转变为另一种状态的过程，可能是由于温度、压力或者外部参数的变化引起的。作者Yen-Kheng Lim来自厦门大学马来西亚分校的数学系，他的研究不仅深化了我们对多势场引力理论的理解，而且对于AdS-CFT对称性（Anti-de Sitter空间与Conformal Field Theory之间的对应关系）以及黑洞物理，特别是它们与量子场论的关联提供了新的见解。文章的发表日期为2020年3月，并且是开放存取的，表明其研究成果可供全球学者自由查阅和引用，且得益于SCOAP3基金的支持。总结来说，这篇论文在理论物理学领域具有重要意义，它不仅扩展了我们对复杂引力系统的认识，还对理解黑洞热力学和AdS/CFT对称性提供了一种新的探索工具，特别是在处理非平凡的势和场配置时，其潜在的实际应用和理论贡献不可忽视。

JHEP03(2020)134

We shall consider potentials V(ψ) where it is the sum of M exponential terms,

V(ψ) =

j=1

2Λ

, (2.2)

where b

and Λ

are constants. Therefore, the case of N > 0 and M = 0 corresponds to

eq. (1.4) under which the Lifshitz black hole was obtained [15], where at least N = 3 is

needed to have a charged Lifshitz black hole with a spherical horizon.

The case N = 1 and

M = 2 corresponds to eq. (1.5) where the dilaton black hole was found [24]. Therefore, we

should consider at least M = N = 3 to be the simplest non-trivial solution to go beyond

the cases already contained in [15] and [24].

Extremising the action gives the Einstein-Maxwell-dilaton equations

µν

∇

ψ∇

ψ +

D − 2

µν

i=1



−2a

)

µλ

)

−

4(D − 2)

−2a

µν



(2.3)

∇



−2a

µν



= 0, (2.4)

∇

ψ = 2

dψ

−

i=1

−2a

. (2.5)

As brieﬂy alluded to in the Introduction, the general form of the Lifshitz and dilaton black

hole share may similar features. In particular, their metrics in both cases take the form

= −f(r)dt

+ h(r)dr

+ r

˜γ

, (2.6)

where f(r) and h(r) are functions depending only on r. The metric ˜γ

is a (D −2)-

dimensional space of constant unit curvature k = 0, ±1. Our present goal can then be

focused into seeking a solution which also takes the form (2.6) and ﬁnd an appropriate

f(r) and h(r) that extremises the action (2.1). We shall also take the ansatz where the

gauge potentials only have ‘electric’ radial components,

= χ

(r) dt, (2.7)

where χ

(r) are scalar functions that depend only on r. Similarly we shall also take the

ansatz where ψ = ψ(r) is a scalar function that depends only on r.

In the following paragraphs we shall show the details of how the solution is derived.

The reader uninterested in such details may skip ahead to the ﬁnal solution shown in

eqs. (2.35), (2.37), and (2.36).

Note the N = 2 case in section 2.1 of [15] gives a black hole with a planar horizon.

– 5 –

JHEP03(2020)134

Substitution of the ansatz into eq. (2.3) gives

−

√



√



−

(D − 2)f

2rfh

j=1

2Λ

2bψ

D − 2

−

i=1

D − 3

2(D − 2)

−2a

, (2.8)

−

√



√



(D − 2)h

2rh

j=1

2Λ

D − 2

−

i=1

D − 3

2(D − 2)

−2a

(2.9)

D − 3



k −



2rh



−



j=1

2Λ

D − 2

i=1

2(D − 2)

−2a

, (2.10)

where primes denote derivatives with respect to r. Furthermore, putting the ansatz into

the Maxwell equation (2.4) and dilaton equation (2.5) give



D−2

√

−2a



= 0, i = 1, . . . , N, (2.11)

√

fhr

D−2

j=1

i=1

−2a

. (2.12)

Taking the diﬀerence between (2.8) and (2.9) leads to

D − 2





= ψ

. (2.13)

To make progress towards obtaining a solution, we shall follow [16, 24, 25, 40] and consider

a speciﬁc ansatz for h as

h =

, (2.14)

for some constant n. While this does not rule out the possibility that there exist solutions

for other forms of h, the ansatz taken here will nevertheless lead to a solution for f that

interpolates between the Lifshitz and dilaton black hole cases. With this ansatz for (2.14),

eq. (2.13) becomes ψ

= 2(D − 2)n/r

and can be solved to give

ψ = −δ ln r + ψ

, δ =

2(D − 2)n, (2.15)

where ψ

is an integration constant. From the above expression for δ, it is clear that

we require n ≥ 0. It turns out that in the context of the hyperscaling-violating solution

in [16], this condition coincides with satisfying the Null Energy Condition, and also for the

Following [15, 25, 40], we shall choose the negative sign upon taking the square root. As we will see

below, solving the equations of motion for this choice of sign will lead to a positive sign for the coupling

parameter a

when the solution to reduced to that of [40] and [15]. Furthermore this will be consistent if

the terms e

−2a

of the action is a result of a dimensional compactiﬁcation [20, 21, 23], in which case a

is a positive number that depends on the number of reduced dimensions.

– 6 –

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