{"title":"A Green’s function for the source-free Maxwell Equations on $AdS^5 \\times \\S^2 \\times \\S^3$","authors":"Damien Gobin, Niky Kamran","doi":"10.4310/atmp.2023.v27.n4.a5","DOIUrl":null,"url":null,"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.","PeriodicalId":50848,"journal":{"name":"Advances in Theoretical and Mathematical Physics","volume":"121 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Theoretical and Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.4310/atmp.2023.v27.n4.a5","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
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.
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.