Vector Horndeski black holes in nonlinear electrodynamics

TL;DR

This work investigates black hole solutions in Einstein-NED-HVT theory, focusing on linear stability in both timelike and spacelike regions. It shows nonsingular backgrounds require $q_M=0$ and that purely electric nonsingular BHs are unstable due to Laplacian modes; among $eta=0$ theories, power-law NED can yield stable singular BHs while Born-Infeld faces strong coupling near the center. Introducing the Horndeski coupling ($\beta\neq0$) generally destabilizes high-curvature BHs: even if electric fields remain finite at $r=0$, Laplacian instabilities and ghosts arise inside or near the horizon, unless $|\beta|$ is tightly suppressed relative to the EFT scale. The stability analysis across Maxwell-HVT, power-law, Born-Infeld, and reconstructed NED theories demonstrates that a UV completion beyond the Horndeski coupling is likely required to obtain fully stable, high-curvature BHs in this framework.

Abstract

On a spherically symmetric and static background, we study the existence of linearly stable black hole (BH) solutions in nonlinear electrodynamics (NED) with a Horndeski vector-tensor (HVT) coupling, with and without curvature singularities at the center ($r=0$). Incorporating the electric charge $q_E$ and the magnetic charge $q_M$, we first show that nonsingular BHs can exist only if $q_M = 0$. We then study the stability of purely electric BHs by analyzing the behavior of perturbations in the metric and the vector field. Nonsingular electric BHs are unstable due to a Laplacian instability in the vector perturbation near the regular center. In the absence of the HVT coupling ($β=0$), singular BHs in power-law NED theories can be consistent with all linear stability conditions, while Born-Infeld BHs encounter strong coupling due to a vanishing propagation speed as $r \to 0$. In power-law NED and Born-Infeld theories with $β\neq 0$, the electric fields for singular BHs are regular near $r=0$, while the metric functions behave as $\propto r^{-1}$. Nevertheless, we show that Laplacian instabilities occur for regions inside the outer horizon $r_h$, unless the HVT coupling constant $β$ is significantly smaller than $r_h^2$. For $β\neq 0$, we also reconstruct the NED Lagrangian so that one of the metric functions takes the Reissner-Nordström form. In this case, there exists a branch where all squared propagation speeds are positive, but the ghost and strong coupling problems are present around the BH center. Thus, the dominance of the HVT coupling generally leads to BH instability in the high-curvature regime.

Figures (6)

• Figure 1: (Left panel) Metric components $h$ and $f$ versus $r/r_h$ for $\beta = 4.589 \times 10^{-2}r_h^2$ and $q_E = 0.214 M_{\rm Pl} r_h$ in Maxwell-HVT theory with the Einstein-Hilbert term. The boundary conditions are chosen as $h(r_i) = -99.999$ and $f(r_i) = -97.413$ at the distance $r_i = 9.726 \times 10^{-3} r_h$, in which case $m>0$. There is a single horizon located at $r = r_h$, where both $h$ and $f$ vanish. (Right panel) Plots of $r_h A_0'/M_{\rm Pl}$, $r_h \Delta$, and $R_{\beta}$ versus $r/r_h$ for the same model parameters and boundary conditions as in the left panel. The effect of the HVT coupling becomes significant when $R_{\beta} > 1$, which corresponds to the region $r < 0.71\, r_h$ in the figure.
• Figure 2: Plots of the same quantities as in Fig. \ref{['fig1']}, but with $\beta=-3.062 \times 10^{-7} r_h^2$ and $q_E=0.210 M_{\rm Pl} r_h$. The boundary conditions are specified as $h(r_i)=1.500 \times 10^3$ and $f(r_i)=7.883 \times 10^3$ at $r_i=7.944 \times 10^{-4}r_h$, for which $m<0$. In this case, two horizons appear, with the outer one corresponding to $r_h$. The quantity $\Delta$ remains negative without crossing 0, whereas $R_\beta$ crosses 0 at $h=1$ (around $r=2.18 \times 10^{-2} r_h$).
• Figure 3: (Left panel) Metric component $h~(=f)$ as a function of $r/r_h$ for the power-law NED with $a_2 = 4.63 \times 10^{-3} r_h^2/M_{\rm Pl}^2$ and $q_E = 2.15 \times 10^{-2} M_{\rm Pl} r_h$, together with the boundary condition $h(r_i) = -1.0 \times 10^3$ at $r_i = 9.7 \times 10^{-4} r_h$. For comparison, we also show $h(r)$ for the RN BH ($a_2=0$ and $q_E=0.155\,M_{\rm Pl} r_h$) with the boundary condition $h(r_i) = 1.0 \times 10^2$ at $r_i = 7.0 \times 10^{-3} r_h$. (Right panel) Radial dependence of $r_h A_0'/M_{\rm Pl}$ and $R_{a_2}$ for $a_2 = 4.63 \times 10^{-3} r_h^2/M_{\rm Pl}^2$ and $q_E = 2.15 \times 10^{-2} M_{\rm Pl} r_h$, with the same boundary condition as in the left panel.
• Figure 4: We plot ${\cal G}_1$, ${\cal G}_2$, and ${\cal G}_3$ (left panel), together with $c_{\Omega 1}^2$, $c_{\Omega 2}^2$, $c_{\Omega 3}^2$, and $c_{\Omega 4}^2$ (right panel), as functions of $r/r_h$, using the same model parameters and boundary conditions as in Fig. \ref{['fig1']}. We observe that ${\cal G}_2$ changes sign at $r = 0.56\, r_h$, where $c_{\Omega 2}^2$, $c_{\Omega 3}^2$, and $c_{\Omega 4}^2$ simultaneously flip their signs.
• Figure 5: Plots of ${\cal G}_1$, ${\cal G}_2$, and ${\cal G}_3$ (left panel), together with $c_{\Omega 1}^2$, $c_{\Omega 2}^2$, $c_{\Omega 3}^2$, and $c_{\Omega 4}^2$ (right panel), as functions of $r/r_h$, obtained with the same model parameters and boundary conditions as in Fig. \ref{['fig2']}. In this case, ${\cal G}_2$ crosses zero at $r = 1.25 \times 10^{-2} r_h$, where $c_{\Omega 2}^2$, $c_{\Omega 3}^2$, and $c_{\Omega 4}^2$ change sign.