A Holographic Derivation of the Weak Gravity Conjecture

Abstract

The Weak Gravity Conjecture (WGC) demands the existence of superextremal particles in any consistent quantum theory of gravity. The standard lore is that these particles are introduced to ensure that extremal black holes are either unstable or marginally stable, but it is not clear what is wrong if this doesn't happen. This note shows that, for a generic Einstein quantum theory of gravity in AdS, exactly stability of extremal black branes is in tension with rigorously proven quantum information theorems about entanglement entropy. Avoiding the contradiction leads to a nonperturbative version of the WGC, which reduces to the usual statement at weak coupling. The argument is general, and it does not rely on either supersymmetry or a particular UV completion, assuming only the validity of Einsteinian gravity, effective field theory, and holography. The pathology is related to the development of an infinite throat in the near-horizon region of the extremal solutions, which suggests a connection to the ER=EPR proposal.

Figures (6)

• Figure 1: Standard illustration of the Euclidean path integral preparing the charged thermofield double state (\ref{['tfd']}). The Euclidean part of the solution is below the dotted line; its boundary is an interval of length $\beta/2$. The fuzzy lines represent the timelike singularities of the subextremal Reissner-Nordstron black brane. We have omitted the transverse $\mathbb{R}^{d-1}$ factor.
• Figure 2: On the left, conformal diagram of the maximal extension of the sub-extremal black brane. At each point we have ommited the transverse $\mathbb{R}^d$ factor. The solid lines represent the two conformal boundaries where each of the CFT copies in the thermofield double state (\ref{['tfd']}) live. Dashed lines represent black brane (outer and inner) horizons. Beyond the inner horizon lie timelike singularities, represented by fuzzy lines. When the black brane becomes extremal, the outer and inner horizons coincide; the left and right pieces disconnect, and the resulting conformal diagram is depicted on the right.
• Figure 3: Illustration of the Ryu-Takayanagi prescription for computing entanglement entropies. The dotted square represents a regularized version of the boundary, in which the entangling surface $\mathcal{S}$ sits. The RT surface is anchored in $\mathcal{S}$, extends into the bulk, and has minimal area.
• Figure 4: In a system with exponential decay of correlations, one could intuitively expect the degrees of freedom in the bulk of region $\mathcal{R}$ to have almost no entanglement with degrees of freedom in $\mathcal{R}^c$. This would mean that only degrees of freedom in a neighbourhood of the boundary (shaded in blue in the picture) are significantly entangled, leading to an area law.
• Figure 5: Illustration of the different entangling surfaces in the thermofield double. The whole setup is taken at $t=0$. The dotted horizontal lines represent the two boundaries, and the solid black line in the middle represents the black brane horizon. The entangling surface consists of two identical pieces, one in each boundary. The red entangling surface is similar to two copies of what one would have in the one-sided black hole, while the blue one goes through the horizon.