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.

Withdrawn because the assumptions made in sections 2.1 and 3.1 about the UV coefficients can be justified only when $g{M}_{pl}\sim \Lambda$.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

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• The String Landscape and the Swampland
• Natural Inflation and Quantum Gravity
• Infrared Consistency and the Weak Gravity Conjecture