A Weak Gravity Theorem

Abstract

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: $Q_{\mathrm{ext}}/M=1+\alpha /M^2$ 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 $\log (\Lambda /m_{\mathrm{gap}})\gg 1$. 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.

Comments: Withdrawn because the assumptions made in sections 2.1 and 3.1 about the UV coefficients can be justified only when $g{M}_{pl}\sim \Lambda$. Therefore the assumption (ii) of section 1.1 is effectively assuming the existence of QFT states with parametrically large charge-to-mass ratio