Spherical non-Abelian solutions in conformal gravity

We study static spherically symmetric solutions of non-Abelian gauge theory coupled to conformal gravity. We find solutions for the self-gravitating pure Yang-Mills case as well as monopolelike solutions of the Higgs system. The former are localized enough to have finite mass and approach asymptotically the vacuum geometry of conformal gravity, while the latter do not decay fast enough to have analogous properties.

Figure 1

Profiles of pure YM solutions with $\alpha =\pm 0.5$. The other parameters are $a_0=-1$ (or $Q_m=0$) and $\kappa =0.1$. (a) The gauge components and their corresponding magnetic fields. Also added is the Abelian field strength. (b) The metric functions: $B^{\prime \prime }$, $m(x)$, and $Q(x)$.

Figure 2

Several properties of pure YM solutions for two opposite values of $\alpha$: $\alpha =\pm 0.5$. $\kappa =0.1$ (a) Dependence on the central magnetic field $b(0)$. Note that all lines cross at $b(0)=0$, which corresponds to the trivial solution with $a_0=1$. (b) Dependence on the magnetic charge $Q_m$.

Figure 3

$\alpha$ dependence of several properties of the pure YM solutions. The other parameters are $a_0=-1$ (or $Q_m=0$) and $\kappa =0.1$. Note that $M_I$ is positive for all $\alpha$, but the signs of both $c_1$ and $c_2$ are correlated with that of $\alpha$.

Figure 4

Monopolelike solutions with two opposite values of $\alpha$: $\alpha =\pm 0.5$. The other parameters are $\lambda =1$ and $\kappa =0.1$. (a) The profiles of the solutions. (b) Demonstration of the asymptotic behavior of $f(x)$, $a(x)$, and $B(x)$.

Figure 5

$\lambda$ dependence of several properties of monopolelike solutions for two opposite values of $\alpha$: $\alpha =\pm 0.5$. $\kappa =0.1$. Note the consistency with Fig. 4; e.g., for $\alpha <0$, $f(x)$ increases towards its asymptotical value, so $f_1<0$, while for $\alpha >0$, $f(x)$ decreases towards its asymptotical value, so $f_1>0$.

Figure 6

$\alpha$ dependence of several properties of monopolelike solutions. We denote $b(x)=-a^{\prime }(x)/x$. The other parameters are $\lambda =1$ and $\kappa =0.1$.