On The Black Hole Weak Gravity Conjecture and Extremality in the Strong-Field Regime

TL;DR

This work shows that the black hole Weak Gravity Conjecture forces sufficiently small extremal black holes to reside in the strong-field regime, thereby probing the UV completion of Maxwell theory. It derives necessary and sufficient decay conditions for extremal black holes and applies them to nonlinear QED models, finding that Euler–Heisenberg and DBI are compatible with the WGC while ModMax is not. The analysis further connects the WGC to the positivity of the $U(1)$ beta function, with implications for colored black holes and conformal hidden sectors, including AdS/CFT realizations where the bound is satisfied. Overall, the results frame extremal BHs as precise probes of UV electrodynamics and constrain IR/UV completions through black hole decay dynamics and renormalization flow.

Abstract

We point out that the Weak Gravity Conjecture (WGC) implies that sufficiently small extremal black holes are necessarily in the strong-field regime of electrodynamics, and therefore probe the UV completion of the Maxwell sector. To investigate the WGC bounds arising from these small extremal black holes, we revisit black hole decay in generic field theories in asymptotically flat space. We derive a necessary and a sufficient condition for any black hole to decay, the latter amounting to a bound on the growth of charge relative to mass. We apply these conditions to extremal black holes derived in various UV completions of the Maxwell sector. We find that the Euler-Heisenberg and DBI effective actions satisfy the sufficient condition for decay, while the ModMax model fails the necessary one, rendering it incompatible with the WGC. Using the decay conditions, we show that the black hole WGC implies positivity of the $U(1)$ gauge coupling beta function. This provides an independent argument that classically stable (embedded-Abelian) colored black holes cannot exist. We also show that the black hole WGC constrains conformal hidden sector models, and is always satisfied in their AdS dual realizations.

Figures (3)

• Figure 1: The regimes of the effective action in the extremal black hole background. We assume $\Lambda=\Lambda_R$. The values in brackets correspond to the Euler-Heisenberg case. The WGC ensures the existence of the intermediate strong-field domain.
• Figure 2: An example of extremality surface for black holes charged under $U(1) \times U(1)^\prime$. If the surface satisfies the monotonicity condition $\frac{\partial \bar{Z}(\hat{\bm n},M)}{\partial M} <0$, then the direction $\hat{n}_0 = \bm Z_0/|\bm Z_0|$ of any extremal black hole BH$_0$ defines a curve along which the norm of the extremal charge-to-mass ratio vectors decreases with the mass $M$ (blue), which is a property used in the proof of the sufficient condition \ref{['eq:WGC_sufficient_condition']}. On the other hand, the decay of extremal black holes allows the construction of a sequence of vectors on the extremality surface whose norm increases as the corresponding mass decreases (red), which is used in the proof of the necessary condition \ref{['eq:WGC_necessary_condition']}.
• Figure 3: Kinematic configurations for a black hole BH$_0$ decaying into two black holes BH$_{1,2}$ charged under $U(1)\times U(1)'$. The charge-to-mass vector ${\bm Z}_0$ of BH$_0$ spans the gray volume, whose boundary corresponds to the extremality surface $Z_0=\bar{Z}_0$. The charge-to-mass vector of BH$_{1,2}$ is encoded in ${\bm Z}_1$, ${\bm Z}_2$. For given ${\bm Z}_{1,2}$, the allowed configurations for the decay of BH$_0$ are given by the intersection of the gray volume with the convex hull of ${\bm Z}_1$, ${\bm Z}_2$, shown in purple. The extremality surfaces of BH$_1$ and BH$_2$ are assumed to be equal for simplicity $Z_{1,2}=\bar{Z}_{1,2}$, which is represented by the dashed line. Left configuration: ${\bm Z}_1$ and ${\bm Z}_2$ are approximately colinear, the decay of extremal BH$_0$ can be symmetric. Right configuration: ${\bm Z}_1$ and ${\bm Z}_2$ are not colinear, the decay of extremal BH$_0$ is asymmetric.