Solitons and hairy black holes in Einstein–non-Abelian–Proca theory in anti–de Sitter spacetime

We present new soliton and hairy black hole solutions of Einstein–non-Abelian–Proca theory in asymptotically anti–de Sitter spacetime with gauge group $\mathfrak{su}(2)$. For static, spherically symmetric configurations, we show that the gauge field must be purely magnetic, and we solve the resulting field equations numerically. The equilibrium gauge field is described by a single function $\omega (r)$, which must have at least one zero. The solitons and hairy black holes share many properties with the corresponding solutions in asymptotically flat spacetime. In particular, all the solutions we study are unstable under linear, spherically symmetric, perturbations of the metric and gauge field.

Figure 1

Regular solutions of ENAP-AdS theory for which $\omega (r)$ has two zeros. Top row: $n=2$ solutions with fixed $\mu =0.02$ and varying $\mathrm{\Lambda }$ (left panel) and fixed $\mathrm{\Lambda }=-0.001$ and varying $\mu$ (right panel). Bottom row: Quasi-$n=1$ solutions with fixed $\mu =0.02$ and varying $\mathrm{\Lambda }$ (left panel) and fixed $\mathrm{\Lambda }=-0.0005$ and varying $\mu$ (right panel).

Figure 2

Comparison of $n=2$ and quasi-$n=1$ soliton solutions with $\mathrm{\Lambda }=-0.003$ and $\mu =0.02$ (left), $\mu =0.03$ (right). The functions for the $n=2$ solitons are shown in red (solid curves) and those for the quasi-$n=1$ solitons are shown in blue (dashed curves).

Figure 3

Gauge field function $1+\omega (r)$ for a selection of $n=2$ and quasi-$n=1$ solitons with fixed $\mathrm{\Lambda }=-0.001$ and varying Proca field mass $\mu =0.00005$, 0.03 and 0.042. Quasi-$n=1$ curves are blue (dashed), while those for $n=2$ are red (solid). The two branches of solutions merge when $\mu ={\mu }_{\max }=0.042$.

Figure 4

$8\pi T_{00}$ (2.5) for two regular solitons with $\mu =0.02$ and $\mathrm{\Lambda }=-0.003$.

Figure 5

Black hole solutions of ENAP-AdS theory for which $\omega (r)$ has one zero. Top row: $n=1$ solutions with fixed $\mu =0.03$ and varying $\mathrm{\Lambda }$ (left panel) and fixed $\mathrm{\Lambda }=-0.01$ and varying $\mu$ (right panel). Bottom row: Quasi-$n=0$ solutions with fixed $\mu =0.03$ and varying $\mathrm{\Lambda }$ (left panel) and fixed $\mathrm{\Lambda }=-0.01$ and varying $\mu$ (right panel). The event horizon radius is fixed to be $r_h=1$.

Figure 6

Black hole solutions of ENAP-AdS theory for which $\omega (r)$ has two zeros. Top row: $n=2$ solutions with fixed $\mu =0.012$ and varying $\mathrm{\Lambda }$ (left panel) and fixed $\mathrm{\Lambda }=-0.0004$ and varying $\mu$ (right panel). Bottom row: Quasi-$n=1$ solutions with fixed $\mu =0.01$ and varying $\mathrm{\Lambda }$ (left panel) and fixed $\mathrm{\Lambda }=-0.0004$ and varying $\mu$ (right panel). The event horizon radius is fixed to be $r_h=1$.

Figure 7

Left: Comparison of $n=1$ and quasi-$n=0$ black hole solutions with $\mathrm{\Lambda }=-0.02$ and $\mu =0.1$. The functions for the $n=1$ black holes are shown in red (solid curves), and those for the quasi-$n=0$ black holes are shown in blue (dashed curves). Right: Gauge field function $1+\omega (r)$ for a selection of $n=1$ and quasi-$n=0$ black holes with fixed $\mathrm{\Lambda }=-0.025$ and varying Proca field mass $\mu =0.00001$, 0.075 and 0.1018. Quasi-$n=0$ curves are blue (dashed), while those for $n=1$ are red (solid). The two branches of solutions merge when $\mu ={\mu }_{\max }=0.1018$.

