Instability of regular planar black holes in four dimensions arising from an infinite sum of curvature corrections

TL;DR

This work investigates four-dimensional planar black holes constructed from an infinite tower of curvature corrections regularized by a conformal rescaling, yielding a shift-symmetric Horndeski effective action. By analyzing linear perturbations around these backgrounds with φ'(r)=1/r, the authors show that odd-parity modes incur ghost and Laplacian instabilities near r=0, while even-parity perturbations exhibit an infinitely strong coupling due to vanishing kinetic terms. The results hold for concrete models (Model 1 and Model 2) and, more generally, for any infinite sum, indicating that regular planar BHs in this framework are not physically viable despite background regularity. The findings emphasize that regularity at r=0 does not guarantee stability and highlight fundamental constraints on regularized 4D gravity theories with higher-curvature corrections. This has implications for the viability of nonsingular planar BHs and motivates further stability analyses of related spherically symmetric solutions in similar theories.

Abstract

In four-dimensional scalar-tensor theories derived via dimensional regularization with a conformal rescaling of the metric, we study the stability of planar black holes (BHs) whose horizons are described by two-dimensional compact Einstein spaces with vanishing curvature. By taking an infinite sum of Lovelock curvature invariants, it is possible to construct BH solutions whose metric components remain nonsingular at $r=0$, with a scalar-field derivative given by $φ'(r)=1/r$, where $r$ is the radial coordinate. We show that such BH solutions suffer from a strong coupling problem, where the kinetic term of the even-parity scalar-field perturbation associated with the timelike coordinate vanishes everywhere. Moreover, we find that these BHs are subject to both ghost and Laplacian instabilities for odd-parity perturbations near $r=0$. Consequently, the presence of these pathological features rules out regular planar BHs with the scalar-field profile $φ'(r)=1/r$ as physically viable and stable configurations.

Figures (2)

• Figure 1: The metric function $f~(=h)$ and the rescaled scalar derivative $\ell^2 X$ are plotted as functions of $r/\ell$ in the range $r \geq 0$ for Model 1 with $M=\ell_{\Lambda}$ and $\ell=\ell_{\Lambda}/2$. The horizon is located at $r_h=2^{1/3}\ell_{\Lambda}=2^{4/3}\ell$, where both $f$ and $X$ vanish. The kinetic term decreases smoothly from the value $X=1/(2\ell^2)$ at $r=0$ to the asymptotic value $X=-1/[2(\ell_{\Lambda}^2-\ell^2)]$, as $r \to \infty$.
• Figure 2: We plot (a) $\ell^2 R$, (b) $\ell^4 R_{\alpha \beta} R^{\alpha \beta}$, and (c) $\ell^4 R_{\alpha \beta \mu \nu} R^{\alpha \beta \mu \nu}$ as functions of $r/\ell$ for Model A with the same values of $M$ and $\ell$ as those used in Fig. \ref{['fig1']}. These curvature invariants are regular for all $r \geq 0$.