Global solutions, and their decay properties, of the spherically symmetric \(\mathfrak {su}(2)\)-Einstein-Yang-Mills-Higgs equations
type: Item
instance of: scholarly article
title: Global solutions, and their decay properties, of the spherically symmetric \(\mathfrak {su}(2)\)-Einstein-Yang-Mills-Higgs equations
zbMATH Open document ID: 1232.35168
author: Calvin Tadmon
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2011-11-02T00:00:00Z
review text: Existence of global solutions of spherically symmetric \({\mathfrak s}{\mathfrak u}(2)\)-Einstein-Yang-Mills-Higgs (EYMH) equations under weaker assumptions on the self-interaction potentials than previous results on Einstein-Maxwell-Riggs equations (cf. \textit{D. Chae} [Ann. Henri Poincaré 4, No. 1, 35--62 (2003; Zbl 1025.83007)]) is proved. Estimates of decay of such solutions are also obtained (Theorem 3.1). The \({\mathfrak s}{\mathfrak u}(2)\)-EYMH equations are \[ R_{\alpha\beta}-{1\over 2} g_{\alpha\beta} R= T_{\alpha\beta},\qquad g^{\lambda\mu} \widehat\nabla_\lambda F^I_{\mu\alpha}= J^I_\alpha,\qquad g^{\lambda\mu} \widehat\nabla_\lambda \widehat\nabla_\mu\Phi= V'(\Phi^dag\Phi)\Phi, \] where \(V'\) is the derivative of the self-interaction potential \(V\), a real function on \([0,\infty)\). To present spherically symmetric ansaetze, the author uses Bondi coordinates \((x^\alpha)= (u,r,\theta,\phi)\) (cf. [Chae, loc. cit.]). Then the metric and Yang-Mills fields are chosen to be \[ g_{\alpha\beta} dx^\alpha dx^\beta= -e^{2\nu} du^2- 2e^{\nu+\lambda} du\,dr+ r^2(d\theta^2+ \sin^2\theta d\phi^2), \] \[ A_0= aT_3\Biggl(=a{i\sigma_3\over 2}\Biggr),\;a= a(u,r),\;A_1= A_2= A_3= 0. \] The spherically symmetric Higgs field is chosen to be \[ \Phi= \begin{pmatrix} 0\\ \psi(u,r)+ i\xi(u,r)\end{pmatrix}. \] Rewritten forms of the EYMH equations by using these ansaetze are described precisely in the text (the Einstein field equations are (4)--(7), the Yang-Mills equations are (8), (9) and the Higgs equation is (10)). Then introducing new functions \(h= (r\psi)'\) and \(k= (r\xi)'\), the Higgs equation is shown to take the form \[ \begin{multlined} DW= {1\over 2r} (g-\widetilde g)(W-\widetilde W)- {Q^2 g\over 4r^3} (W-\widetilde W)+ {Qg\over 4r} \sigma_1\overline W+\\ {a\over 2} i\sigma_2 W-{r\over 2} [Vg(W-\overline W)+ V'g\overline W],\end{multlined} \] \[ W={h\choose k},\;g= e^{\lambda+\nu},\;\widetilde g= e^{\nu-\lambda},\;D= {\partial\over\partial u}- {\overline g\…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 35Q75
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81T13
Mathematics Subject Classification ID: 83C05
zbMATH DE Number: 5966328
zbMATH Keywords: \({\mathfrak s}{\mathfrak u}(2)\)-Einstein-Yang-Mills-Higgs equation
zbMATH Keywords: spherically symmetric field
zbMATH Keywords: contraction method
MaRDI profile type: Publication
full work available at URL: j.crma.2011.09.004
OpenAlex ID: W2089182873
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DOI: 10.1016/J.CRMA.2011.09.004
description: scientific article; zbMATH DE number 5966328