Global solutions, and their decay properties, of the spherically symmetric \(\mathfrak {su}(2)\)-Einstein-Yang-Mills-Higgs equations
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 \[ \begin{multlined} DW= {1\over 2r} (g-\widetilde g)(W-\widetilde W)- {Q^2 g\over 4r^3} (W-\widetilde W)+ {Qg\over 4r} \sigma_1\overline W+\\ {a\over 2} i\sigma_2 W-{r\over 2} [Vg(W-\overline W)+ V'g\overline W],\end{multlined} \] \[ W={h\choose k},\;g= e^{\lambda+\nu},\;\widetilde g= e^{\nu-\lambda},\;D= {\partial\over\partial u}- {\overline g\over 2}{\partial\over\partial r}. \] The author remarks that the Einstein equations (4)--(7) are integrated under the asymptotic condition \(\lambda+\nu\to 0\), \(r\to\infty\) and the Yang-Mills equation (9) is integrated by classical tools. Thus the problem is to solve the above Higgs equation. Then he proves the following Theorem 3. If \(V\) is \(C^2\)-class and \[ |V(t)|+ t|V'(t)|\leq K_0 t^{p+1},\quad t\geq 0,\text{ for some } p\geq {3\over 2}, \] \[ W_0\in C^1([0,\infty)),\quad W_0= O(r^{-2}),\quad W_0'= O(r^{-3}), \] where \(W_0= W(0, r)\), then there exists \(d> 0\) such that if \[ \sup_{r\geq 0}\{(1+ r)^2|W_0|+ (1+ r)^3|W'|\}< d, \] there exists a unique global solution \(W\in C^1([0,\infty)\times [0,\infty))\) of the Higgs equation, satisfying \(W(0,r)= W_0(r)\). In addition, this solution satisfies the decay condition \[ |W(u,r)|\leq C(1+ u+ r)^{-2},\quad |W'(u,r)|\leq C(1+ u+ r)^{-3}. \] In [Chae, loc. cit.], this result for the Einstein-Maxwell-Higgs equations is proven under the stronger assumption \(p\geq 3\). The proof of this theorem is done by contraction method as in [Chae, loc. cit.] correcting some key estimates. The contraction map is defined by \({\mathcal P}: W\to w\), where \(w\) is the solution of the first-order linear initial value problem \[ \begin{multlined} Dw= {1\over 2r}(g- \widetilde g)(w-\overline W)- {Q^2g\over 4r^3} (w-\overline W)+ {Qg\over 4r} \sigma_1\overline W+{a\over 2} i\sigma_2 w-\\ {r\over 2} [Vg(w-\overline W)+ V'g\overline W],\quad w(0, r)= W_0(r).\end{multlined} \] The author remarks that, due to non-vanishing of the local charge, the solution \(W\) obtained in this paper decays slower than the solutions of the spherically-symmetric massless Einstein scalar-field system obtained in [\textit{D. Christodolou}, Commun. Math. Phys. 105, No. 3, 337--361 (1986; Zbl 0608.35039)]. This answers some questions raised in [Chae, loc. cit.].
\({\mathfrak s}{\mathfrak u}(2)\)-Einstein-Yang-Mills-Higgs equation
scientific article; zbMATH DE number 5966328