Local existence proofs for the boundary value problem for static spherically symmetric Einstein–Yang–Mills fields with compact gauge groups

Abstract

We prove local existence and uniqueness of static spherically symmetric solutions of the Einstein-Yang-Mills equations for an arbitrary compact semisimple gauge group in the so-called regular case. By this we mean the equations obtained when the rotation group acts on the principal bundle on which the Yang-Mills connection takes its values in a particularly simple way (the only one ever considered in the literature). The boundary value problem that results for possible asymptotically flat soliton or black hole solutions is very singular and just establishing that local power series solutions exist at the center and asymptotic solutions at infinity amounts to a nontrivial algebraic problem. We discuss the possible field equations obtained for different group actions and solve the algebraic problem on how the local solutions depend on initial data at the center and at infinity. * Supported in part by NSERC grant A8059.

On the existence of soliton and hairy black hole solutions of {\mathfrak {su}}(N) Einstein–Yang–Mills theory with a negative cosmological constant

“…We need to derive the boundary conditions at each of these singular points. For Λ = 0, local existence of solutions of the field equations near these singular points has been rigorously proved [21,22]. The extension of these results to Λ < 0 will be the focus of Section 3.…”

“…Local existence near the origin has been proved in the asymptotically flat case for su(N) gauge fields in [22], using a method which is different from that employed here, and for general compact gauge groups in [21], whose method we follow. As might be expected, the inclusion of a negative cosmological constant Λ does not significantly change the analysis.…”

“…The dashed line is ω 1 (r h ) = ω 2 (r h ), along which lie the embedded su(2) solutions. In each case, our method follows that employed in [17] for asymptotically flat su(2) EYM, and subsequently generalized in asymptotically flat space to any compact gauge group in [21]. For su(N) EYM in asymptotically flat space, the paper [22] used a different approach to prove local existence.…”

“…Without loss of generality (due to (17)), we take the positive square root in (21). The expansions (19) are substituted into the field equations (12,15) to determine the values of the remaining constants in the expansions.…”

“…It turns out that this is the most lengthy and technically intricate part of the whole existence proof, particularly for local solutions near the origin. Our method in Section 3 follows a well-established approach [17,20,21,22] which involves casting the field equations in an appropriate form, so that a standard theorem (Theorem 1 [17]) can be applied. It is deriving a suitable form for the equations near the origin which is the most complicated part of the analysis.…”