Evading Derrick’s theorem in curved space: Static metastable spherical domain wall

A recent analysis by one of the authors [L. Perivolaropoulos, Gravitational interactions of finite thickness global topological defects with black holes, Phys. Rev. D 97, 124035 (2018).] has pointed out that Derrick’s theorem can be evaded in curved space. Here we extend that analysis by demonstrating the existence of a static metastable solution in a wide class of metrics that include a Schwarzschild-Rindler-anti–de Sitter spacetime (Grumiller metric) defined as $d{s}^2=f(r)d{t}^2-f(r{)}^{-1}d{r}^2-r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)$ with $f(r)=1-\frac{2Gm}{r}+2br-\frac{\mathrm{\Lambda }}{3}{r}^2$ ($\mathrm{\Lambda }<0$ $b<0$). This metric emerges generically as a spherically symmetric vacuum solution in a class of scalar-tensor theories [D. Grumiller, Model for Gravity at Large Distances, Phys. Rev. Lett. 105, 211303 (2010); D. GrumillerPhys. Rev. Lett.106, 039901(E) (2011).] as well as in Weyl conformal gravity [P. D. Mannheim and D. Kazanas, Exact vacuum solution to conformal Weyl gravity and galactic rotation curves, Astrophys. J. 342, 635 (1989).] It also emerges in general relativity in the presence of a cosmological constant and a proper spherically symmetric perfect fluid. We demonstrate that this metric supports a static spherically symmetric metastable soliton scalar field solution that corresponds to a spherical domain wall. We derive the static solution numerically and identify a range of parameters $m$, $b$, $\mathrm{\Lambda }$ of the metric for which the spherical wall is metastable. Our result is supported by both a minimization of the scalar field energy functional with proper boundary conditions and by a numerical simulation of the scalar field evolution. The metastable solution is very well approximated as $\varphi (r)=\mathrm{Tanh}[q(r-r_0)]$, where $r_0$ is the radius of the metastable wall that depends on the parameters of the metric and $q$ determines the width of the wall. We also find the gravitational effects of the thin spherical wall solution and its backreaction on the background metric that allows its formation. We show that this backreaction does not hinder the metastability of the solution even though it can change the range of parameters that correspond to metastability.

Figure 1

Three different behaviors of the metric function $r^{2}f(r)$ and the corresponding forms of the field configuration after energy functional minimization. Only the red dotted lines correspond to metastable solutions. It is the only one where $r^{2}f(r)$ has three extrema and two roots. In the other cases we get instabilities either towards collapse (blue lines) or ghost instabilities (purple lines) where the gradient terms diverge.

Figure 2

The field configuration that minimizes the energy functional for $b=-0.21$ and $\mathrm{\Lambda }=-0.14$, $m=0$ (blue line). An analytic fit (red dashed line) of the form $\mathrm{\Phi }(r)=\mathrm{Tanh}(q(r-r_0))$ is also shown to provide an excellent fit to the numerically obtained metastable solution.

Figure 3

The field configuration of the metastable spherical wall solution for $m=0$ $b=-0.255$, $\mathrm{\Lambda }=-0.2$ (left panel). The spherically symmetric thin shell corresponding to the energy density is also shown (right panel).

Figure 4

The yellow area corresponds to the stability region in the parameter space $b$, $\mathrm{\Lambda }$ for $m=0$ where there is a minimum of the energy functional with no gradient instabilities at the location of the wall. For $\overline{\kappa }=8\pi G{\eta }^2\ll \frac{|b|}{\eta }$ and $\overline{\kappa }=8\pi G{\eta }^2\ll \frac{|\mathrm{\Lambda }|}{{\eta }^2}$ (here we restored $\eta$ for clarity) we anticipate negligible backreaction of the wall metric on the background metric (left panel). Points above the top dashed line correspond to nonexistence of a minimum of $r^{2}f(r)$ ($b$ is too small) while points below the lower dashed line have a deep minimum with $f(r_{\min })<0$ and thus they correspond to gradient (ghost) instabilities ($b$ is too low). The middle and left panels show how the stability region changes as the level of backreaction increases. As discussed in Sec. 4 backreaction tends to lower the energy minimum and lead to $f(r_{\min })<0$. Thus the yellow region tends to decrease from below (see also Fig. 8 of Sec. 4).

Figure 5

Same as Fig. 4 for $Gm\eta =0.1$.

Figure 6

Simulation of the field evolution with parameter values $m=0$, $b=-0.25$, $\mathrm{\Lambda }=-0.2$ and initial wall approximated by $\mathrm{Tanh}[q(r-r_0)]$ with initial radius $r_0$ equal to the radius to the static solution (left panel $r_0=3.3$), slightly smaller (middle panel $r_0=3.1$), and slightly larger (right panel $r_0=3.5$) than the radius of the static solution. The initial configuration remains static in the left panel but it collapses in both other panels.

Figure 7

The static wall solution in the absence and in the presence of backreaction. Notice that backreaction changes not only the depth but also the position of the minimum of the energy functional (see also 8).

Figure 8

The effects of a small backreaction. ($\overline{\kappa }=0.02$ on the static solution and on the energy functional. We have used the parameter values $m=0$, $b=-0.25$, $\mathrm{\Lambda }=-0.2$).