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

TL;DR

This paper investigates the stability of certain four-dimensional planar black holes with scalar fields, revealing they suffer from instabilities and strong coupling issues that make them physically unviable.

Contribution

It demonstrates that black holes constructed via an infinite sum of curvature invariants with a specific scalar profile are unstable and suffer from strong coupling problems.

Findings

Black holes exhibit ghost and Laplacian instabilities near the horizon.

Scalar-field perturbations have vanishing kinetic terms, indicating strong coupling.

Pathological features invalidate these black holes as stable physical solutions.

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 $\phi'(r)=1/r$, 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 $\phi'(r)=1/r$ as physically viable and stable configurations.