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.Comment: 35 pages, 2 figure
