Entropy Bounds on Effective Field Theory from Rotating Dyonic Black Holes

We derive new bounds on higher-dimension operator coefficients in four-dimensional Einstein-Maxwell theory. Positivity of classically-generated corrections to the Wald entropy of thermodynamically stable, rotating dyonic black holes implies a multiparameter family of field basis invariant inequalities that exhibit electromagnetic duality and are satisfied by examples from field and string theory. These bounds imply that effective operators modify the extremality condition of large black holes so as to permit their decay to smaller ones, thus satisfying the weak gravity conjecture.

Introduction.-The swampland program [1][2][3] is founded on the intriguing possibility that an infrared (IR) consistent effective field theory (EFT) may nevertheless be incompatible with quantum gravitational ultraviolet (UV) completion. An archetype of this approach is the weak gravity conjecture (WGC) [3], which states that any Abelian gauge theory coupled consistently to gravity must contain a state whose charge exceeds its mass in Planck units. Other notable results from the swampland program include the distance conjecture [2], and AdS [4] and dS [5,6] conjectures.
The very notion of the swampland is predicated on a complete knowledge of the space of IR consistent EFTs. After all, if a putative swampland theory is also pathological from IR considerations, then UV completion has little to do with its sickness. For example, gauge anomalies are absent in grand unified theories, but they are also ruled out by low-energy reasoning. In parallel with the swampland program, there has emerged a bottom-up approach to constraining EFTs using IR properties like unitarity, causality, and analyticity of scattering amplitudes [7]. These tools can be used to derive new positivity conditions on higher-dimension operator coefficients in EFTs [7][8][9][10][11], in some cases with broad implications, e.g., the proof of the four-dimensional a-theorem [12].
At the same time, the study of black hole thermodynamics has produced many elegant and powerful constraints on entropy with important implications for both the IR and UV [13][14][15]. In this paper we bring together these three lines of inquiry-the swampland program, EFT bounds, and entropy inequalities-to use black hole entropy to constrain the landscape of consistent EFTs.
We consider rotating dyonic black holes in the EFT of gravitons and photons in four spacetime dimensions, e-mail: clifford.cheung@caltech.edu, jliu2@caltech.edu, grant.remmen@berkeley.edu with action´d 4 x √ −g(L + ∆L), where the Einstein- in the sign conventions of Ref. [16] and where κ 2 = 8πG. This action encodes all leading-order parity-conserving interactions of gravitons and photons at low energies. In the absence of charged currents, any operator with derivatives on F µν can be recast into those already in Eq. (1) using the Bianchi identities [17].
In Ref. [18] it was proven that when the higherdimension operators in Eq. (1) are generated at tree level by quantum field theoretic dynamics, they induce a positive shift to the Wald entropy of a thermodynamically stable black hole at fixed mass and charge, This bound applies for black holes sufficiently small so that the classical Wald entropy shift dominates over quantum corrections, which is always achievable within the regime of validity of a weakly coupled EFT [18]. The physical origin of positivity follows from an intimate linkage between the shift in entropy and the variation of the Euclidean action away from its local minimum, which is positive for thermodynamically stable systems. The extensive details can be found in Ref. [18]. As shown in Ref. [19], higher-derivative operators in the EFT modify the extremality condition for black holes. In Ref. [18] it was shown that ∆S > 0 implies constraints on c i that precisely tip the extremality condition so that large electrically charged black holes are unstable to decay to smaller ones. Consequently, the WGC is automatically satisfied since there exist states-namely, black holes-with charge-to-mass ratio greater than unity. More generally, Ref. [18] showed that ∆S > 0 implies the WGC for any number of Abelian forces and in arbitrary spacetime dimension ≥ 4.
