The simplest model of a scalarized black hole in the Einstein-Klein-Gordon theory

TL;DR

This paper demonstrates the existence of scalarized black holes in a minimal Einstein-Klein-Gordon framework with a negative potential, showing they can form without non-minimal coupling but are generally unstable.

Contribution

It introduces a simple model of scalarized black holes with a negative potential, challenging the necessity of non-minimal coupling for scalarization.

Findings

Scalarized black holes exist in the minimal Einstein-Klein-Gordon theory.

All scalarized solutions on a single branch are unstable.

Linear analysis does not predict scalarization due to absence of tachyonic instability.

Abstract

We investigate scalarized black holes in the Einstein-minimally coupled scalar theory with a negative potential $V(\phi)=-\alpha^2\phi^6$. The tachyonic instability is absent from analyzing the linearized scalar equation, which could not allow for spontaneous scalarization. However, we obtain the black hole solutions with scalar hair by solving three full equations because this scalar potential violates the weak energy condition. This shows clearly that scalarized black holes can be obtained without introducing a non-minimal scalar coupling term. We perform the stability analysis for scalarized black holes by adopting radial perturbations, implying that all scalarized black holes belonging to a single branch are unstable.