The Weak Gravity Conjecture from Entropy

Generalized entropy and endpoint regularity.

Let $\Sigma \left(\lambda \right)$ be a family of null cuts along a complete horizon generator with affine parameter $\lambda$. We assume that

$$S_{\mathrm{gen}}\left(\lambda \right)={\displaystyle \frac{A\left(\lambda \right)}{4G}}+S_{\mathrm{out}}\left(\lambda \right)$$

(3)

is well defined and twice differentiable in the distributional sense under smooth null deformations. Thus all statements involving $S_{\mathrm{out}}^{\prime \prime }$ are understood after smearing against nonnegative test functions of compact support. We also assume that QNEC, Eq. (2), holds as a distributional inequality along the generator.

Definition 1 

(Regular QNEC endpoint). The endpoint $\lambda =\infty$ is called a regular QNEC endpoint, within a fixed geometric-affine modular class, if the following data extend to the endpoint: (i) the geometric null-cut family $\Sigma \left(\lambda \right)$ with λ as affine parameter; (ii) the associated exterior von Neumann algebras $\mathcal{A}\left(\lambda \right)$ with a normal limiting state and modular data; (iii) the relative-entropy functional matched to $S_{\mathrm{gen}}$ up to state-independent local counterterms; and (iv) the local semiclassical stress tensor, with $\langle T_{kk}\rangle$ locally bounded on arbitrarily late finite segments. In this class QNEC continues to hold and the area term has no positive endpoint curvature singularity.

Remark 1 

(Inherited endpoint class). In applying Definition 1 to an extremal horizon, the fixed geometric-affine modular class is the class inherited from the one-sided non-extremal evolution approaching extremality. A regular endpoint is an extension of the same null-cut algebraic system $\{\Sigma \left(\lambda \right),\mathcal{A}\left(\lambda \right),{\Omega }_{\lambda },{\Delta }_{\lambda }\}$, not a replacement of that system by a different modular realization defined only at the endpoint. In particular, allowing the extremal horizon to choose an independent affine-translation modular class would not provide a regular endpoint of the non-extremal entropy flow; it would change the modular problem at $\lambda =\infty$. The obstruction in Lemma 1 therefore concerns precisely the failure of the non-extremal Kay–Wald/Bisognano–Wichmann/HSMI realization to extend continuously to the extremal boundary of the one-sided charged state space.

Assumption A1 

(Semiclassical endpoint regularity). A complete null generator that remains in the semiclassical regime, has no geometric singularity at finite affine parameter, and has locally bounded $\langle T_{kk}\rangle$ on arbitrarily late finite segments admits a regular QNEC endpoint in the sense of Definition 1.

Remark 2 

(Semiclassical motivation for Assumption A1). Assumption A1 is the endpoint analogue of the usual semiclassical regularity requirement used in generalized-entropy arguments. The generalized entropy is constructed so that the short-distance divergences of $S_{\mathrm{out}}$ are absorbed into local gravitational couplings, leaving a finite renormalized entropy functional for smooth cuts. Moreover, QNEC is naturally formulated as a distributional statement about null deformations of this renormalized entropy functional, and algebraic derivations relate these null variations to relative entropy and modular flow.

The assumption is not that every fixed-background extremal solution automatically supplies a regular endpoint. Rather, it is a regularity criterion for the semiclassical entropy evolution considered here: if a null generator remains complete and nonsingular, the local stress tensor is bounded on arbitrarily late finite segments, and no Planckian or finite-affine-parameter geometric singularity is encountered, then a finite modular termination of the QNEC entropy flow should be regarded as a failure of endpoint extendibility, not as an admissible regular endpoint. Assumption A1 therefore excludes such terminal endpoints as allowed endpoints of semiclassical entropy evolution within the same geometric-affine null-cut class.

This expectation is also compatible with entropy-based singularity theorems. Bousso and Shahbazi-Moghaddam showed, assuming the covariant entropy bound, that if the inward lightsheet of a hyperentropic region initially contracts, then at least one null generator is incomplete [27]. This suggests that genuine entropy obstructions in semiclassical gravity are naturally associated with null incompleteness or a breakdown of semiclassical evolution, rather than with a finite-entropy modular endpoint along an otherwise complete and nonsingular generator. In this sense, Assumption A1 is the corresponding regularity condition in the present QNEC/modular setting.

Theorem 1 

(Entropy–modular Weak Gravity theorem). Assume QNEC, the endpoint regularity principle of Assumption A1, and local boundedness of $\langle T_{kk}\rangle$ on arbitrarily late finite segments of complete null generators. Then the charged spectrum contains a state satisfying

$${\displaystyle \frac{q}{m}}\ge 1.$$

(4)

Equivalently, a purely subextremal charged sector is inconsistent with QNEC-compatible semiclassical entropy regularity.

