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首页AdS 3中爱因斯坦-麦克斯韦理论的保守电荷黑洞新审视
本文探讨了阿尔弗雷多·佩雷斯、米格尔·里克尔梅、大卫·泰波和里卡多·特龙科索在《现代物理》杂志(JHEP)2015年10月期上发表的一篇论文,题目是"爱因斯坦-麦克斯韦理论在AdS 3中的保守电荷和黑洞的重新审视"。文章聚焦于在三维时空中研究具有负宇宙学常数且与麦克斯韦场耦合的广义相对论的静态圆对称解。关键发现在于,作者考虑了一个特殊场景,即电磁场的衰减速度较物质局部分布的标准衰减更慢,这使得通过选择合适的电磁拉格朗日乘数,能够确保作用量达到真正的极值。 在这样的条件下,电磁场的边界行为满足非平凡的可积性条件,导致质量(通常是通过曲率度量的)和角动量自然成为有限值。这与高维情况不同,质量和角动量的具体值依赖于所选的边界条件,而非通常需要的正则化程序。值得注意的是,这种分析揭示了与常规情况下的病态行为明显的区别:对于特定的全息边界条件,带电(旋转)黑洞的能量谱是正向的,而且质量保持恒定,电荷受到上界的限制。 全息边界条件的引入,要求与洛伦兹不变性相兼容,并且要求存在一种固定解的缩放对称性。这意味着可以通过一个参数来描述这些边界条件,形成了一类特殊的条件。作者强调,这种选择不仅解决了能量谱的正定性问题,还对物理性质产生了显著的影响,例如质量与电荷的关系。 这篇论文深入研究了AdS 3背景下的爱因斯坦-麦克斯韦理论,揭示了新的保守电荷和黑洞特性,对于理解三维引力系统以及量子重力的全息原则提供了新的见解。这项工作不仅扩展了我们对基本物理定律在特定环境下的理解,也为黑洞物理和量子引力理论的未来发展奠定了基础。
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JHEP10(2015)161
two-form fields, for a generic choice of boundary conditions, the mass and also the angular
momentum become automatically finite, and acquire explicit contributions from the matter
field. In section 2.1, it is shown that requiring compatibility of the boundary conditions
with the scaling symmetry of the class of stationary solutions, singles out a precise set of
“holographic boundary conditions” that is described by an arbitrary function of a single
variable. It can also be seen that if one further requires the holographic boundary conditions
to be compatible with Lorentz symmetry, selects a very special subset that is parametrized
just by a single arbitrary fixed constant. In section 3 we compare the global charges that
are obtained in the case of an electrically charged rotating black hole solution for different
choices of boundary conditions. It is shown that the standard result in [2] corresponds
to the simplest choice of Lorentz invariant boundary conditions. Noteworthy, it is found
that if the boundary conditions are compatible with both Lorentz and scaling symmetries,
the energy spectrum of an electrically charged (rotating) black hole is nonnegative, and
also, for a fixed value of the mass, the electric charge becomes bounded from above. We
conclude with further remarks in section 4.
2 Reduced action principle: stationary solutions, integrability conditions
and conserved charges
Let us consider the electromagnetic field minimally coupled to General Relativity with
negative cosmological constant in three spacetime dimensions. The action reads
I =
ˆ
d
3
x
√
−g
1
2κ
(R − 2Λ) −
0
4
F
αβ
F
αβ
, (2.1)
where the Newton constant G and the AdS radius l are defined through κ = 8πG and
Λ = −l
−2
, respectively. The “vacuum permeability”
0
has units of length, and is assumed
to be fixed as
0
= 1. A wide family of exact stationary circularly symmetric solutions has
already been found in the literature [1, 2, 7–22]. Hereafter we deal with generic stationary
circularly symmetric configurations, so that the spacetime metric can be written as
ds
2
= −N (r)
2
F (r)
2
dt
2
+
dr
2
F (r)
2
+ R(r)
2
N
φ
(r) dt + dφ
2
, (2.2)
and the gauge field can be chosen to be given by
1
A = A
t
(r) dt + A
φ
(r) dφ . (2.3)
It is then clear that the form of (2.2) and (2.3) is mapped into itself under the action of
a Lorentz boost in the “t − φ cylinder”. It is also worth pointing out that the metric and
the gauge field are invariant under scalings of the form
r → λr , t → λ
−1
t , φ → λ
−1
φ , (2.4)
1
Here the gauge field has been assumed to be stationary and circularly symmetric. Had we made the
same assumption for the field strength, one would obtain a wider class of configurations, so that (2.3)
corresponds to one of the two possible branches (see e.g., [22, 23]).
– 2 –
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