Cylindrically symmetric solitons in Einstein-Yang-Mills theory

Recently new Einstein-Yang-Mills (EYM) soliton solutions were presented which describe superconducting strings with Kasner asymptotics [D. V. Gal’tsov, E. A. Davydov, and M. S. Volkov, hep-th/0610183.]. Here we study the static cylindrically symmetric SU(2) EYM system in more detail. The ansatz for the gauge field corresponds to superposition of the azimuthal $B_{\phi }$ and the longitudinal $B_z$ components of the color magnetic field. We derive sum rules relating data on the symmetry axis to asymptotic data and show that generic asymptotic structure of regular solutions is Kasner. Solutions starting with vacuum data on the axis generically are divergent. Regular solutions correspond to some bifurcation manifold in the space of parameters which has the low-energy limiting point corresponding to string solutions in flat space (with the divergent total energy) and the high-curvature point where gravity is crucial. Some analytical results are presented for the low-energy limit, and numerical bifurcation curves are constructed in the gravitating case. Depending on the parameters, the solution looks like a straight string or a pair of straight and circular strings. The existence of such nonlinear superposition of two strings becomes possible due to self-interaction terms in the Yang-Mills action which suppress contribution of the circular string near the polar axis.

Figure 1

Bifurcation of solutions in flat space. Dashed lines—the two singular solutions corresponding to the parameter values $p=-0.9076$ and $p=-0.9078$ respectively. Solid lines correspond to the regular solution with the bifurcational value $p=-0.90777863505\dots$.

Figure 2

Bifurcation curves for $k=1$, 2, 3, 4 in the flat space.

Figure 3

Bifurcation curves for $k=1$, 2, 3 in the self-gravitating case (solid lines) and in the flat space (dashed lines).

Figure 4

Kasner asymptotic parameters corresponding to the solutions with $k=1$.

Figure 5

The metric and field functions for the self-gravitating solutions. Dashed lines—the singular solution $(q=0.39,p=-0.5)$, solid lines—the regular solution $(q=0.3862649505\dots ,p=-0.5)$.

Figure 6

The square root of the metric tensor $\sqrt{-g}$ and the energy density $\sigma =T_0^0\sqrt{-g}$ for self-gravitating solutions. Dashed lines—the singular solution $(q=0.39,p=-0.5)$, solid lines—the regular solution at bifurcation $(q=0.3862649505\dots ,p=-0.5)$.

Figure 7

The components of the stress-energy tensor $\sigma =T_0^0\sqrt{-g}$, ${\tau }_i=T_i^i\sqrt{-g}$ for the self-gravitating configuration.

Figure 8

The metric of the regular solution with $p=-1.85$. Solid lines correspond to the effective exponents of metric functions ${\tilde{a}}_i=(\ln N_i{)}^{\prime }r$ where $i$ labels $a$, $b$, $c$ and $N$, $L$, $K$ respectively. When ${\tilde{a}}_i\approx \mathrm{const}$, the metric functions $N_i\sim r^{{\tilde{a}}_i}$. Dashed line—the energy density $\sigma =T_0^0\sqrt{-g}$.

Figure 9

Continuation of the previous plot Fig. 8 with an enhanced (factor of 100) scale for $\sigma$.

Figure 10

The geometry of the section $z=\mathrm{const}$ of the solution carrying the metric $d{l}^2=d{r}^2+L^2(r)d{\phi }^2$ as embedded into the three-dimensional flat space (the desired surface is the figure of rotation of this curve around the vertical axis). The coordinate along the horizontal axis is $r_{\mathrm{flat}}=\sqrt{g}\phi \phi$, the vertical coordinate is defined by $d{z}_{\mathrm{flat}}^2=d{r}^2-d{r}_{\mathrm{flat}}^2$. Three regions correspond to the nearly flat metric, and that of the Melvin straight and circular strings, respectively.

Figure 11

Plot of the metric function $K(r)$ showing an effective longitudinal thickness of the configuration along the $z$-axis as a function of the radial. Three regions are shown which correspond to the nearly flat metric, the straight string core and the circular string.