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

## TL;DR

This paper demonstrates the existence of static metastable spherical domain walls in curved spacetime metrics, extending Derrick's theorem evasion to a broad class of solutions including Schwarzschild-Rindler-AntideSitter spacetime.

## Contribution

It extends Derrick's theorem evasion to curved space by showing metastable scalar field solutions in a wide class of metrics, including those from scalar-tensor theories and Weyl gravity.

## Key findings

- Existence of metastable spherical domain walls in specified metrics.
- Numerical solutions approximate the scalar field as a hyperbolic tangent profile.
- Backreaction effects do not prevent metastability of the solutions.

## Abstract

A recent analysis by one of the authors\cite{Perivolaropoulos:2018cgr} 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-AntideSitter spacetime (Grumiller metric) defined as $ds^2= f(r) dt^2 - f(r)^{-1} dr^2 - r^2 (d\theta^2 +\sin^2\theta d\phi^2)$ with $f(r)=1-\frac{2Gm}{r}+2br-\frac{\Lambda}{3} r^2$ ($\Lambda<0\; b<0$). This metric emerges generically as a spherically symmetric vacuum solution in a class of scalar-tensor theories\cite{Grumiller:2010bz} as well as in Weyl conformal gravity\cite{Mannheim:1988dj}. It also emerges in General Relativity (GR) 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 $m, b, \Lambda$ 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 $\phi(r) = Tanh\left[q (r-r_0)\right]$ 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.