We present a semiclassical proof of the weak gravity conjecture in @math spacetime dimensions for scalar matter gauged under a @math gauge group. We compute the non-perturbative macroscopic entropy of a scalar field in an extremal black hole background at the level of linearized backreaction on the metric. The scalar field is assumed to violate or saturate the weak gravity conjecture. The scalar contributes a logarithmic correction to the entropy in the black hole geometry that outgrows the classical contribution. We demonstrate that the entropy of the gauged scalar violates the generalized second law in the limit of large black hole charge. Our result suggests that entropy inequalities may directly discriminate between effective field theories that live in the landscape versus the swampland.
A separate application of the second law towards understanding the weak gravity conjecture appears in @cite_23 , which appeared during the preparation of this manuscript. However, their calculation is orthogonal to ours. Their paper argues for the weak gravity conjecture using a bound on relaxation rates of quasinormal modes of near-extremal black holes. Although related to the second law, the connection is indirect: the second law implies the relaxation rate bound, which in turn implies the weak gravity conjecture. In this paper, we present a a more direct link between the second law and the weak gravity conjecture.
@cite_23
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The weak gravity conjecture suggests that, in a self-consistent theory of quantum gravity, the strength of gravity is bounded from above by the strengths of the various gauge forces in the theory. In particular, this intriguing conjecture asserts that in a theory describing a U(1) gauge field coupled consistently to gravity, there must exist a particle whose proper mass is bounded (in Planck units) by its charge: m mP < q. This beautiful and remarkably compact conjecture has attracted the attention of physicists and mathematicians over the last decade. It should be emphasized, however, that despite the fact that there are numerous examples from field theory and string theory that support the conjecture, we still lack a general proof of its validity. In the present paper, we prove that the weak gravity conjecture (and, in particular, the mass–charge upper bound m mP < q) can be inferred directly from Bekenstein’s generalized second law of thermodynamics, a law which is widely believed to reflect a fundamen...