黑洞熵揭示弱重力猜想：高维算子的正向贡献

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本文主要探讨了"利用黑洞熵证明弱引力猜想"这一主题，由Cliﬀord Cheung、Junyu Liu和Grant N. Remmen三位作者共同完成，发表于2018年10月的《Journal of High Energy Physics》(JHEP)第10期。他们研究的核心是探究在固定质量和电荷的条件下，高维算子如何对热力学稳定黑洞的熵产生正向影响。这个结果是基于量子场理论的树级动力学主导的修正，这意味着这些理论对于黑洞熵的计算在某种程度上是可靠的。更深入地，他们的研究表明，高维算子的正贡献实际上与黑洞的自由能相关的一系列不等式紧密相连。这些不等式要求对高维算子的系数设定新的正定边界条件。其中一个条件尤其有趣，它指出随着黑洞质量的增加，极端黑洞的质荷比将趋向于一个特定的上限，这被称为"统一极限"。这个发现暗示着大型末梢黑洞（质量较大的黑洞）由于稳定性问题会逐渐衰变为较小的末梢黑洞，从而自然满足了弱重力猜想。值得注意的是，这个理论不仅适用于四维时空，也扩展到了任意维度的时空以及包含多个尺度场的系统。此外，论文还指出其结论在多种物理框架下都是有效的，其中包括弦论结构，特别是当这些弦论结构在弦尺度以下保持稳定时。作者们通过严谨的数学证明，展示了黑洞熵与量子场论的深层次联系，以及它们如何在引力理论的强弱猜想中扮演关键角色。这一工作对于理解黑洞的微观性质，特别是在弦理论和量子引力的框架内，具有重要的理论价值和物理意义。

JHEP10(2018)004

charge exceeds its mass in Planck units [7], so

> 1.

(1.11)

The motivation for the WGC is to forbid the existence of an inﬁnite number of stable

states not protected by symmetry. The main result of this paper is that this bound is

automatically satisﬁed by small black holes. Though mysterious at ﬁrst glance, the con-

nection between entropy and extremality in eq. (1.10) actually follows immediately from

the near-horizon properties of the metric for a charged black hole. As we will show, this

connection enables us to straightforwardly extend all of our arguments to the multi-charge

generalization of the WGC [8] in arbitrary dimension D.

The WGC is satisﬁed in numerous concrete examples and is strongly motivated by

folk theorems forbidding exact global symmetries that arise in the vanishing-charge limit.

Moreover, the WGC is a celebrated avatar of the so-called swampland program [7, 9–11],

whose ultimate aim is to systematically delineate regions in eﬀective ﬁeld theory space

consistent with quantum gravitational ultraviolet completion.

Strictly speaking, a bona ﬁde swampland condition distinguishes between low-energy

eﬀective ﬁeld theories that from an infrared standpoint are otherwise entirely consistent.

From this perspective it is unclear whether a theory that fails the WGC is in the swampland

or merely pathological in a sense that can be diagnosed purely from low-energy consider-

ations. For this reason, a related eﬀort has sought to exclude regions of the low-energy

parameter space using infrared consistency, e.g., constraints from causality, unitarity, and

analyticity of scattering amplitudes [11–15]. While the WGC has previously eluded a

formal general proof, the present work demonstrates that it is mathematically equivalent

to a certain well-motivated — and in many circumstances provable — property of black

hole entropy.

The remainder of this paper is organized as follows. In section 2 we prove, given certain

assumptions, that corrections to the Bekenstein-Hawking entropy are positive. Afterwards,

in section 3 we discuss how various contributions to the black hole entropy arise and esti-

mate their relative size. In order to constrain the coeﬃcients of higher-dimension operators,

we restrict to black holes within a certain mass range. Next, in section 4 we present the

perturbative solution for a charged black hole in the presence of higher-dimension oper-

ators. We then compute the black hole entropy in section 5 and translate the positivity

condition on entropy into a new class of bounds on higher-dimension operator coeﬃcients

in section 6. After discussing the implications of these results for the WGC, we conclude

in section 7.

2 Proof of ∆S > 0

In this section, we study corrections to the Bekenstein-Hawking entropy of a thermody-

namically stable black hole at ﬁxed mass and charge. We prove that these contributions

are positive whenever they come from higher-dimension operators generated at tree level

by quantum ﬁelds.

– 4 –

JHEP10(2018)004

2.1 Assumptions

Let us state our assumptions explicitly. First, we assume the existence of quantum ﬁelds

φ at a characteristic mass scale m

 Λ, where Λ is the energy scale at which quantum

ﬁeld theory breaks down. This parametric separation is required so that quantum ﬁeld

theory has some regime of validity. By the usual rules of eﬀective ﬁeld theory, the higher-

dimension operators in eq. (1.4) receive small contributions suppressed by the cutoﬀ Λ. In

general, Λ can be parametrically smaller than the Planck scale.

Second, we assume that the φ degrees of freedom couple to photons and gravitons such

that integrating them out generates the higher-dimension operators in eq. (1.4) classically,

i.e., at tree level. Since m

 Λ, these states induce the dominant contributions to the

higher-dimension operators in eq. (1.4). Speciﬁcally, the corresponding operator coeﬃcients

scale as c

∝ 1/m

 1/Λ

since tree-level φ exchange is always accompanied by a single

factor of 1/m

coming from the propagator denominator. Thus, eﬀects arising from the

cutoﬀ Λ will be negligible in any context in which quantum ﬁeld theory is applicable.

As noted previously, states like φ are a common feature in string theory, whose low

energy spectrum includes particles like the dilaton and moduli, which are massless in the

supersymmetric limit. In the presence of supersymmetry breaking, these ﬂat directions are

lifted, thus inducing masses m

 Λ, where Λ is the string scale.

While our arguments are perfectly consistent with a scale Λ far below the Planck

scale, we will frequently refer to pure Einstein-Maxwell theory and the pure Reissner-

Nordstr¨om solution as a baseline of comparison. We do so entirely out of convenience

and not because any component of our argument requires that quantum ﬁeld theory be

applicable up to the Planck scale. Hence, in an abuse of nomenclature, we hereafter refer to

the higher-dimension operator contributions of order 1/m

as corrections to pure Einstein-

Maxwell theory, bearing in mind that we actually mean pure Einstein-Maxwell theory plus

contributions of order 1/Λ

, which are parametrically smaller than all the contributions

of interest.

Third, we focus on black holes that are thermodynamically stable, i.e., have positive

speciﬁc heat. As we will see, this is necessary for technical reasons so that we can exploit

certain properties of the Euclidean path integral.

2.2 Positivity argument

Consider a positively charged black hole of mass M and charge Q perturbed by higher-

dimension operator corrections in general spacetime dimension D. As we will show in detail

in section 4, the perturbed metric g

µν

= eg

µν

+ ∆g

µν

can be computed from the perturbed

Lagrangian L =

L + ∆L, where unless otherwise stated all quantities are expressed as

perturbations on a Reissner-Nordstr¨om black hole of the same mass M and charge Q in

pure Einstein-Maxwell theory.

From the perturbed entropy we can deﬁne the corresponding inverse temperature β =

∂

S, which we write as

β =

β + ∆β,

(2.1)

– 5 –

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