三维Bondi-Metzner-Sachs对称性：解析黑洞熵的钥匙

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本文讨论的主题是"近视邦迪-梅茨纳-萨克斯对称性（Near-horizon Bondi-Metzner-Sachs symmetry）在黑洞熵计算中的作用，以及它与维度缩减的关系"。作者S. Carlip在2018年的《物理评论快报》中提出，对于一个通用黑洞，其保持视界不变的微扰变换（horizon-preserving diffeomorphisms）可以提升到三维的邦迪-梅茨纳-萨克斯对称性（Bondi-Metzner-Sachs symmetry），这一对称性对于确定黑洞的贝肯斯坦-霍金熵（Bekenstein-Hawking entropy）具有关键作用。在本文中，作者详尽阐述了这一观点的细节和扩展。首先，他探讨了如何放宽黑洞边界条件，这在理论探讨中至关重要，因为它允许更广泛的物理情景被纳入分析。其次，文章深入剖析了维度缩减（dimensional reduction）的概念，即在黑洞附近，由于极端的引力环境，时空结构可能展现出不同于全局的特性，这对理解黑洞熵的微观起源提供了新的视角。所谓的"近水平对称"（near-horizon symmetry）是指在黑洞事件视界附近的对称性，这里的对称性不是通常意义上的全空间对称，而是受限于边界条件下的局部对称。这种对称性的存在有助于简化黑洞附近的物理问题，并可能揭示出黑洞熵背后的统计学原理。文章最后指出，黑洞熵的"问题的普遍性"（problem of universality），即所有黑洞不论大小或质量，其熵遵循相同的公式，这是一个尚未完全解决的重要课题。通过三维邦迪-梅茨纳-萨克斯对称性的研究，作者试图提供一种理论框架，以期解答这个问题，从而增进我们对量子引力和黑洞热力学的理解。总结来说，本文的核心内容围绕着黑洞的对称性提升、边界条件的处理、维度缩减效应以及这些因素如何影响黑洞熵的计算，特别是在寻找黑洞熵背后微观机制方面的重要作用。通过这些概念的深入分析，作者为黑洞物理学提供了一种新的研究途径。

The reason I have given such a complicated description

of a relatively simple procedure is that most of the steps

generalize quite broadly. Let Δ be a nonexpanding horizon

[41,42], that is, a null (D − 1)-dimensional manifold with

null normal l such that:

(1) Δ has the topology ð0; 1Þ ×

Δ, where

Δ is usually

taken to be compact (a sphere for a black hole, a

torus for a black ring, etc.).

(2) The expansion θ

ðlÞ

of the null normal vanishes. If

the stress-energy tensor satisfies the null energy

condition for l, that is, T

≥ 0, then this

condition further implies [42] that l has vanishing

shear and that

¼ 0; ð3:1Þ

where L is the Lie derivative and q

is the (degenerate)

induced metric on Δ.

A n onexpanding horizon generali zes the notion of a

Killing horizon, dropping the requirement of a Killing

vector but retaining a time translation symmetry on the

horizon itself. As in the Schwarzschild case, the integral

curves of l

on Δ are the null geodesic generators of Δ.

In the absence of a Killing vector, there is no preferred

time coordinate, but it may be shown that a generic non-

expanding horizon has a pr eferred foliation, a set of

“good cuts,” that generalize the constant time slices of

the Schwarzschild metric [43]. Hence, we c an again

construct an r–v plane and a Gaussian null coordinate

system near Δ.

The main change from the Schwarzschild case is that the

cross section

Δ of a nonexpanding horizon need not have

any symmetries, so conventional Kaluza-Klein methods no

longer apply. As Yoon has shown, though, there is a

generalized Kaluza-Klein reduction even in the absence

of symmetries [44,45]. Start by writing the metric in the

general form

¼ g

þ ϕ

μν

ðdy

þ A

Þðdy

þ A

Þ;

ð3:2Þ

where lowercase roman indices (a,b ,…) run fr om 0 t o 1,

lowercase greek indices (μ; ν; …)runfrom2toD − 1,

and uppercase roman indices (A; B; …) run from 0 to

D − 1.The“x” coord inates label our preferr ed two -

dimensional manifold, while the “y” coordinates are the

transverse directions. Define the Kaluza-Klein-like

derivatives

∂

¼ ∂

− A

∂

;

μν

∂

μν

− ð∂

Þϕ

ρν

− ð∂

Þϕ

ρμ

; ð3:3Þ

connections

∂

−

∂

Þ;

μν

ρσ

ð∂

σν

þ ∂

σμ

− ∂

μν

Þ; ð3:4Þ

and curvatures

∂

−

∂

−

;

μν

¼ ∂

μν

− ∂

νρ

μν

ρσ

−

μσ

νρ

;

∂

−

∂

;

R ¼ g

; R ¼ ϕ

μν

: ð3:5Þ

A straightforward calculation of the Einstein-Hilbert

action then gives

I ¼

D−2

ð3:6Þ

with

16πG

ﬃﬃﬃﬃﬃﬃ

−g

ﬃﬃﬃﬃ



R þ

μν

ρσ

ðD

μρ

νσ

− D

μν

ρσ

μν

ð∂

∂

− ∂

∂

ÞþR



;

ð3:7Þ

where g and ϕ are the determinants of g

and ϕ

μν

The first two lines in (3.7) look like an ordinary Kaluza-

Klein reduction, and can in fact be viewed as the action of a

Kaluza-Klein theory whose gauge group is the group of

diffeomorphisms of the transverse manifold [44]. A two-

dimensional interpretation of the third line is less obvious.

We can cure this, though, with a partial gauge fixing. First,

we can always choose local coordinates in which



−2h 1



: ð3:8Þ

Yoon calls this “Polyakov gauge,” after a similar choice in

two-dimensional field theories [46], while from the D-

dimensional point of view, it is essentially Bondi gauge

[47] or Gaussian null coordinates [39]. It is easy to see that

with this choice, even if h depends on the y

μν

ð∂

∂

− ∂

∂

Þ¼0: ð3:9Þ

For our purposes, Eq. (3.8) is too restrictive a gauge

choice—it hides a piece of the symmetry we are

trying to understand. But if we now allow an arbitrary

S. CARLIP PHYS. REV. D 101, 046002 (2020)

046002-4

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