# On Derrick's theorem in curved spacetime

## TL;DR

This paper generalizes Derrick's theorem to curved spacetimes, demonstrating that static scalar field stars cannot exist, and provides a new tool for analyzing the stability of scalar field solutions in various gravitational theories.

## Contribution

It extends Derrick's theorem to curved spacetimes, enabling stability analysis of scalar fields and compact objects in non-minimally coupled gravity theories.

## Key findings

- Static scalar field stars are impossible in static curved spacetimes.
- The generalized theorem aids in stability checks for scalar field models.
- Applicable to various theories of gravity with scalar degrees of freedom.

## Abstract

We extend Derrick's theorem to the case of a generic irrotational curved spacetime adopting a strategy similar to the original proof. We show that a static relativistic star made of real scalar fields is never possible regardless of the geometrical properties of the (static) spacetimes. The generalised theorem offers a tool that can be used to check the stability of localised solutions of a number of types of scalar fields models as well as of compact objects of theories of gravity with a non-minimally coupled scalar degree of freedom.