A Green’s function for the source-free Maxwell Equations on $AdS^5 \times \S^2 \times \S^3$

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Damien Gobin, Niky Kamran
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引用次数: 0

Abstract

$\def\D{\mathcal{D}}$We compute a Green’s function giving rise to the solution of the Cauchy problem for the source-free Maxwell’s equations on a causal domain $\D$ contained in a geodesically normal domain of the Lorentzian manifold $AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$, where $AdS^5$ denotes the simply connected $5$-dimensional anti-de-Sitter space-time. Our approach is to formulate the original Cauchy problem as an equivalent Cauchy problem for the Hodge Laplacian on $\D$ and to seek a solution in the form of a Fourier expansion in terms of the eigenforms of the Hodge Laplacian on $\mathbb{S}^3$. This gives rise to a sequence of inhomogeneous Cauchy problems governing the form-valued Fourier coefficients corresponding to the Fourier modes and involving operators related to the Hodge Laplacian on $AdS^5 \times \mathbb{S}^2$, which we solve explicitly by using Riesz distributions and the method of spherical means for differential forms. Finally we put together into the Fourier expansion on $\mathbb{S}^3$ the modes obtained by this procedure, producing a $2$-form on $\D \subset AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$ which we show to be a solution of the original Cauchy problem for Maxwell’s equations.
A Green's function for the source-free Maxwell Equations on $AdS^5 \times \S^2 \times \S^3$
$def\D\{mathcal{D}}$我们计算了一个格林函数,它给出了无源麦克斯韦方程组在因果域$\D$上的考奇问题的解,该因果域包含在洛伦兹流形$AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$的大地法域中,其中$AdS^5$表示简单连接的5$维反德-西特时空。我们的方法是将原始的考奇问题表述为$\D$上霍奇拉普拉奇的等效考奇问题,并根据$\mathbb{S}^3$上霍奇拉普拉奇的特征形式寻求傅里叶展开形式的解。这就产生了一系列非均质考奇问题,它们支配着与傅里叶模式相对应的形式值傅里叶系数,并涉及与 $AdS^5 \times \mathbb{S}^2$ 上霍奇拉普拉斯相关的算子,我们利用里兹分布和微分形式的球面手段方法明确地解决了这些问题。最后,我们把通过这个过程得到的模合并到 $\mathbb{S}^3$ 上的傅里叶展开中,在 $\D \subset AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$ 上产生了一个 2$ 形式,我们证明它是麦克斯韦方程的原始考奇问题的解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Advances in Theoretical and Mathematical Physics
Advances in Theoretical and Mathematical Physics 物理-物理:粒子与场物理
CiteScore
2.20
自引率
6.70%
发文量
0
审稿时长
>12 weeks
期刊介绍: Advances in Theoretical and Mathematical Physics is a bimonthly publication of the International Press, publishing papers on all areas in which theoretical physics and mathematics interact with each other.
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