Spontaneous scalarization of a conducting sphere in Maxwell-scalar models

We study the spontaneous scalarization of a standard conducting charged sphere embedded in Maxwell-scalar models in flat spacetime, wherein the scalar field $\varphi$ is nonminimally coupled to the Maxwell electrodynamics. This setup serves as a toy model for the spontaneous scalarization of charged (vacuum) black holes in Einstein-Maxwell-scalar (generalized scalar-tensor) models. In the Maxwell-scalar case, unlike the black hole cases, closed-form solutions exist for the scalarized configurations. We compute these configurations for three illustrations of nonminimal couplings: one that exactly linearizes the scalar field equation, and the remaining two that produce nonlinear continuations of the first one. We show that the former model leads to a runaway behavior in regions of the parameter space and neither the Coulomb nor the scalarized solutions are stable in the model; but the latter models can heal this behavior producing stable scalarized solutions that are dynamically preferred over the Coulomb one. This parallels reports on black hole scalarization in the extended-scalar-Gauss-Bonnet models. Moreover, we analyze the impact of the choice of the boundary conditions on the scalarization phenomenon. Dirichlet and Neumann boundary conditions accommodate both (linearly) stable and unstable parameter space regions, for the scalar-free conducting sphere; but radiative boundary conditions always yield an unstable scalar-free solution and preference for scalarization. Finally, we perform numerical evolution of the full Maxwell-scalar system, following dynamically the scalarization process. They confirm the linear stability analysis and reveal that the scalarization phenomenon can occur in qualitatively distinct ways.

Figure 1

${\overline{C}}_0$ as a function of $\overline{Q}$ for the scalar field solution (40), in the model with an inverse quartic coupling (37), and for the boundary condition ($D$) (solid curves) or the boundary condition ($N$) (dashed curves). Red, blue, green, and purple curves correspond to solutions with 0-, 1-, 2- and 3-nodes, respectively. Each curve is truncated at ${\overline{C}}_0=0.9995$. Inset: enlargement of the 0-node solutions around the origin.

Figure 2

$C_1$ as a function of $\overline{Q}$ for the scalar field solution (48), in the model with an inverse quartic coupling (45), and for the boundary condition ($D$) (solid curves) or the boundary condition ($N$) (dashed curves). Red, blue, green, and purple curves correspond to solutions with 0-, 1-, 2-, and 3-nodes, respectively. Each curve is truncated at $C_1=1.999$. Inset: enlargement of the 0-node solutions around the origin.

Figure 3

${\overline{\varphi }}_0$ (left panel) and $V_{\mathrm{eff}}$ (right panel) as functions of $\overline{r}$ for the model (37) and the boundary condition ($N$). We set $\overline{Q}=3.22738$ and ${\overline{C}}_0=0.995$. The inset shows an enlargement of $V_{\mathrm{eff}}$ to clearly exhibit the negative region.

Figure 4

Same as Fig. 3 but now for the model (45) and the boundary condition ($N$). We set $\overline{Q}=3.24484$ and $C_1=1.95$.

Figure 5

(Un)stable region of the scalarized solution in the two considered nonlinear models: Eq. (40) (left panels) and Eq. (48) (right panels). The boundary condition for the perturbation, at the conductor, is taken as the ingoing condition (top panels), Dirichlet condition (middle panels), and Neumann condition (bottom panels). The gray dashed lines (“energy line”) correspond to the critical value of $\overline{Q}$ above which the scalarized solution has less energy than the Coulomb solution. The curved lines correspond to the static solutions satisfying the boundary conditions ($D$) (solid curves), or ($N$) (dashed curves), or ($R$) [also enforcing a potential minimum Eqs. (38) and (46) at the surface $\overline{r}=1$] (dot-dashed curves). The correspondence between the line color and the node of the solution is the same as Figs. 1 and 2 for the boundary condition ($D$) and the boundary condition ($N$). For the potential minimum condition, we can write the solution of Eq. (46) as $\overline{\varphi }=n\pi /2$, where $n$ is an integer, and red, blue, green, purple... lines correspond to the solution with $n=1,2,3\dots$.

Figure 6

Two illustrations of time evolution in the inverse quartic polynomial model (37) with a radiative boundary condition (68)–(69). The initial data are a momentarily static Gaussian initial data (75) with $w=4r_{\mathrm{s}}$ and (top panels) $A\sqrt{\frac{a{k}^2}{2}}=0.01$, $\overline{Q}=\sqrt{2}$ or (bottom panels) $A\sqrt{\frac{a{k}^2}{2}}=0.07$, $\overline{Q}=0.1\sqrt{2}$. Left panels: contour plot of the time evolution for $r\overline{\varphi }/r_{\mathrm{s}}$. Right panels: radial profiles from snapshots of the time evolution at different times and of the final scalarized solution (blue dashed line) which has ${\overline{C}}_0\simeq 1.559$ (top right panel) or ${\overline{C}}_0\simeq 100$ (bottom right panel).

