Stability and critical behavior of gravitational monopoles

I dynamically evolve spherically symmetric spacetimes containing gravitational ’t Hooft–Polyakov monopoles and determine the stable end states of the evolutions. I do so to study stability and critical behavior of the well-known static gravitational monopole solutions. For the static solutions, there exist regions of parameter space where two static monopole black holes and the static Reissner-Nördstrom black hole have the same mass. I find strong evidence that one of the static monopole black hole solutions is a critical solution, to which near-critical solutions are dynamically attracted before evolving to one of the other two static solutions as end states. I also discuss the no-hair conjecture for this model in the context of collapse.

Figure 1

(a) Regular monopole solutions for $\lambda =0$ and (from right to left for $w$ and bottom to top for $\phi$) $\overline{v}=0.2$ (purple), 0.3 (blue), and 0.4 (black). (b) Black hole monopole solutions for $\lambda =0$, ${\overline{r}}_h=1$, and the same values of $\overline{v}$ as in (a). (c) and (d) are the same plot with (d) on a log scale to see more easily the region just outside the black hole. Both show regular and black hole monopole solutions for $\lambda =0$, $\overline{v}=0.4$, and ${\overline{r}}_h=0$ (black), 1 (medium gray), and 1.5 (light gray).

Figure 2

Horizon radius ${\overline{r}}_h$ as a function of mass $\overline{M}$ for static solutions. In (a) the solid lines from top to bottom are for $\overline{v}=0.2$ (purple), 0.3 (blue), and 0.4 (black). The dashed line is the outer horizon of the Reissner-Nordström (RN) black hole which only exists for $\overline{M}\ge 1$ and ${\overline{r}}_h\ge 1$. (b–d) are magnifications of the regions where three static solutions exist with the same mass. In each, $A$ marks the cusp where the two monopole solution branches meet and $B$ indicates the bifurcation where the two branches first appear and specifically marks the edge of the bottom branch. The values of $({\overline{M}}_A,{\overline{M}}_B)$ for $v=0.2$ are (1.802, 1.593), for $v=0.3$ are (1.338, 1.224), and for $v=0.4$ are (1.144, 1.082).

Figure 3

The $L_2$ norm across the computational grid for $c_a=\dot{a}-(\cdots )$ and $c_K={{\dot{K}}^r}_r-(\dots )$, where the dots represent the respective right hand sides of (8). The results are shown for three spatial resolutions: (from top to bottom) $\mathrm{\Delta }r=0.02$ (purple), 0.01 (blue), and 0.005 (black). That the results are small indicates that the constraints $c_a=0$ and $c_K=0$ are obeyed and that the results drop by a factor of 4 (for spatial resolutions that drop by a factor of 2) indicates second order convergence.

Figure 4

“Phase” or “end state” diagrams indicating the stable end state (regular monopole, monopole black hole, or RN black hole) of an evolution starting from initial data (44) with $\overline{\lambda }=0$ and ${\overline{s}}_{\phi }=10$. Each column is for the value of $\overline{v}$ indicated. Circles mark the threshold of collapse and squares mark the transition between monopole and RN black holes. The middle row zooms in on the squares and the bottom row plots the squares as a function of mass $\overline{M}$ (with ${\overline{s}}_w$ suppressed).

Figure 5

Typical time evolution of the scalar field $\overline{\phi }$ (blue) and gauge field $w$ (purple) for initial data (44) in which the end state is the static regular solution (dashed lines). The initial data use $(\overline{v},\overline{\lambda },x_w,s_w,s_{\phi })=(0.4,0,1.6,0.4,10)$. All frames have the same axes. The coordinate time $\overline{t}$ for each frame is given in the corner.

Figure 6

The same as Fig. 5 except with the end state being a black hole monopole. The initial data use $(\overline{v},\overline{\lambda },x_w,s_w,s_{\phi })=(0.4,0,1.6,0.25,10)$ and the end state has mass $\overline{M}=0.7856$.

Figure 7

The same as Figs. 5 and 6 except with the end state being the RN black hole [see (40)]. The initial data use $(\overline{v},\overline{\lambda },x_w,s_w,s_{\phi })=(0.4,0,1.6,0.07,10)$ and the end state has mass $\overline{M}=1.181$. Solutions at three different times are plotted in the final frame.

Figure 8

$w(\overline{t},{\overline{r}}_{*})$ and $\overline{\phi }(\overline{t},{\overline{r}}_{*})$ are the same dynamic solutions shown in Fig. 6, with $w_{\mathrm{st}}$ and ${\overline{\phi }}_{\mathrm{st}}$ the corresponding static solutions. These plots are for ${\overline{r}}_{*}=9.005$ where $w_{\mathrm{st}}({\overline{r}}_{*})=0.1289$ and ${\overline{\phi }}_{\mathrm{st}}({\overline{r}}_{*})=0.3117$. Analogous plots for different values of ${\overline{r}}_{*}$ and different evolutions look very similar.

Figure 9

Time evolution similar to Fig. 5, but for two near-critical dynamic solutions with initial data using $(\overline{v},\overline{\lambda },{\overline{x}}_w,{\overline{s}}_{\phi })=(0.3,0,2.0,10)$. The ${\overline{s}}_w>{\overline{s}}_w^{*}$ solutions are the dashed blue curves and the ${\overline{s}}_w<{\overline{s}}_w^{*}$ are the dotted black curves, which begin on top of each other since for both $|{\overline{s}}_w-{\overline{s}}_w^{*}|/{\overline{s}}_w^{*}\approx {10}^{-15}$. The solid green lines introduced in the $\overline{t}=16$ frame are a static monopole black hole solution from the lower branch in Fig. 2 with $(\overline{M},{\overline{r}}_h)=(1.280,2.077)$ and is an intermediate attractor since both dynamic solutions evolve toward it. Eventually the dynamic solutions leave the green lines and separate as seen starting in the $\overline{t}=1450$ frame. The yellow lines introduced in the $\overline{t}=1600$ frame are a static monopole black hole solution from the upper branch in Fig. 2 with $(\overline{M},{\overline{r}}_h)=(1.285,2.098)$, which one of the dynamic solutions evolves to as its stable end state. The other dynamic solution evolves to the RN black hole (40) as its stable end state. Dynamic solutions for three different times are shown in the final frame.

Figure 10

Critical solutions sitting between stable black hole monopole and stable RN black hole end states obey the time scaling $\mathrm{\Delta }\overline{t}=-\lambda \ln |{\overline{s}}_w-{\overline{s}}_w^{*}|$ where $\mathrm{\Delta }\overline{t}$ is the elapsed time measured by an observer at infinity and $\lambda$ (not to be confused with the scalar field self-coupling) is the characteristic time scale for decay of the unstable critical solution or the inverse of the Lyapounov exponent for the unstable mode. This time scaling expresses that the closer you get to the critical solution, i.e. the closer ${\overline{s}}_w$ gets to ${\overline{s}}_w^{*}$, the longer the dynamic solutions spend near the critical solution before evolving away. I define $\mathrm{\Delta }\overline{t}$ for the figures here to be the time from the beginning of the evolution until $|\overline{\phi }(\overline{r})-\overline{v}|<{10}^{-3}$ for all unexcised values of $\overline{r}$ (I have found that once this inequality is satisfied an RN black hole end state is assured to occur). Each figure is for the indicated values of $\overline{v}$ and ${\overline{x}}_w$. Also shown in the figure is $\lambda$ as obtained from a least-squares fit.