Charged qOS-extremal black hole and its scalarization by entropy function approach

TL;DR

This work investigates charged qOS-extremal black holes within the Einstein-Gauss-Bonnet-scalar theory with nonlinear electrodynamics, focusing on spontaneous scalarization driven by the Gauss-Bonnet coupling. Using linearized analysis around the extremal background and its Bertotti-Bobinson near-horizon geometry, together with Sen's entropy function, it identifies onset conditions and constructs scalar clouds corresponding to two branches, with AdS$_2\times$S$^2$ geometry playing a crucial role. The entropy analysis shows the scalarized branch with $\lambda>0$ is entropically favored over the negative-$\lambda$ branch, suggesting a possible phase transition from cqOSe to scalarized cqOSe. Overall, the BR near-horizon structure provides seeds for hair formation and offers a thermodynamic criterion for extremal black hole scalarization in this setup.

Abstract

We investigate scalarization of charged quantum Oppenheimer-Snyder extremal (cqOSe)-black hole in the Einstein-Gauss-Bonnet-scalar theory with a nonlinear electrodynamics term. This black hole is described by quantum parameter $α$ and magnetic charge $P$. It is equivalent to the qOS-extremal black hole whose action is still unknown when imposing a relation of $(3αP^2)^{1/4}\to 3M/2$. Focusing on the onset of scalarization, we find the single branch of scalarized cqOS extremal (scqOSe)-black holes. To obtain a scalar cloud (seed) for the single branch, however, we have to consider its near-horizon geometry of the Bertotti-Bobinson (BR) spacetime. In this case, two scalar clouds for positive and negative coupling constant $λ$ are found to represent two branches. Applying Sen's entropy function approach to this theory, we obtain the entropy which is the only physical quantity to describe the scqOSe-black holes. We find that the positive branch is preferred than the negative branch.

Figures (5)

• Figure 1: (a) Two horizons $r_{\pm}(M=1,\alpha=1,P)$ are function of $P\in[0,P_e=1.299]$, showing the upper bound for the charge $P$. Here, $r_e(\alpha=1,P)$ as a function of $P$ represents the extremal horizon, starting from $P=0$. The bounce radius $r_b(M=1,\alpha=1,P)$ is located inside the inner horizon. (b) Heat capacity $C(M=1,\alpha=1,P)/|C_S(1,0)|$ with $|C_S(1,0,0)|=25.13$. Heat capacity blows up at Davies point ($P_D=0.9077,$ red dot) where the temperature $T(1,1,P)$ has the maximum. The shaded region denotes $C>0$ for cqOS-black holes. The heat capacity and temperature are always zero at the cqOSe-black line.
• Figure 2: (a) $\mathcal{R}^2_{\rm GB}(r,M=1,\alpha=1,P)<0$ as a function of $(r\in [r_+(M=1,\alpha=1,P),2.2],P\in[0,1.5])$ in the near-horizon for cqOS black holes. Its sign changes from positive ($0<P<P_c~(=1.1329)$) to negative ($P_c<P<P_e~(=1.299)$) with the critical onset point $[(P_c,r_+(1,1,P_c)$, black dot]. (b) $\bar{\mathcal{R}}^2_{\rm GB}(r,\alpha=1,P)<0$ as a function of $(r\in [r_e(\alpha=1,P),3],P\in[0,1.5])$ in the whole near-horizon for cqOSe-black holes. Hence, one finds that $-\lambda \bar{\mathcal{R}}^2_{\rm GB} <0$ in the whole near-horizon for $\lambda <0$, leading to tachyonic instability.
• Figure 3: (a) Extremal scalar potentials $V_e(r,\alpha=1,P=0.6,\lambda)$ with $\lambda=0,-0.44,-1,-2$ as a function of $r\in[r_+=1.019,5]$ for GB$^{\rm e}$ scalarization. Here, one has the integration $I=0.39(\lambda=0),~0(\lambda=-0.44),~-0.5(\lambda=-1),~-1.39(\lambda=-2)$. (b) Sufficient unstable (shaded) region of cqOSe-black hole for GB$^{\rm e}$ scalarization. A single branch for $\lambda<0$ is allowed for $0<P<P_{\rm cqOSe}~(=-0.7879\lambda)$.
• Figure 4: Three different scalar clouds. (a) Tachyonic scalar cloud of $\delta \phi(\rho,\mu^2=-8)$ and its negative potential $V_{\rm GB}=-8\rho^2$ with negative $\lambda$ for GB$^{\rm BR}$ scalarization. This has many nodes but it has a large value as 100 at $\rho=10^{-4}$. (b) Regular scalar clouds $\delta \phi_n(r,r_+=2)$ with $n=0,1,2,3$ for GB$^+$ scalarization Myung:2018iyq. Here, $n$ represents number of nodes (number of zero-crossings at $r$-axis). (c) Scalar cloud of $\delta \phi(\rho,\mu^2=8)$ which blows up as $\rho\to 0$ and its positive potential $V_{\rm GB}=8\rho^2$ with positive $\lambda$.
• Figure 5: Entropy function $\mathcal{E}_f(\alpha=1,p,\zeta=\pm 2,\lambda)$ with $\lambda=1.01,0,-1.01$ as a function of magnetic charge $p$ for scqOSe- and cqOSe-black holes. The dashed line is located at $p=0.82$. For $p=0.82$, one has $\mathcal{E}_f(1,0.82,2,1.01)=7.658$, 4.462 for $\lambda=0$, and $\mathcal{E}_f(1,0.82,-2,-1.01)$=1.266.