Spherically symmetric Yang-Mills solutions in a $(4+n)$-dimensional space-time

We consider the Einstein-Yang-Mills Lagrangian in a $(4+n)$-dimensional space-time. Assuming the matter and metric fields to be independent of the $n$ extra coordinates, a spherical symmetric Ansatz for the fields leads to a set of coupled ordinary differential equations. If we assume the metric to be diagonal, we find that either $n-1$ of the gauge potentials associated with the $n$ extra dimensions must vanish or the solutions are embedded abelian solutions. These latter are the Einstein-Maxwell-dilaton solutions in $(4+n)$ dimensions. We present generic solutions of the effective 4-dimensional Einstein-Yang-Mills-Higgs-dilaton model, which possesses $n$ Higgs triplets coupled in a specific way to $n$ independent dilaton fields. These solutions are the abelian Einstein-Maxwell- dilaton solutions and analytic nonabelian solutions, which have diverging Higgs fields. In addition, we construct numerically asymptotically flat and finite energy solutions for $n=2$. If we choose the metric to have off-diagonal components, generic solutions of the full model are possible. We present the numerical solutions for the case $n=2$.

Figure 1

The values $A(0)$, $N_m$, $-\psi (0)$ are shown as functions of $\alpha$ for $\rho =1$. The indices “1”, “2”, “3”, “4”, respectively, correspond to the 1., 2., 3. and 4. branch of solutions (see also Fig. 2).

Figure 2

The mass $M$ of the solutions is shown as function of $\alpha$ for $\rho =1$ and $\rho =2$. “1”, “2” and “3”, respectively, denote the 1., 2. and 3. branch of solutions.

Figure 3

The values of $-{\psi }_1(0)$ and $-{\psi }_2(0)$ as well as the difference ${\psi }_2(0)-{\psi }_1(0)$ are shown as functions of $\alpha$ for $\rho =2$. “1”, “2”, “3” and “4”, respectively denote the 1., 2., 3. and 4. branch of solutions.

Figure 4

The profiles of the functions ${\psi }_1(x)$, ${\psi }_2(x)$, $K(x)$, $A(x)$ and $N(x)$ are shown for $\rho =2$ and $\alpha$ close to the critical value ${\alpha }_{cr}\approx 0.179$.

Figure 5

The profiles of the metric functions $A(x)$, $J(x)$, $N(x)$, ${\xi }_1(x)={\xi }_2(x)\equiv \xi (x)$ are shown for two different values of $\alpha$.

Figure 6

The dependence of the values $J(0)$ and $-{\xi }_1(0)=-{\xi }_2(0)\equiv -\xi (x)$ as well as the ADM mass are shown as functions of $\alpha$.

Figure 7

The dependence of $N(x_{min})=N_m$ and $A(0)$ are shown as functions of $\alpha$.