全息复杂性研究：Vaidya时空与薄壳黑洞

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"这篇论文是关于Vaidya时空中的全息复杂性研究，主要探讨了两种不同的复杂性度量——复杂性=体积(CV)和复杂性=行动(CA)提案，以及它们如何应用于时间依赖性的Vaidya时空。作者们关注了一个特殊的物理场景：零流体薄壳塌缩成空的AdS空间并形成一个单侧黑洞的过程。在CA方法中，他们提出了无效流体的作用原理，以此来描述Vaidya几何。同时，他们强调了在Wheeler-DeWitt片的边界上添加特定对立项的重要性，以准确反映引力作用下的边界状态复杂性。研究结果表明，对于CV和CA提案（在考虑特定边界条件的情况下），单面黑洞的全息复杂性增长速率在后期与永恒黑洞的情况一致。" 这篇学术文章发表在JHEP06(2018)046期刊上，由Springer出版，于2018年5月7日提交，5月26日接受，6月8日发布。作者包括Shira Chapman、Hugo Marrochio和Robert C. Myers，他们均来自加拿大滑铁卢的Perimeter研究所和滑铁卢大学物理与天文学系。文章的核心概念是全息复杂性，这是量子引力理论中的一个重要研究领域，它试图将黑洞内部的复杂性与边界上的理论复杂性相联系。Vaidya时空是一个动态的AdS时空模型，常用于研究黑洞形成和演化的过程。在这个模型中，复杂性=体积(CV)提案认为黑洞的复杂性与其边界上的最大体积区域成正比，而复杂性=行动(CA)提案则认为复杂性与该区域的欧几里得行动（即无边界条件下的引力作用）相关。在CA方法的应用中，作者引入了无效流体的概念，这种流体可以产生Vaidya几何，而且分析了无效流体薄壳对引力作用的贡献。此外，Wheeler-DeWitt片是量子宇宙学中的一个关键概念，它代表了量子引力背景下的一类时空区域。在该区域内，作者发现必须在边界上加入特定的对立项，才能确保引力作用能正确地反映系统的复杂性。最终，他们的研究揭示了在某些条件下，单面黑洞的全息复杂性增长规律与永恒黑洞相似。这一发现加深了我们对黑洞复杂性和全息原理的理解，可能为理解量子引力提供新的视角。

JHEP06(2018)046

Figure 1. The null shell has a ﬁnite thickness 2ε around the null ray v = v

. The portion enclosed

by the WDW patch is shaded in orange. The contribution of the two joints indicated by red dots

exactly cancels the surface term for the portion of the null boundary connecting the joints, where

we have a time dependent κ(v).

diﬀerent boundary time slices in a given bulk geometry. Hence our evaluation of the WDW

action in the following and in the next section will use both of these choices.

As discussed above, we want to consider an AdS-Vaidya spacetime (2.1) where the

shell of null ﬂuid is narrow but still has a ﬁnite width. In particular, the shell will extend

from v

min

= v

−ε to v

max

= v

+ε, as shown in ﬁgure 1. Further, the shell will separate two

stationary

spacetime regions characterized by the mass parameter ω

d−2

inside the shell

and by ω

d−2

outside the shell. The details of the proﬁle f

(v) in the metric will not be

important but we assume that it is continuous (and smoothly increasing). Of course, from

integrating eq. (2.10) across the shell, the proﬁle must also satisfy

+ ε) − f

− ε) =

shell

dv f

(v) = ω

d−2

− ω

d−2

. (2.17)

With these choices, in the limit ε → 0, the proﬁle reduces to that given in eq. (2.3).

Now we will evaluate the contribution of the null shell to the WDW action, but we will

be particularly interested in the limit where the shell becomes inﬁnitely thin, i.e., ε → 0.

Let us examine the various terms in eq. (2.16). First, of course, the ﬂuid action I

ﬂuid

vanishes

on-shell, as we showed in the previous section. The bulk term in I

grav

is (approximately)

That is, ∂

satisﬁes the usual Killing equations in either region.

Our calculations here are general enough to accommodate both black hole formation which we examine

in section 3, and null shocks in an eternal black hole background which we will study in [60].

– 8 –

JHEP06(2018)046

proportional to the volume of the shell and so vanishes in the limit that ε → 0. Similarly

evaluating the GHY term at the r = 0 singularity (following the prescription in [34]) yields

a result which vanishes as ε → 0. The Hayward joint terms are not relevant for this

particular region and hence we turn to the null surface and null joint terms.

First, we must introduce (outward-directed) normals for the upper and lower null

boundaries,

v = v

+ ε : k

= β dv ,

v = v

− ε : k

s–

= −β dv . (2.18)

With this choice, these null normals are aﬃnely parameterized and therefore the null surface

term vanishes, i.e., κ = 0, for these two boundaries. We might add that the null joint terms

vanish where these boundaries meet the singularity at r = 0 because there the transverse

volume vanishes for these two joints.

The ﬁnal boundary for the shell region is a portion of the past null boundary of the

WDW patch. From the metric (2.1), we can see that the normal to this boundary can be

written as

∂

= H(r, v)



F (r, v)

∂

+ ∂



, (2.19)

where F (r, v) is the usual metric function — see eq. (2.1). Note that with eq. (2.19), we

are describing the null normal for the entire past null boundary B

past

. Hence in the regions

beyond the shell, the metric function F simpliﬁes to

F (r, v) = f

(r) =

+ k −

d−2

, (2.20)

with i = 1 and 2 denoting the region inside (v < v

min

) and outside (v > v

max

) of the

shell, respectively. Of course, across the shell, F depends on both r and v as shown in

eq. (2.1). Further, we have introduced an overall factor H(r, v) in eq. (2.19) to allow for

the possibility that the normalization of the null normal changes when the past boundary

crosses the shell of null ﬂuid. For v > v

max

, we will set H to be a ﬁxed constant, i.e.,

H(r, v) = α to match the asymptotic boundary condition k ·

t = −α (see discussion above).

As we will see below, this simple choice also ensures that κ = 0 on this outer portion of

the past boundary. Similarly for v < v

min

, we set H(r, v) = ˜α which is again a positive

constant in order for the null generators to be aﬃnely parametrized on the inner portion of

the null boundary. However, we have taken the liberty to choose an independent constant

˜α since this portion of the boundary never reaches asymptotic inﬁnity. Expressing this

normal (2.19) as a form, we have

= H(r, v)



−dv +

F (r, v)



. (2.21)

We have chosen the same normalization constant β for the two normals to simplify the ﬁnal result,

i.e., this choice ensures that the null joint terms exactly cancel with the surface term below. Of course,

another choice would yield the same result for the total action of the WDW patch after summing with the

relevant boundaries and joints for the portions of the WDW patch above/below the null shell, since these

are all inner boundaries of the WDW patch that we have introduced. However, we note that if the two

normals in eq. (2.18) were not normalized with the same constant, the null shell would make a nonvanishing

contribution to the total action.

– 9 –

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