Schwinger stability of T-duality-inspired extremal black holes

Technical Summary: Schwinger Stability of T-Duality-Inspired Extremal Black Holes

Problem Statement
The paper investigates the stability of near-extremal charged black holes against Schwinger pair production of charged scalars. While near-extremal Reissner–Nordström (RN) black holes possess an $AdS_2 \times S^2$ throat where the Hawking temperature vanishes, they remain unstable to charged particle production due to a non-vanishing near-horizon electric field. The authors examine whether this instability persists in "regular" black hole geometries inspired by string T-duality and zero-point length ($l_0$) prescriptions. Specifically, they analyze whether the regularization of the source, which removes the central singularity, leads to a stable extremal endpoint where Schwinger pair production is suppressed.

Methodology
The authors employ a semi-classical probe approximation within a specific effective geometry derived from T-duality:

1. Effective Geometry: They utilize the static, spherically symmetric metric obtained from the zero-point-length prescription. This geometry includes a non-trivial radial form factor, $F(r)$, which arises from the gravitational response to a regularized charge distribution. The metric is given by $f(r) = 1 - 2m(r)/r$, where the mass function $m(r)$ incorporates both the smeared mass and the electromagnetic energy density modified by $F(r)$.
2. Physical Potentials: Crucially, the authors distinguish between the gravitational form factor and the gauge potential. They maintain that the physical electromagnetic potential $A_t(r)$ remains the regularized form $-Q/\sqrt{r^2 + l_0^2}$, consistent with the original T-duality prescription. The form factor $F(r)$ is treated as a property of the mass function (packaging the radial contribution of the charge sector relative to the ADM mass) rather than a redefinition of the gauge field.
3. Extremal Analysis: They solve the extremality conditions $f(r_{ext}) = 0$ and $f'(r_{ext}) = 0$ for a general form factor to derive the extremal mass ($M_{ext}$), charge ($Q_{ext}$), and the radius of the near-horizon $AdS_2$ throat ($L_2$).
4. Instability Criterion: The stability is analyzed by coupling a minimally charged scalar field $\Psi$ to the background. The radial equation in the near-horizon $AdS_2$ region is mapped to an effective Breitenlohner–Freedman (BF) bound violation. The instability occurs if an effective parameter $\mu_*^2 > 0$. The authors derive the threshold condition for the charge-to-mass ratio ($q/m$) required to trigger this instability.
5. Comparative Analysis: The results are contrasted with the Ayón–Beato–García (ABG) Einstein–nonlinear-electrodynamics (NLED) black hole. The ABG metric is algebraically similar to the T-duality metric under specific parameter replacements, but it possesses a distinct electromagnetic field derived from its own NLED Lagrangian.

Key Results

• Extremal Branch Structure: For the explicit T-duality form factor, the extremal branch extends down to a lower endpoint at $r_{ext} = \sqrt{2} l_0$.
• Vanishing Electric Field: At this endpoint ($r_{ext} = \sqrt{2} l_0$), the extremal charge $Q_{ext}$ and the near-horizon electric field $E_H$ both vanish. However, the $AdS_2$ radius $L_2$ remains finite, approaching $\sqrt{3} l_0$.
• Schwinger Stability:
  • At the endpoint, the exact instability parameter $\mu_*^2$ is strictly negative (specifically $\mu_*^2 \to -3m^2 l_0^2 - \dots < 0$). Consequently, the local charged-scalar instability is absent at the endpoint.
  • By continuity, for any fixed massive species with a finite charge-to-mass ratio ($q/m$), there exists a neighborhood near the endpoint where the instability condition is not met.
  • In the semiclassical regime ($mL_2 \gg 1$), the threshold for instability, defined by $\Xi_F = 1/(E_H^2 L_2^2)$, diverges as $\alpha \to \sqrt{2}$ (where $\alpha = r_{ext}/l_0$). This divergence is driven by the vanishing electric field $E_H$, which cannot be compensated by the finite $AdS_2$ radius.
• Comparison with ABG: In contrast, the ABG extremal black hole retains a non-vanishing near-horizon electric field ($|E_H^{ABG}| \approx 0.816/|Q|$) and a finite $AdS_2$ radius. The corresponding Schwinger threshold $\Xi_{ABG}$ is finite ($\approx 0.378$), meaning the instability persists. This demonstrates that the divergent Schwinger barrier found in the T-duality model is not a generic consequence of regularity but is specific to the T-duality branch where the physical horizon electric field vanishes.

Significance and Claims
The paper claims to identify a specific mechanism within T-duality-inspired regular black holes that stabilizes the extremal endpoint against local Schwinger pair production. The primary significance lies in the decoupling of the gravitational form factor from the gauge potential: the vanishing of the electric field is a result of the specific extremal branch of the T-duality geometry, not merely the regularization of the singularity.

The authors explicitly state limitations to their findings:

• The analysis is a local probe calculation and does not constitute a proof of complete black hole evaporation or the dynamical evolution of an evaporating black hole to this endpoint.
• The treatment of the scalar field uses an ordinary local propagator; a full microscopic minimum-length theory might modify the pair-production criterion.
• The stability against Schwinger production does not address other potential instabilities, such as the classical angular Laplacian instability in NLED models identified by De Felice and Tsujikawa, which is a distinct issue from the quantum Schwinger effect.

Ultimately, the work establishes that for the specific T-duality extremal branch, the near-horizon electric field vanishes while the $AdS_2$ throat remains finite, rendering the exact instability parameter negative and eliminating the local charged-scalar instability at the endpoint.