Vector Horndeski black holes in nonlinear electrodynamics

Technical Summary: Vector Horndeski Black Holes in Nonlinear Electrodynamics

Problem Statement
The existence of black hole (BH) solutions in Einstein-Maxwell theory is well-established, with the Reissner-Nordström (RN) metric describing charged, non-rotating BHs. While linear stability analyses confirm the stability of RN solutions, attempts to construct nonsingular BHs (where curvature scalars remain finite at $r=0$) within Einstein-Nonlinear Electrodynamics (NED) have historically failed. Previous studies indicated that regular electric BHs in Einstein-NED suffer from Laplacian instabilities in the vector field perturbations near the center, rendering them unstable.

This paper investigates whether the introduction of a Horndeski vector-tensor (HVT) coupling, which couples the vector field directly to the curvature tensors while maintaining second-order field equations and $U(1)$ gauge invariance, can resolve these instabilities. Specifically, the authors ask: (1) Can linearly stable, nonsingular BHs exist in Einstein-NED-HVT theory? (2) If not, under what conditions can singular BHs (with curvature singularities at $r=0$) remain linearly stable in both timelike and spacelike regions?

Methodology
The authors analyze the action $S = \int d^4x\sqrt{-g} [M_{Pl}^2 R/2 + L(F) + \beta \mathcal{L}_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}]$, where $L(F)$ represents a general NED Lagrangian and $\beta$ is the HVT coupling constant. They consider a spherically symmetric and static (SSS) background with both electric ($q_E$) and magnetic ($q_M$) charges.

1. Background Analysis: The authors derive the background field equations and expand the metric functions $f(r)$ and $h(r)$ near the center ($r=0$) to test for nonsingular solutions. They examine the behavior of the electric field $A'_0(r)$ and the metric functions for various NED models, including Maxwell-HVT, Power-law NED ($L = F + a_p F^p$), and Born-Infeld theory.
2. Perturbation Theory: A comprehensive linear perturbation analysis is performed on the SSS background. The perturbations are decomposed into odd-parity and even-parity sectors. The authors derive the second-order action for the four dynamical degrees of freedom (two from gravity, two from the vector field) in both timelike ($f>0, h>0$) and spacelike ($f<0, h<0$) regions.
3. Stability Criteria: The stability of the solutions is assessed by checking for:
   • Ghost instabilities: Ensuring the kinetic matrices have positive eigenvalues (no negative energy modes).
   • Laplacian instabilities: Ensuring the squared propagation speeds ($c^2_{\Omega}$) in the angular direction are positive.
   • Strong coupling: Checking if propagation speeds vanish, leading to a breakdown of the effective field theory (EFT).

Key Contributions and Results

• Impossibility of Nonsingular BHs: The authors prove that for a nonsingular BH solution to exist at the background level in Einstein-NED-HVT theory, the magnetic charge must vanish ($q_M = 0$). Even for purely electric BHs with regular centers, the linear stability analysis reveals that the squared angular propagation speed of the even-parity vector perturbation ($c^2_{\Omega4}$) is always negative near $r=0$. This leads to a Laplacian instability, ruling out the existence of linearly stable nonsingular BHs in this framework.

• Stability of Singular BHs: The paper shifts focus to singular BHs (where curvature diverges at $r=0$) and analyzes their stability across five theoretical subclasses:

  • (A) Maxwell-HVT Theory: Singular BHs exhibit both ghost and Laplacian instabilities near the center. To avoid these, the region of instability ($r < r_g$) must lie below the EFT scale ($r_{EFT}$). This imposes a stringent constraint on the coupling: $|\beta| \lesssim r_{EFT}^3 / r_h$. Since $r_{EFT} \ll r_h$, this implies $|\beta| \ll r_h^2$, effectively suppressing the HVT effects outside the horizon.
  • (B) Power-law NED ($\beta=0$): For $L = F + a_p F^p$ with $p \ge 2$ and $a_p > 0$, all linear stability conditions (no ghosts, no Laplacian instabilities) are satisfied in both timelike and spacelike regions.
  • (C) Born-Infeld Theory ($\beta=0$): While free of ghosts and Laplacian instabilities for $r>0$, the squared propagation speed $c^2_{\Omega4}$ approaches zero as $r \to 0$, indicating a strong coupling problem. This requires the coupling parameter $b$ to be constrained such that $b < 2r_{EFT}^4/q_E^2$.
  • (D) Power-law/Born-Infeld with $\beta \neq 0$: The presence of the HVT coupling dominates the NED terms near $r=0$, causing $c^2_{\Omega4}$ to become negative. Consequently, these theories suffer from Laplacian instabilities unless $\beta$ is extremely small (similar to case A).
  • (E) Reconstructed NED Theories: The authors reconstruct NED Lagrangians such that the metric function $h(r)$ takes the form $1 - 2M/r + b_1/r^2$. In this specific branch (with $\beta > 0, b_1 < 0$), Laplacian instabilities are absent for all $r>0$. However, this solution suffers from a ghost instability (due to $G_2 < 0$) and strong coupling ($c^2_{\Omega1}, c^2_{\Omega4} \to 0$) near the center.

Significance and Claims
The paper concludes that the dominance of the HVT coupling generally leads to BH instability in the high-curvature regime. Specifically:

1. No Regular Stable BHs: The inclusion of HVT coupling does not rescue nonsingular BHs from the Laplacian instabilities found in pure Einstein-NED; they remain linearly unstable.
2. Constraints on HVT Coupling: For singular BHs to be stable, the HVT coupling constant $\beta$ must be significantly smaller than the square of the horizon radius ($|\beta| \ll r_h^2$). If $\beta$ is large enough to significantly modify the background geometry near the horizon, it inevitably triggers ghost or Laplacian instabilities inside the horizon.
3. Observational Implications: Because the stability constraints require $\beta$ to be very small, the observational signatures of the HVT coupling (such as modifications to quasinormal modes) are likely to be difficult to detect, as the background solution remains nearly indistinguishable from the standard RN or power-law NED solutions outside the horizon.
4. Theoretical Implication: The results suggest that the HVT Lagrangian is not a suitable ultraviolet completion for stabilizing electrically charged BHs in the high-curvature regime. Alternative vector-tensor theories or UV completions may be required to achieve stable, regular BH solutions without pathological behavior.

The authors explicitly state that their analysis is limited to linear perturbations and that a full numerical simulation incorporating nonlinear effects would be necessary to determine if these unstable BHs can form via gravitational collapse, though such simulations are beyond the scope of this work.