Kerr-Newman (A)dS黑洞的近层对称性与通用熵公式验证

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本文主要探讨了Kerr-Newman(A)dS黑洞的近层对称性和熵公式在四维爱因斯坦-麦克斯韦理论中的普遍性。在这个理论框架下，存在一个宇宙学常数，研究者考虑了在高斯零坐标系中描述的Kerr-Newman(A)dS黑洞的近似度量。文章的核心内容聚焦于分析黑洞的几何结构和U(1)规范场在近地平线区域的行为。作者首先明确了研究背景，即在四维纯爱因斯坦引力之外寻找熵公式4πJ0 + J0−的更广泛适用性。这个公式对于理解黑洞熵具有重要意义，因为它超越了传统理论的边界。他们考虑了公制（描述时空曲率的数学工具）和U(1)规范场（描述电磁场的理论元素）在近黑洞区域的衰减条件，这些条件对于确定黑洞的物理性质至关重要。作者发现了一个重要的组合对称生成器，它由微分同构（描述空间和时间连续变换的数学概念）和U(1)规范变换组成。这个生成器能够保持下降条件，即在近层区域的物理性质的不变性。这揭示了黑洞的三种基本对称模式：超平移、超旋转以及多电荷模式。超旋转模式在此过程中遇到了挑战，但通过引入新的组合对称发生器，这一问题得到了解决，使得熵公式能够得到证明。具体来说，超平移表示黑洞的局部平移不变性，而超旋转则涉及旋转对称性的增强，反映了黑洞的自旋特性。多电荷模式则对应于黑洞的电荷分布对称性。这些对称性是理解黑洞熵背后深层次物理原理的关键。这篇发表于《Journal of High Energy Physics》(JHEP04, 2018)的文章，不仅验证了熵公式在Kerr-Newman(A)dS黑洞的适用性，而且对扩展理论的普适性进行了探索。它为黑洞物理学的研究提供了新的视角，并可能对黑洞信息悖论和量子引力理论等领域产生深远影响。总结来说，本文的贡献在于通过精确的数学分析，揭示了Kerr-Newman(A)dS黑洞的内在对称性与熵公式的关联，进一步推动了我们对四维广义相对论和量子引力的深入理解。

JHEP04(2018)133

Equations (2.4) are known as Einstein’s ﬁeld equations, where G

µν

is the Einstein tensor

and T

µν

is the electromagnetic energy-momentum tensor

µν

= F

µα

−

µν

αβ

. (2.7)

Also, equations (2.5) together with ∇

[λ

µν]

= 0 are Maxwell ﬁeld equations in the curved

spacetime.

Now consider two arbitrary variations δ

and δ

. Suppose these two variations do not

commute δ

6= δ

. By varying eq.(2.3), we ﬁnd second order variation of the Lagrangian

L[Φ] = δ

[Φ]δ

Φ + E

[Φ]δ

Φ + ∂

[Φ, δ

Φ]. (2.8)

Similarly, one can write

L[Φ] = δ

[Φ]δ

Φ + E

[Φ]δ

Φ + ∂

[Φ, δ

Φ]. (2.9)

By subtracting eq.(2.9) from eq.(2.8), we have

[1,2]

L[Φ] = δ

[Φ]δ

Φ − δ

[Φ]δ

Φ + E

[Φ]δ

[1,2]

Φ + ∂

(δ

[Φ, δ

Φ] − δ

[Φ, δ

Φ]) ,

(2.10)

where δ

[1,2]

= δ

−δ

is commutator of two variations δ

and δ

. By using eq.(2.3), and

replacing δ → δ

[1,2]

, we can write eq.(2.10) as

∂

[Φ; δ

Φ, δ

Φ] = −

16π

(δ

[Φ]δ

Φ − δ

[Φ]δ

Φ) , (2.11)

where

[Φ; δ

Φ, δ

Φ] =

16π



[Φ, δ

Φ] − δ

[Φ, δ

Φ] − Θ

[Φ, δ

[1,2]

Φ]



, (2.12)

is the Lee-Wald symplectic current. Since the symplectic potential is linear in δΦ then the

terms containing δ

Φ, δ

Φ and δ

[1,2]

Φ eliminate each other and ω

is a skew-symmetric

bilinear in δ

Φ and δ

Φ. The Lee-Wald symplectic current is conserved when equations of

motion and linearized equations of motion are satisﬁed. In other words, if Φ is a solution of

= 0 and δ

Φ and δ

Φ are solutions of δE

= 0, then the Lee-Wald symplectic current

is conserved

∂

[Φ; δ

Φ, δ

Φ] ' 0. (2.13)

The sign ' indicates that the equality is held on-shell. We can deﬁne symplectic 2-form

on solution space through the Lee-Wald symplectic current

Ω

[Φ; δ

Φ, δ

Φ] =

[Φ; δ

Φ, δ

Φ]d

, (2.14)

where C is a codimension-1 spacelike surface. Solution phase space can be constructed

by factoring out the degeneracy subspace of conﬁguration space (see ref. [15] for detailed

discussion). Hence Ω

will be a symplectic form on solution phase space and it is closed,

skew-symmetric and non-degenerate.

– 3 –

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