论文标题
$ ADS^5 \ times \ Mathbb {s}^2 \ times \ times \ mathbb {s}^3 $上的green的green函数
A Green's function for the source-free Maxwell equations on $AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$
论文作者
论文摘要
我们计算了绿色的功能，从而引发了凯奇问题的解决方案，用于无源的麦克斯韦方程$ \ mathcal {d} $在Lorentzian歧管的地理上正常域中包含在lorentzian歧管$ ads^5 \ times^5 \ times^times \ times \ times \ mathb {s} s}^2 \ times \ mathbb^$ dENOT $ dENOT $ dENOT $ dENOT $ dENOTS $ ner $ 5 $ - 维度的反de-sitter时空。我们的方法是将原始的Cauchy问题作为$ \ Mathcal {D} $上的Hodge laplacian的等效库奇问题，并以Hodge laplacian的特征性在$ \ Mathbb {s}}^3 $上寻求傅立叶laplacian的特征形式的解决方案。这引起了一系列不均匀的库奇问题，该问题管理了与傅立叶模式相对应的形式值的傅立叶系数，并且与hodge laplacian相关的涉及运营商$ ads^5 \ times^5 \ times \ mathbb {s}^2 $，我们通过使用riessical和Spherical的形式来求明上词，并在这些方面求解。 Finally we put together into the Fourier expansion on $\mathbb{S}^3$ the modes obtained by this procedure, producing a $2$-form on $\mathcal{D}\subset AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$ which we show to be a solution of the original Cauchy problem for Maxwell's equations.
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 $\mathcal{D}$ 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 $\mathcal{D}$ 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 $\mathcal{D}\subset AdS^5 \times \mathbb{S}^2 \times \mathbb{S}^3$ which we show to be a solution of the original Cauchy problem for Maxwell's equations.
