A Weak Gravity Theorem

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: $Q_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{/}M=1+\alpha \mathrm{/}{M}^2$ for some constant $\alpha$. We will show that $\alpha >0$ if below the quantum gravity scale $\mathrm{\Lambda }$ there are many degrees of freedom with a hierarchically small mass gap $\log (\mathrm{\Lambda }\mathrm{/}{m}_{\mathrm{g}\mathrm{a}\mathrm{p}})\gg 1$. 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.