Lemma 1 

(Kay–Wald/Bisognano–Wichmann identification and extremal obstruction). Let ${\left\{\mathcal{A}\left(\lambda \right)\right\}}_{\lambda \ge 0}$ be the exterior algebras associated with a nested family of null cuts $\Sigma \left(\lambda \right)$ along a fixed horizon generator. Suppose that the state Ω is locally Hadamard and cyclic separating for these algebras, and that the null-cut family admits a half-sided modular inclusion (HSMI) realization. Thus there are von Neumann algebras $N\subset M$, a cyclic separating vector Ω, and a strongly continuous unitary representation

$$U\left(\lambda \right)=e^{i\lambda P},P\ge 0,$$

(5)

such that

$$\mathcal{A}\left(\lambda \right)\cong M\left(\lambda \right)=U\left(\lambda \right)MU{\left(\lambda \right)}^{*},$$

(6)

and the modular group of $(M,\Omega )$ obeys the Borchers relation

$${\Delta }_M^{it}U\left(\lambda \right){\Delta }_M^{-it}=U\left(e^{-2\pi t}\lambda \right).$$

(7)

Let the horizon be stationary with Killing parameter v and surface gravity κ. For $\kappa >0$, the Kay–Wald KMS relation and the local Bisognano–Wichmann theorem identify the modular flow with Killing flow,

$${\sigma }_t^{\Omega }\left(A\right)={\Delta }^{it}A{\Delta }^{-it}={\alpha }_{-{\beta }_{H}t}\left(A\right),{\beta }_H={\displaystyle \frac{2\pi }{\kappa }},$$

(8)

where ${\alpha }_s$ denotes Killing evolution. This non-extremal geometric identification cannot be continued to $\kappa =0$ while preserving the same affine cut parameter and the same HSMI modular action. At extremality, the Kay–Wald/Bisognano–Wichmann identification of modular flow with Killing flow fails for algebraic, not merely coordinate, reasons.

Proof. 

For a non-extremal horizon,

$$\lambda =C{e}^{\kappa v},C>0.$$

(9)

A Killing translation $v\mapsto v+s$ therefore acts as

$$\lambda \mapsto e^{\kappa s}\lambda .$$

(10)

Using the KMS normalization $s=-{\beta }_{H}t=-2\pi t/\kappa$, this is precisely the HSMI dilation $\lambda \mapsto e^{-2\pi t}\lambda$ appearing in Eq. (7). Thus, on a non-extremal bifurcate Killing horizon, the modular flow is geometrically the Killing flow in the KMS sense.

At extremality, $\kappa =0$, the Killing generator is affinely parametrized on the horizon. Up to an affine rescaling and shift,

$$\lambda =av+b,a\ne 0.$$

(11)

The Killing flow therefore acts by translations of the affine cut parameter,

$$\lambda \mapsto \lambda +as.$$

(12)

If the same geometric modular identification survived at extremality, the modular automorphism group would have to be both the Killing translation on the affine cut parameter and the HSMI dilation of Eq. (7). Infinitesimally, the same modular vector field would have to be identified with both

$$a{\partial }_{\lambda }\mathrm{and}-2\pi \lambda {\partial }_{\lambda }.$$

(13)

These are not related by any constant KMS normalization on a nonempty interval of $\lambda$. The contradiction is not that translations and dilations cannot coexist abstractly; the affine group and the Borchers relations precisely allow them to coexist. The contradiction is that the same modular flow cannot simultaneously be the extremal Killing translation and the HSMI dilation on the same affine null-cut parameter. Hence the non-extremal geometric modular identification terminates at extremality. □

Definition 2 

(Geometric-modular terminality). An endpoint is geometrically modular terminal if there is no nontrivial continuation of $\left\{\mathcal{A}\right(\lambda \left)\right\}$ beyond it that preserves the same null-cut family $\Sigma \left(\lambda \right)$, the same affine parameter λ, and the Kay–Wald/Bisognano–Wichmann identification of modular flow with horizon Killing flow in the HSMI realization.

Lemma 2 

(Extremal terminality excludes a regular QNEC endpoint). For an extremal horizon satisfying the hypotheses of Lemma 1, the endpoint $\lambda =\infty$ is geometrically modular terminal and therefore cannot be a regular QNEC endpoint in the sense of Definition 1.

Proof. 

Lemma 1 shows that the geometric-affine modular realization of the non-extremal horizon cannot be continued to an extremal horizon while preserving the affine cut parameter and the modular/Killing identification. This is precisely geometric-modular terminality.

