  In a gravitational theory with a massless photon the maximum charge-to-mass
ratio of black holes approaches the prediction of the Einstein-Maxwell theory
as black hole mass increases: $Q_{\rm ext}/M =1+ \alpha/M^2$ for some constant
$\alpha$. We will show that $\alpha>0$ if below the quantum gravity scale
$\Lambda$ there are many degrees of freedom with a hierarchically small mass
gap $\log(\Lambda/m_{\rm gap})\gg 1$. In this regime one can treat gravity as a
non-dynamical background field and derive field-theoretic sum-rules for the
coefficients of the leading corrections to the Einstein-Maxwell theory. The
positivity of $\alpha$ follows from the sum-rules. As a consequence,
gravitational attraction gets weaker than the electric force among maximally
charged black holes as they become lighter, and large extremal black holes can
decay into smaller black holes.
