d we find that the discharge rate is fast if the WGC is satisfied by a light particle of mass $m\ll1/M$. For sufficiently large black holes $M\gtrsim {\rm max}(e^{|\Delta\phi|},1/m)$ the rate is slow.

We conclude with a loose conjecture: quantum gravity in asymptotically flat space requires a general bound on large localized moduli space excursions of the form $ |\Delta\phi|\lesssim |\log(R\Lambda)|$, where $R$ is the size of the minimal region enclosing the excitation and $\Lambda^{-1}$ is a short-distance cutoff. Both neutral and charged KK bubbles have finite $R$ and infinite excursions, but are strongly unstable. Dilatonic black holes in a controlled EFT also satisfy the bound. KK monopoles provide another example: they are stable and sample an infinite distance in moduli space, but only at a single point, so the visible excursion is limited by the short-distance cutoff.

Charged KK Bubbles
==================

We begin by discussing a representative class of Kaluza-Klein bubbles perturbatively stabilized by flux. We then add matter of mass $m$ and charge $q$ to the system and demonstrate the existence of a rapid pair-production instability for $q/m\gtrsim 1$.

Classical Solutions
-------------------

A number of static charged bubble solutions were obtained in [@Gibbons:1994vm; @Horowitz:2005vp]. In [@Horowitz:2005vp], a 6D bubble stabilized by electric and magnetic 3-form flux was constructed from a family of 5D zero-momentum initial data characterizing bubbles of different sizes. This method is particularly convenient for assessing bubble stability against radial perturbations. We review this construction here, simplifying to case of purely electric 3-form flux.

A family of five-dimensional spatial metrics is given by $$\begin{aligned}
ds^2_{spatial}=U(\rho)d\chi^2+\frac{d\rho^2}{U(\rho) h(\rho)}+\rho^2d\Omega_3
\label{eq:spatial}\end{aligned}$$ with $$\begin{aligned}
U(\rho)\equiv 1-\frac{\rho_0^2}{\rho^2}.
\label{eq:U}\end{aligned}$$ The function $h$ will be determined by the Hamiltonian constraint, a