Figure 8

Left: Comparison of $n=2$ and quasi-$n=1$ black hole solutions with $\mathrm{\Lambda }=-0.0005$ and $\mu =0.014$. The functions for the $n=2$ black holes are shown in red (solid curves), and those for the quasi-$n=1$ black holes are shown in blue (dashed curves). Right: Gauge field function $1+\omega (r)$ for a selection of $n=2$ and quasi-$n=1$ black holes with fixed $\mathrm{\Lambda }=-0.0005$ and varying Proca field mass $\mu =0.00001$, 0.009 and 0.01442. Quasi-$n=1$ curves are blue (dashed), while those for $n=2$ are red (solid). The two branches of solutions merge when $\mu ={\mu }_{\max }=0.01442$.

Figure 9

Unstable perturbations ${\omega }_1$ for $n=2$ equilibrium solitons. Top row: nodeless perturbations with $N=0$. Bottom row: perturbations having a single node, $N=1$. Left-hand plots: Fixed Proca field mass $\mu =0.02$ and varying cosmological constant $\mathrm{\Lambda }$. Right-hand plots: Fixed $\mathrm{\Lambda }=-0.001$ and varying $\mu$.

Figure 10

Unstable perturbations ${\omega }_1$ for quasi-$n=1$ equilibrium solitons. All perturbations shown are nodeless, with $N=0$. Left-hand plot: Fixed $\mu =0.02$ and varying $\mathrm{\Lambda }$. Right-hand plot: Fixed $\mathrm{\Lambda }=-0.001$ and varying $\mu$.

Figure 11

Unstable perturbations ${\omega }_1$ for $n=2$ equilibrium black holes. Top row: nodeless perturbations with $N=0$. Bottom row: Perturbations having a single node, $N=1$. Left-hand plots: Fixed Proca field mass $\mu =0.01$ and varying cosmological constant $\mathrm{\Lambda }$. Right-hand plots: Fixed $\mathrm{\Lambda }=-0.0004$ and varying $\mu$. The event horizon radius is fixed to be $r_h=1$.

Figure 12

Unstable perturbations ${\omega }_1$ for quasi-$n=1$ equilibrium black holes. All perturbations have $N=0$. Left-hand plot: Fixed Proca field mass $\mu =0.01$ and varying cosmological constant $\mathrm{\Lambda }$. Right-hand plot: Fixed $\mathrm{\Lambda }=-0.0004$ and varying $\mu$. The event horizon radius is fixed to be $r_h=1$.

Figure 13

Unstable perturbations ${\omega }_1$ for $n=1$ equilibrium black holes. All perturbations have $N=0$. Left-hand plot: Fixed Proca field mass $\mu =0.03$ and varying cosmological constant $\mathrm{\Lambda }$. Right-hand plot: Fixed $\mathrm{\Lambda }=-0.01$ and varying $\mu$. The event horizon radius is fixed to be $r_h=1$.

Figure 14

$N=1$ (left) and $N=2$ (right) unstable perturbations of a quasi-$n=0$ black hole solution with $\mu =0.03$ and $\mathrm{\Lambda }=-0.075$. A subplot in the left-hand panel shows the behavior of the perturbations near the event horizon. The quantities ${\chi }_1$ and ${\tilde{\mathrm{\Theta }}}_1$ are denoted by red and blue colors, respectively. The quantity ${\tilde{\mathrm{\Theta }}}_1$ has a deeper trough than ${\chi }_1$. The event horizon radius is fixed to be $r_h=1$.

Figure 15

$N=1$ (left) and $N=2$ (right) unstable perturbations of quasi-$n=0$ black hole solutions with fixed $\mu =0.03$ and varying $\mathrm{\Lambda }$. A subplot in the left-hand panel shows the behavior of the perturbations near the event horizon. The quantities ${\chi }_1$ and ${\tilde{\mathrm{\Theta }}}_1$ are denoted by red and blue colors, respectively. The quantity ${\tilde{\mathrm{\Theta }}}_1$ has a deeper trough than ${\chi }_1$. The event horizon radius is fixed to be $r_h=1$.

Figure 16

$N=2$ unstable perturbations of quasi-$n=0$ black hole solutions with fixed $\mathrm{\Lambda }=-0.01$ and varying $\mu$. The quantities ${\chi }_1$ and ${\tilde{\mathrm{\Theta }}}_1$ are denoted by red and blue colors, respectively. The quantity ${\tilde{\mathrm{\Theta }}}_1$ has a deeper trough than ${\chi }_1$. For these solutions we do not find any $N=1$ perturbation modes. The event horizon radius is fixed to be $r_h=1$.