Figure 7

Two illustrations of time evolution in the inverse quartic polynomial model (37) with the boundary condition ($D$) (71). The initial data are a momentarily static Gaussian initial data (75) with $w=4r_{\mathrm{s}}$, $r_0=8r_{\mathrm{s}}$, and (top panel) $A\sqrt{\frac{a{k}^2}{2}}=0.1$, $\overline{Q}=\sqrt{2}$ or (bottom panels) $A\sqrt{\frac{a{k}^2}{2}}=0.01$, $\overline{Q}=4\sqrt{2}$. Top and bottom left panels: contour plot of the time evolution for $r\overline{\varphi }/r_{\mathrm{s}}$. Bottom right panel: radial profiles from snapshots of the time evolution at different times and of the final scalarized solution (blue dashed line) which has ${\overline{C}}_0\simeq 0.978$. In the simulation of the top panel, the final state is the Coulomb solution.

Figure 8

Two illustrations of time evolution in the inverse quartic polynomial model (37) with the boundary condition ($N$) (73). The initial data are a momentarily static Gaussian initial data (75) with $r_0=r_{\mathrm{s}}$ and (top panel) $A\sqrt{\frac{a{k}^2}{2}}=0.1$, $w=4r_{\mathrm{s}}$, $\overline{Q}=1$ or (bottom panels) $A\sqrt{\frac{a{k}^2}{2}}=0.1$, $w=4r_{\mathrm{s}}$, $\overline{Q}=5$. Top and bottom left panels: contour plot of the time evolution for $r\overline{\varphi }/r_{\mathrm{s}}$. Bottom right panel: radial profiles from snapshots of the time evolution at different times and of the final scalarized solution (blue dashed line) which has ${\overline{C}}_0\simeq 0.99995$. In the simulation of the top panel, the final state is the Coulomb solution.

Figure 9

Two illustrations of time evolution in the inverse cosine model (45) with a radiative boundary condition (68)–(69). The initial data are a momentarily static Gaussian initial data (75) with $r_0=r_{\mathrm{s}}$, $w=8.0r_s$, $\overline{Q}=4\sqrt{2}$ and (top panels) $A\sqrt{\frac{a}{2}}=3.0$, or (bottom panels) $A\sqrt{\frac{a}{2}}=3.5$. Left panels: contour plot of the time evolution for $r\overline{\varphi }/r_{\mathrm{s}}$. Right panels: radial profiles from snapshots of the time evolution at different times and of the final scalarized solution (blue dashed line) which has $C_1\simeq 2.0004$ (top right panel) or $C_1\simeq 69.0$ (bottom right panel).

Figure 10

Two illustrations of time evolution in the inverse cosine model (45) with the boundary condition ($D$) (71). The initial data are a momentarily static Gaussian initial data (75) with $r_0=8r_{\mathrm{s}}$ and (top panel) $A\sqrt{\frac{a}{2}}=0.1$, $w=4r_{\mathrm{s}}$, $\overline{Q}=\sqrt{2}$ or (bottom panels) $A\sqrt{\frac{a}{2}}=0.1$, $w=4r_{\mathrm{s}}$, $\overline{Q}=4\sqrt{2}$. Top and left bottom panels: contour plot of the time evolution for $r\overline{\varphi }/r_{\mathrm{s}}$. Right bottom panel: radial profiles from snapshots of the time evolution at different times and of the final scalarized solution (blue dashed line) which has $C_1\simeq 1.88$. In the simulation of the top panel, the final state is the Coulomb solution.

Figure 11

Two illustrations of time evolution in the inverse cosine model (45) with the boundary condition ($N$) (73). The initial data are a momentarily static Gaussian initial data (75) with $r_0=r_{\mathrm{s}}$, $A\sqrt{\frac{a}{2}}=0.1$, $w=4r_{\mathrm{s}}$ and (top panel) $\overline{Q}=\sqrt{2}$ or (bottom panels) $\overline{Q}=4\sqrt{2}$. Top and left bottom panels: contour plot of the time evolution for $r\overline{\varphi }/r_{\mathrm{s}}$. Right bottom panel: radial profiles from snapshots of the time evolution at different times and of the final scalarized solution (blue dashed line) which has $C_1\simeq 1.9996$. In the simulation of the top panel, the final state is the Coulomb solution.