Proof of the Weak Gravity Conjecture from Black Hole Entropy

TL;DR

This paper proves that higher-dimension operators increase black hole entropy and establishes positivity bounds that imply the weak gravity conjecture, showing large extremal black holes are unstable and decay, with results applicable across various theories.

Contribution

It provides a proof that positivity bounds on higher-dimension operators ensure the weak gravity conjecture holds, generalizing to multiple dimensions and gauge fields.

Findings

Higher-dimension operators contribute positively to black hole entropy.

Positivity bounds imply extremal black holes are unstable to decay.

The results are valid across various spacetime dimensions and string theory scenarios.

Abstract

We prove that higher-dimension operators contribute positively to the entropy of a thermodynamically stable black hole at fixed mass and charge. Our results apply whenever the dominant corrections originate at tree level from quantum field theoretic dynamics. More generally, positivity of the entropy shift is equivalent to a certain inequality relating the free energies of black holes. These entropy inequalities mandate new positivity bounds on the coefficients of higher-dimension operators. One of these conditions implies that the charge-to-mass ratio of an extremal black hole asymptotes to unity from above for increasing mass. Consequently, large extremal black holes are unstable to decay to smaller extremal black holes and the weak gravity conjecture is automatically satisfied. Our findings generalize to arbitrary spacetime dimension and to the case of multiple gauge fields. The assumptions of this proof are valid across a range of scenarios, including string theory constructions with a dilaton stabilized below the string scale.