AdS中振荡壳与球的动态分析及其多态方程影响

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本文主要探讨了AdS（Anti-de Sitter空间）中的特殊物理现象，即“摆动壳”和“摆动球”。近期的研究发现，在AdS环境中，存在一种薄的、类似于时间的壳体会经历周期性的振荡运动。这种振荡性壳模型对于理解量子引力和宇宙学中的边界动态具有重要意义。作者们在测地近似下计算了探测场的两点函数，这是一种在量子场论中用于研究粒子间相互作用的重要工具。他们发现，两点函数在摆动壳运动后呈现出明显的振荡模式，这揭示了壳的振动如何影响周围的量子场。这项工作扩展了我们对AdS背景下的非静态时空结构的理解。特别值得注意的是，当壳上的完美流体遵循多态状态方程时，也能够产生类似的振荡动力学。这表明了不同物质性质对时空波动的影响，可能为探索新的物理现象提供线索。此外，文章还关注了球形配置在AdS中的振荡行为。这些球状结构同样展现出周期性的运动，而且作者们讨论了如何与外部静态或准静态的解进行平滑匹配，确保了解的全局一致性。这有助于建立更全面的理论框架，尤其是在黑洞物理和引力波研究中。最后，研究者们证明了这些振荡配置满足弱能量条件，这是广义相对论中的一个基本要求，保证了物理系统的稳定性。这一发现进一步加强了这些振荡解作为有效物理模型的合理性。这篇论文不仅深化了我们对AdS中非稳定几何结构的认识，还展示了如何通过特定的物质状态和几何形态来实现振荡现象，这对于理解宇宙早期阶段的快速扩张和量子引力效应具有重要的理论价值。

JHEP07(2017)026

Let the position of the shell be speciﬁed by

r = r

(τ) , t

= t

±,s

(τ). (2.2)

Then we get

∂

= u

∂

±,s

∂

+ ˙r

∂

, (2.3)

∂

= δ

∂

, i = 1, . . . , (d − 1). (2.4)

Here, the overhead dot denotes derivative w.r.t. τ, and u

is the four velocity of the shell.

The four velocity is canonically normalised, u

= −1, which yields,

±,s

) + ˙r

)

. (2.5)

Since the derivatives

do not match at the location of the shell, t

is not continuously

related to t

−

. This will be carefully taken into account when we discuss spacelike geodesics

crossing the shell in the next subsection.

The induced metric on the shell is:

shell

= h

= −dτ

+ r

dΩ

d−1

. (2.6)

The unit normalised vector normal to the shell in ± coordinates is

µ,±



− ˙r

±,s

, 0, 0



=⇒ n



−1

) ˙r

, f

)

±,s

, 0, 0



. (2.7)

It satisﬁes u

µ,±

= 0 and n

µ,±

= 1. An overall (positive) sign choice has been made in

writing the above normal vector, so it points from M

−

to M

Einstein equations become a set of matching conditions on U

∩U

−

. These are known as

the Israel junction conditions [21]. For writing down these conditions, we need to evaluate

the extrinsic curvature and also assign a stress-tensor to the thin-shell matter ﬁeld. The

extrinsic curvature, deﬁned as K

= e

∇

, has the following non-zero components,

ττ,±

= −

˙r

, K

,±

= β

, (2.8)

,±

= (K

,±

)

, i 6= 1. (2.9)

Equivalently,

τ,±

˙r

, K

i,±

. (2.10)

For simplicity, we can take the stress-tensor of the thin-shell to be of the perfect ﬂuid form,

= diag(−σ, p, p, . . .

| {z }

(d−1) terms

), (2.11)

where σ and p are the energy density and the pressure of the corresponding matter on the

shell, related via a suitable equation of state.

– 4 –

JHEP07(2017)026

The two Israel junction conditions are (i) continuity of metric across the shell, and (ii)

jump in the extrinsic curvature is related to the stress-tensor of the thin-shell,

] − h

[K] = −κS

, or [K

] = −κ



−

d − 1



, (2.12)

where κ = 8πG

, K ≡ h

and S ≡ h

are the traces of the corresponding tensors.

The bracket, denoted by [] represents the jump from M

−

to M

[O] ≡ O

− O

−

(2.13)

for some ﬁeld O. This deﬁnition is tied to our convention of choosing the direction of the

normal vector in (2.7).

Together with (2.10) and (2.11), the junction conditions (2.12) become

[β]

= −

κσ

d − 1

, (2.14)

[

β]

˙r

= κ



p + σ

d − 2

d − 1



. (2.15)

Since, f

≤ f

−

, we have β

≤ β

−

, and by virtue of (2.14), we conclude σ ≥ 0. The

inequality here is saturated for the trivial junction where the extrinsic curvature has no

jump, and the shell does not exist.

To make further progress, one needs to input an equation of state. A suﬃciently

general choice is the polytropic equation of state: p =

d−1

, where γ is the polytropic

exponent. The overall constant α ﬁxes the normalization of e.g. the trace of the shell energy-

momentum tensor. With a polytropic equation of state, equation (2.15) takes the form,

[

β]

˙r

= −

[β]



ασ

γ−1

+ d − 2



, (2.16)

which can be integrated using (2.14) to yield,

− [β] =

−

(d − 1)

γ−2

γ−1

α r

1−γ

+ Mr

(d−2)(γ−1)

1−γ

, γ ∈ Z \ {1}, (2.17)

where M is a constant of motion.

It is also possible to obtain analytical solutions for equation (2.16) with non-integer

values

of γ, however, those seem valid case-by-case and we were not able to obtain one

compact expression for all possible values of γ. The special case of γ = 1 can be worked

out separately, yielding,

[β] = −Mr

2−d−α

, (2.18)

where M is an integration constant.

In this case we have a linear equation of state

p =

d−1

σ. There are two cases of special interest, α = 0 and α = 1. α = 0 corresponds to

This is certainly of physical importance, see e.g., [22].

For σ to be positive, the constant M in equation (2.18) needs to be positive, cf. (2.14). For the

polytropic equation of state, the relation between the integration constant M and σ is not direct. Since a

physical interpretation of M is not transparent, one can consider both positive and negative values of M

for the polytropic equation of state. In this paper we only consider M > 0.

– 5 –

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AdS中振荡壳与球的动态分析及其多态方程影响

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