Nonsingular black holes and spherically symmetric objects in nonlinear electrodynamics with a scalar field

TL;DR

The paper systematically analyzes linear stability of static, spherically symmetric objects in GR with scalar and nonlinear electrodynamics across several Lagrangian classes L(F, φ, X). By deriving the second-order perturbation action and identifying ghost-free and Laplacian-instability conditions, it shows that none of the studied models yield linearly stable nonsingular black holes; angular instabilities of vector or scalar perturbations generally destroy regular centers. However, in certain L = X + μ(φ) F^n theories, horizonless regular compact objects can exist for specific ranges of n, though still without horizons and with careful tuning to avoid ghosts and Laplacian instabilities. The work highlights the difficulty of achieving stable regular BHs in local four-dimensional classical field theories and points to potential future directions involving rotation or nonlocal/gravity extensions. Overall, the results strongly constrain the viability of nonsingular BHs in broad classical field-theory frameworks and identify narrow windows for stable horizonless compact alternatives.

Abstract

In general relativity with vector and scalar fields given by the Lagrangian ${\cal L}(F,φ,X)$, where $F$ is a Maxwell term and $X$ is a kinetic term of the scalar field $φ$, we study the linear stability of static and spherically symmetric objects without curvature singularities at their centers. We show that the background solutions are generally described by either purely electrically or magnetically charged objects with a nontrivial scalar-field profile. In theories with the Lagrangian $\tilde{\cal L}(F)+K(φ, X)$, which correspond to nonlinear electrodynamics with a k-essence scalar field, angular Laplacian instabilities induced by vector-field perturbations exclude all the regular spherically symmetric solutions including nonsingular black holes. In theories described by the Lagrangian ${\cal L}=X+μ(φ)F^n$, where $μ$ is a function of $φ$ and $n$ is a constant, the absence of angular Laplacian instabilities of spherically symmetric objects requires that $n>1/2$, under which nonsingular black holes with apparent horizons are not present. However, for some particular ranges of $n$, there are horizonless compact objects with neither ghosts nor Laplacian instabilities in the small-scale limit. In theories given by ${\cal L}=X κ(F)$, where $κ$ is a function of $F$, regular spherically symmetric objects are prone to Laplacian instabilities either around the center or at spatial infinity. Thus, in our theoretical framework, we do not find any example of linearly stable nonsingular black holes.

Figures (7)

• Figure 1: Metric component $h(r)$ versus $r/r_0$ for electric SSS objects present in theories given by the Lagrangian (\ref{['EMD']}). We choose $N(r)$ of the form (\ref{['Nexample']}) with $N_0=0.2$. Each case corresponds to (a) $n=1/2$, (b) $n=1$, (c) $n=2$, (d) $n=5$, and (e) $n \gg 1$. For $n$ in the range $n>1/2$, the theoretical lines of $h(r)$ are between (a) and (e), so that $h(r)$ is always positive at any distance $r$.
• Figure 2: ADM mass $M$ (unit of $G=1$) versus $r/r_0$ for electric SSS objects present in theories given by the Lagrangian (\ref{['EMD']}). We choose $N(r)$ to be (\ref{['Nexample']}) with $N_0=0.2$. From bottom to top, each line corresponds to $n=1/2, 4/5, 1, 5/4, 3/2, 2, 4$. For $n<3/2$, $M$ asymptotically approaches a constant, while, for $n \geq 3/2$, $M$ grows in the regime $r \gg r_0$.
• Figure 3: We plot $r_0\phi'/M_{\rm Pl}$, $q_E A_0'/M_{\rm Pl}^2$, $f$, and $h$ versus $r/r_0$ for the electric BH in theories given by the Lagrangian (\ref{['EMD']}) with $n=1$. We choose $N(r)$ to be (\ref{['Nexample']}) with $N_0=0.2$. We observe that $\phi'$ and $A_0'$ approach 0 as $r \to 0$. The two metric components $f$ and $h$ are different around $r=0$, but both converge to 1 at spatial infinity.
• Figure 4: The coupling $\mu$ versus $r/r_0$ for the electric SSS object with the choice $N_0=0.2$ in Eq. (\ref{['Nexample']}). We choose five different powers: $n=0.6, 0.7, 0.9, 1, 2$. For $n>0.887$, $\mu$ is positive at any distance $r$.
• Figure 5: The allowed region of parameter space for electric compact objects, represented by the light-blue shaded area. The lower boundary corresponds to the values of $n_{\rm min}$, below which the no-ghost condition is violated for a given $N_0$ in the range $0<N_0<1$. The upper boundary, represented by $n=3/2$, marks the limit beyond which the solutions no longer describe compact objects. Thus, the allowed range for $n$ is $n_{\rm min}\leq n<3/2$. The curve of $n_{\rm min}(N_0)$ is obtained by interpolating over a set of 99 numerically computed values of $n_{\rm min}$ for $N_0$ in the range $0.01 \leq N_0 \leq 0.99$.