Abstract

The mild form of the Weak Gravity Conjecture states that quantum or higher-derivative corrections should decrease the mass of large extremal charged black holes at fixed charge. This allows extremal black holes to decay, unless protected by a symmetry (such as supersymmetry). We reformulate this conjecture as an integrated condition on the effective stress tensor capturing the effect of quantum or higher-derivative corrections. In addition to charged black holes, we also consider rotating BTZ black holes and show that this condition is satisfied as a consequence of the c-theorem, proving a spinning version of the Weak Gravity Conjecture. We also apply our results to a five-dimensional boosted black string with higher-derivative corrections. The boosted black string has a BTZ×S2 near-horizon geometry and, after Kaluza-Klein reduction, describes a four-dimensional charged black hole. Combining the spinning and charged Weak Gravity Conjecture we obtain positivity bounds on the five-dimensional Wilson coefficients that are stronger than those obtained from charged black holes alone.

Highlights

  • The swampland conjecture that is the focus of this paper is the Weak Gravity Conjecture (WGC) [4], which in its original form states that any theory with a U(1) gauge field must include at least one state whose charge-to-mass ratio exceeds that of extremal black holes in that theory

  • Given an extremal charged black hole perturbed by quantum or higher-derivative corrections, it is of interest to understand under what conditions the charge-to-mass ratio increases in a canonical ensemble, such that the mild form of the WGC is satisified

  • As a particular application we evaluated this condition for four-dimensional Reissner-Nördstrom and rotating BTZ black holes perturbed by higher-derivative corrections, but it can be applied to any stationary black hole and more general corrections

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Summary

Loose motivation

Given an extremal charged black hole perturbed by quantum or higher-derivative corrections, it is of interest to understand under what conditions the charge-to-mass ratio increases in a canonical ensemble (fixed temperature and charge), such that the mild form of the WGC is satisified. We are comparing two different black holes (one with and one without higher-derivative corrections) to each other and not having a positive real shift of the outer horizon, i.e. δr < 0, at odds with the WGC only implies that an extremal black hole in the uncorrected theory is not a regular solution in the corrected theory. When this correction to the extremality bound is induced by additional matter (for example heavy matter that is integrated out, generating higher-derivative terms), it is natural to expect that whenever that matter is “healthy” it leads to a correction compatible with the WGC. We show that the correct condition (1.1) takes into account a correction to the stress tensor of the gauge field

Deriving the general relation
BTZ black hole
Reissner-Nordström black hole
Spinning WGC from holographic RG
Example: scalar perturbation
Five-dimensional black string
Four-dimensional black hole
Including higher-derivative corrections
Four-dimensional black hole solution
WGC bounds
Discussion
A Covariant phase space formalism
Notation and conventions
Iyer-Wald formalism
Einstein-Maxwell gravity
B Five-dimensional black string with higher-derivative corrections
Full Text
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