Suppose nevertheless that $\lambda =\infty$ were a regular QNEC endpoint. By Definition 1, the same null-cut family, exterior algebra family, normal limiting state, and modular data would extend to the endpoint while maintaining the relative-entropy/generalized-entropy matching in the same geometric-affine modular class. That is exactly the continuation excluded by terminality. Hence an extremal terminal endpoint cannot be regular. □

Proposition 1 

(QNEC excludes the curvature-singular branch). Let $S_{\mathrm{out}}^{\prime \prime }$ denote the second null variation of the outside entropy in the distributional sense. If $\langle T_{kk}\rangle$ is locally bounded on arbitrarily late finite segments and QNEC holds, then $S_{\mathrm{out}}^{\prime \prime }$ cannot develop an upward distributional divergence at late affine parameter. In particular, for every compact interval I on which $\langle T_{kk}\rangle \le M$ almost everywhere and every nonnegative $\phi \in C_c^{\infty }\left(I\right)$ with $\int \phi =1$,

$$\langle S_{\mathrm{out}}^{\prime \prime },\phi \rangle \le 2\pi M.$$

(14)

Proof. 

Testing QNEC, Eq. (2), against $\phi$ gives

$$\langle S_{\mathrm{out}}^{\prime \prime },\phi \rangle \le 2\pi \langle \langle T_{kk}\rangle ,\phi \rangle \le 2\pi M.$$

(15)

Thus positive distributional unboundedness of $S_{\mathrm{out}}^{\prime \prime }$ is incompatible with QNEC whenever the null stress tensor is bounded on the segment used to test the endpoint. □

Proof of Theorem 1.

Assume, for contradiction, that all charged states in the semiclassical charged sector satisfy $q/m<1$. Consider an extremal charged horizon with extremality parameter $\epsilon =M-Q$ in units where the extremal bound is $M=Q$. Any admissible charged excitation crossing the horizon obeys

$$\delta M-\delta Q\ge 0$$

(16)

under the purely subextremal assumption. Hence $ϵ$ cannot be driven below zero by any such semiclassical process. The extremal surface $ϵ=0$ is therefore a one-sided endpoint of the semiclassical charged state space generated by these perturbations. By endpoint regularity we require regularity in this inherited null-cut modular class, since the extremal surface is reached as the boundary of the one-sided semiclassical charged evolution.

By Lemma 1, the geometric-affine modular realization of null entropy variations becomes terminal at this extremal endpoint. By Lemma 2, a terminal extremal endpoint cannot be a regular QNEC endpoint. But Assumption A1 requires precisely such a regular endpoint for complete nonsingular null generators that remain semiclassical and have locally bounded $\langle T_{kk}\rangle$ on arbitrarily late finite segments. The alternative in which endpoint failure is represented by an upward divergence of $S_{\mathrm{out}}^{\prime \prime }$ is excluded by Proposition 1. Thus the purely subextremal charged sector is inconsistent with QNEC-compatible semiclassical entropy regularity. The contradiction is avoided only if the charged spectrum contains a state with $q/m\ge 1$.

Discussion.

The result identifies the WGC as an endpoint-regularity condition on semiclassical entropy evolution. The role of superextremal states is not merely to provide a decay product for extremal black holes; it is to prevent extremal charged horizons from becoming one-sided modularly terminal boundaries of the null entropy flow. The proof is local in the sense that it uses null cuts, relative entropy, modular flow, and QNEC along complete generators, rather than a global decay channel or asymptotic holography.

This also clarifies the relation to previous entropy arguments. Earlier work used higher-derivative corrections to black-hole entropy and showed that entropy positivity shifts the extremality bound in the superextremal direction [17,18]. Other approaches use string modular invariance, unitarity, causality, or amplitude positivity to constrain the same charged spectrum [19,20,21,22]. The present argument instead uses the algebraic modular structure of the horizon entropy functional itself. The obstruction appears exactly at extremality because the non-extremal KMS relation turns Killing translations into affine dilations, while the extremal Killing flow translates the affine parameter. This makes the modular endpoint structure of extremal horizons sensitive to whether the charged sector is purely subextremal.

The assumptions used here are semiclassical: generalized entropy must be well defined under null deformations, QNEC must hold in the distributional sense, and complete nonsingular null generators with bounded local stress must admit regular QNEC endpoints. Under these assumptions, a purely subextremal charged sector cannot coexist with entropy extendibility and modular consistency. In this sense, the WGC follows from the requirement that semiclassical quantum information have no finite-entropy modularly terminal boundary along complete null directions. Further details on the modular normalization, endpoint regularity, distributional QNEC, an extremal Reissner–Nordström endpoint diagnostic, and the logical form of the theorem are given in the Supplemental Material.