Static axially symmetric solutions of Einstein-Yang-Mills equations with a negative cosmological constant: Black hole solutions

We investigate static axially symmetric black hole solutions in a four-dimensional Einstein-Yang-Mills-SU(2) theory with a negative cosmological constant $\Lambda$. These solutions approach asymptotically the anti-de Sitter spacetime and possess a regular event horizon. A discussion of the main properties of the solutions and the differences with respect to the asymptotically flat case is presented. The mass of these configurations is computed by using a counterterm method. We note that the $\Lambda =-3$ configurations have a higher dimensional interpretation in context of $d=11$ supergravity. The existence of axially symmetric monopole and dyon solutions in a fixed Schwarzschild-anti-de Sitter background is also discussed. An exact solution of the Einstein-Yang-Mills equations is presented in the appendix.

Figure 1

(a) The gauge function $H_1$ is shown as a function of the radial coordinate $r$ for the angles $\theta =0$, $\pi /4$, and $\pi /2$ for three $n=3$ axially symmetric monopole solutions in a fixed SAdS background with $\Lambda =-0.01,r_h=1$. The magnetic charge and total mass of these YM solutions are ($Q_M=-1.39$, $M=1.09$), ($Q_M=3$, $M=0.33$), and ($Q_M=-11.73$, $M=0.91$). (b) Same as Fig. 1a for the gauge function $H_2$. (c) Same as Fig. 1a for the gauge function $H_3$. (d) Same as Fig. 1a for the gauge function $H_4$. (e) Same as Fig. 1a for the energy density $-T_t^t$ (in units $4\pi /e^2$).

Figure 2

The total mass $M$ (in units $4\pi /e^2$) is plotted as a function of magnetic charge $Q_M$ for monopole solutions in a fixed Schwarzschild-anti-de Sitter background with $\Lambda =-3,r_h=1$. The winding number $n$ is also marked.

Figure 3

Typical nongravitating spherically symmetric dyon solutions in a SAdS background, for $\Lambda =-1,r_h=1$, the same value of ${\omega }_h=0.9$, and $u_h=0,0.2,0.4$. The solution with $u_h=0$ corresponds to a magnetic monopole. The energy density $ϵ(r)$ is given in units $4\pi /e^2$.

Figure 4

(a) The gauge function $H_1$ is shown as a function of the radial coordinate $r$ at $\theta =0$, $\pi /4$, and $\pi /2$ for three different static dyon solutions. Here the solutions have winding numbers $n=1$, 2, and 3, the same ratio $Q_M/n=-0.391$, the same value of the electric potential at infinity $u_0=0.04$, and masses $M(n=1)=1.541$, $M(n=2)=1.714$, and $M(n=3)=2.071$. (b) Same as Fig. 4a for the gauge function $H_2$. (c)  Same as Fig. 4a for the gauge function $H_3$. (d) Same as Fig. 4a for the gauge function $H_4$. (e) Same as Fig. 4a for the gauge function $H_5$. (f) Same as Fig. 4a for the gauge function $H_6$. (g) Same as Fig. 4a for the energy density $ϵ=-T_t^t$ (in units $4\pi /e^2$).

Figure 5

The energy $E$ and the angular momentum $J$ (in units $4\pi /e^2$) for rotating YM dyon solutions in fixed SAdS black hole geometry with $\Lambda =-3,r_h=1$ are shown as a function on the parameter $V$ for a fixed ${\omega }_0=2.15$. Also shown are the electric charge $Q_E$ and the ratio $E_e/E$.

Figure 6

(a) The magnetic gauge function $H_1$ is shown as a function of the coordinates $\rho =r\sin \theta ,z=\rho \cos \theta$ for a rotating dyon solution with ${\omega }_0=2.15,V=2,n=1$. The energy and the angular momentum of this configuration (in units $4\pi /e^2$) are $E=9.29$ and $J=4.414$ respectively, while $Q_E=6.833$. The results are obtained in a SAdS background with $\Lambda =-3,r_h=1$. (b) Same as Fig. 6a for the gauge function $H_2$. (c) Same as Fig. 6a for the gauge function $H_3$. (d) Same as Fig. 6a for the gauge function $H_4$. (e) Same as Fig. 6a for the magnitude of the electric gauge functions $|A_t|$. (f) Same as Fig. 6a for the component $T_{\phi }^t$ of the energy-momentum tensor associated with rotation. (g) Same as Fig. 6a for the component $T_t^t$ of the energy-momentum tensor. (h) Surfaces of constant energy density $ϵ=-T_t^t=0.16$ are plotted for the same solution. (i) Same as Fig. 6h for the $ϵ=-T_t^t=0.22$. (j) Same as Fig. 6a for the $ϵ=-T_t^t=0.24$.

Figure 7

(a) The gauge function $H_1$ is shown as a function of the radial coordinate $r$ for the angles $\theta =0$, $\pi /4$, and $\pi /2$. The parameters of these gravitating solutions are $n=1$, 2, and 3, $k=0$, $r_h=1,\Lambda =-0.1$, the node number $k=0$, $Q_M/n=-2.027$, $M(n=1)=1.541$, $M(n=2)=1.714$, and $M(n=3)=2.071$. (b) Same as Fig. 7a for the gauge function $H_2$. (c) Same as Fig. 7a for the gauge function $H_3$. (d)  Same as Fig. 7a for the gauge function $H_4$. (e) Same as Fig. 7a for the metric function $f$. (f) Same as Fig. 7a for the metric function $l$. (g) Same as Fig. 7a for the metric function $m$. (h)  Same as Fig. 7a for the energy density $ϵ=-T_t^t$ of the solutions.

Figure 8

The total mass $M$ is plotted as a function of magnetic charge $Q_M$ for black hole gravitating monopole solutions at $\Lambda =-0.1,r_h=1$. The winding number $n$ is also marked.

Figure 9

(a) The energy density $ϵ=-T_t^t$ is shown as for a gravitating monopole solution with ${\omega }_0=0.49,n=2$. Here the cosmological constant is $\Lambda =-1$ and the black hole radius is $r_h=1$. (b) A surface of constant energy density $ϵ=-T_t^t=0.014$ are plotted for the solution of Fig. 9a. (c) Same as Fig. 9b for $ϵ=0.019$. (d) Same as Fig. 9b for $ϵ=0.03$.

Figure 10

(a) The gauge function $H_i$ is shown as a function of the radial coordinate $r$ for the angles $\theta =0$, $\pi /4$, and $\pi /2$. The results correspond to EYM black hole solutions with winding number $n=2$, magnetic charge $Q_M=1.52$, horizon radius $r_h=1$, and cosmological constants $\Lambda =-1,-0.1$ and $-0.001$. (b) Same as Fig. 10a for the gauge function $H_2$. (c) Same as Fig. 10a for the gauge function $H_3$. (d) Same as Fig. 10a for the gauge function $H_4$. (e) Same as Fig. 10a for the metric function $f$. (f) Same as Fig. 10a for the metric function $l$. (g) Same as Fig. 10a for the metric function $m$. (h) Same as Fig. 10a for the energy density $ϵ=-T_t^t$.

Figure 11

The mass-temperature diagrams for black hole monopole solutions at $\Lambda =-0.1,r_h=1$ and three winding numbers. The value of the magnetic potential at infinity is the control parameter which varies along each curve.