Newly scalarization of the Einstein-Euler-Heisenberg black hole

TL;DR

This paper investigates spontaneous scalarization of EEH black holes within the EEHS framework using an exponential scalar coupling to the Maxwell term. By analyzing the linearized scalar equation and conducting full nonlinear constructions, the authors map out onset thresholds $α_{\mathrm{th}}(1,q)$ as a function of the magnetic charge and demonstrate infinite scalarized branches with a stable $n=0$ branch and unstable higher branches. They also extend the analysis to dual couplings to both $\mathcal{F}$ and $\mathcal{F}^2$, showing increased thresholds but preserved existence of the fundamental scalarized branch. The results broaden the landscape of no-hair theorem evasion, allow for unlimited magnetic charge scenarios, and may have implications for the astrophysical phenomenology of scalarized black holes.

Abstract

Th spontaneous scalarization of the Einstein-Euler-Heisenberg (EEH) black hole is performed in the EEH-scalar theory by introducing an exponential scalar coupling (with $α$ coupling constant) to the Maxwell term.Here, the EEH black hole as a blad black hole is described by mass $M$ and magnetic charge $q$ with an action parameter $μ$. A choice of $μ=0.3$ gurantees a single horizon with unrestricted magnetic charge $q$. The onset scalarization of this black hole appears for a positive coupling $α$ with an unlimited magnetic charge $q$. However, there exists a difference between $q\le1$ and $q>1$ onset scalarizations. We notify the presence of infinite branches labeled by the number of $n=0,1,2,\cdots$ of scalarized charged black holes by taking into account the scalar seeds around the EEH black hole. We find that the $n=0$ fundamental branch of all scalarized black holes is stable against the radial perturbations, while the $n=1$ excited branch is unstable.

Figures (10)

• Figure 1: (Left) Two outer horizons $r_+(1,q)>r_{\rm RN+}(1,q\in[0,1])$ with an inner horizon $r_{RN-}(1,q\in[0,1])$. $r_+(1,q)$ has the minimum of $r_+(1,1.39)=0.83$ and then, it is an increasing function of $q$. We note $r_+(1,100)=5.88$. (Right) Metric functions $f(r,M=1,q,\mu=0.3)$ as functions of $r\in[0.5,20]$ with $q=0.5,2,20$. They cross $r$-axis at $r=1.87(q=0.5),~ 0.89(q=2),~2.63(q=20)$, representing three event horizons $r_+(M=1,q)$ at $q=0.5,2,20$.
• Figure 2: Potential $V_{\rm EEH}(r,M=1,q,\alpha)$ and its integration $I(M=1,q,\alpha)$ with $q=0.5,2,20$. (Left) Three $\alpha$-dependent potentials $V_{\rm EEH}(r,M=1,q=0.5,\alpha)$ as functions of $r\in [r_+(1,0.5)=1.866,10]$ with $\alpha=18.4(=\alpha_{\rm in}),~19.5102(=\alpha_{\rm th}),~20.4(=\alpha_{\rm sEEH})$. (Middle) Three potentials $V_{\rm EEH}(r,1,q=2,\alpha)$ as functions of $r\in [r_+(1,2)=0.893,5]$ with $\alpha=0.437(=\alpha_{\rm th}),~0.61(=\alpha_{\rm sEEH}),~0.97(=\alpha_{\rm in})$. (Right) Three potentials $V_{\rm EEH}(r,1,q=20,\alpha)$ as functions of $r\in [r_+(1,20)=2.63,10]$ with $\alpha=~0.1169(=\alpha_{\rm th}),~~0.572(=\alpha_{\rm in}),~0.602(=\alpha_{\rm sEEH})$. If $q>1.14,17.5$, their roles of $\alpha_{\rm in}$, $\alpha_{\rm th}$, and $\alpha_{\rm sEEH}$ are exchanged.
• Figure 3: (Left) Sufficient condition for tachyonic instability $\alpha_{\rm sEEH}(1,q)$, threshold of instability $\alpha_{\rm th}(1,q)$, and instability condition $\alpha_{\rm in}(1,q)$ as functions of $q\in[0,22]$ with $\alpha_{\rm sRN}(1,q\in[0,1])$. It implies a conventional inequality of $\alpha_{\rm in} \le \alpha_{\rm th}\le \alpha_{\rm sEEH}$ for the green shaded region ($q\in[0,1]$). The whole shaded region represents unstable region of $\alpha(1,q)\ge \alpha_{\rm th}(1,q)$. (Right) Their enlarged picture for $\alpha\in[0,2]$. One finds positive regions for $\alpha$ with two dashed lines at $q=1.14,17.5$ (crossing points), $\alpha_{\rm sEEH}(1,1000)=0.58$, and $\alpha_{\rm in}(1,1000)=0.22$. One expects to have other inequality of $\alpha_{\rm th}\le\alpha_{\rm sEEH}\le\alpha_{\rm in}$ for $1.14<q<17.5$, while $\alpha_{\rm th}\le\alpha_{\rm in}\le\alpha_{\rm sEEH}$ for $q>17.5$.
• Figure 4: (Left) Three curves of $\Omega$ in $e^{\Omega t}$ as a function of $\alpha$ are used to determine the thresholds of tachyonic instability $[\alpha_{\rm th}(1,q)]$ around the EEH black holes. We find that $\alpha_{\rm th}(1,q) =19.5102(q=0.5),~ 0.437 (2),~ 0.1169(20)$ when three curves cross $\alpha$-axis. (Right) $\varphi(r)=u(r)/r$ as a function of $r\in[r_+=1.87,30]$ for representing the first three scalar seeds with $q=0.5$. $\varphi_n(r)$ is classified by the order number $n=0,1,2$ which is also identified by the number of nodes (zero crossings).
• Figure 5: (Left) Existence curves $q_{\rm sEEH}(1,\alpha),~q_{\rm th}(1,\alpha)$, and $q_{\rm in}(1,\alpha)$. The green shaded region [$q_{\rm sRN}(1,\alpha)\le q(1,\alpha)\le 1$] denotes a convential existence region for RN black hole. The whole shaded region represents existence region of $q(1,\alpha)\ge q_{\rm th}(1,\alpha)$, implying unlimited magenetic charge $q$. (Right) Their enlarged picture for $\alpha\in[0,100]$ vs $q\in[0,2]$.