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: $Q_{\rm ext}/M =1+ \alpha/M^2$ 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.