In this paper we consider four-dimensional rotating dyonic black holes, generalizing the results of Ref. [18] to obtain a family of constraints on the coefficients c i , labeled by the angular momentum and electric and magnetic charge-to-mass ratios of the black hole. As before, ∆S > 0 exactly guarantees that the extremality condition tips so that large black holes are unstable, thus establishing a dyonic rotating version of the WGC.
Metric and Action.-To begin, we compute the Wald entropy for a black hole of fixed electric charge Q, magnetic charge Q, angular momentum J, and mass M at leading order in the higher-dimension operator coefficients c i in Eq. (1). For later convenience we introduce natural quantities m = κ 2 M/8π, q = κQ/4 √ 2π, q = κ Q/4 √ 2π and rescaled coefficients d 1 The unperturbed Kerr-Newman (KN) metric g µν in Boyer-Lindquist coordinates is given by where ρ 2 = r 2 + a 2 cos 2 θ, a = J/M = κ 2 J/8πm, and ∆ = r 2 − 2mr + a 2 + q 2 + q 2 . It will be convenient to define ξ = m 2 − (q 2 + q 2 + a 2 )/m, so that ξ → 0 is the extremal limit. The event horizon is located at rcoordinate r H = m(1 + ξ). We also define parameters µ = (q 2 − q 2 )/(q 2 + q 2 ) and ν = a/r H to characterize the charge and spin, so µ = 1 and µ = −1 correspond to pure electric and pure magnetic black holes, respectively, ν = 0 to nonrotating charged black holes, and ν = (1 − ξ)/(1 + ξ) to uncharged rotating black holes. Next, we employ a relation between entropy and the Euclidean action that was discussed in Ref. [18] and recently generalized and proven in Ref. [20] via a simple argument from basic thermodynamic identities: where ∆S is the entropy shift at fixed mass, charge, and angular momentum, and ∆I denotes the higherdimension operator contributions to the action evaluated on the leading-order black hole solution.
We will compute the entropy shift ∆S to O(d i ) using Eq. (4) for a rotating dynonic black hole. Since KN is a stationary spacetime with Killing vector ∂ t , we have ∆I = −β´d 3 x √ −g∆L| KN , with the right-hand side evaluated on the Lorentzian KN black hole background at fixed t, where the integration is over all (r, θ, φ) outside the horizon and β is the inverse temperature.
Note that the Gauss-Bonnet term L GB = R 2 − 4R µν R µν + R µνρσ R µνρσ corrects black hole entropy [21] even though it is a total derivative in four dimensions. This is not contradictory since the black hole horizon provides a boundary on which total derivatives have support. This is similar in spirit to what happens to F µν F µν in electrodynamics, which is a total derivative but comes into play in the background of a magnetic monopole.
Nevertheless, one can show that are no other total derivative or boundary operators that correct the entropy at the leading order of interest. For example, R and (F µν F µν ) do not affect the Wald entropy nor the equations of motion. Meanwhile, higher-derivative boundary terms at asymptotic infinity, analogous to the Gibbons-Hawking-York term, are subleading in 1/r at large distances and can therefore be ignored [18,20].

Black Hole Entropy and Consistency
Checks.-Any physical constraint on the action must be invariant under change of field basis, e.g., the metric transformation Consequently, a given coefficient d i is not individually meaningful because it mixes with others under a field redefinition. The four field-redefinition invariant linear combinations of coefficients are (d 0 , d 3 , d 6 , d 9 ), where [18] Note that Eq. (4) is automatically field-basis invariant because those combinations of higher-dimension operators that are removable by a field redefinition are proportional to equations of motion and thus vanish when evaluated on the leading-order solutions. We find that ∆S is indeed field-basis invariant:  Table I, d3 = d6 = 0 and allowed values of d0,9 are indicated by the black dashed lines in the leftmost panel.
While Eq. (6) carries a formal divergence at ξ = 0, as shown in Ref. [18] we can take the near-extremal, i.e., small-ξ, limit consistent with perturbative control. Requiring that the near-extremal shifts in entropy and temperature be much smaller than their unperturbed values mandates ξ κΛ, but κΛ is parametrically small for a field-theoretic completion. For later convenience we will sometimes use ξ = 0 to indicate this limiting value.
As an independent check of our calculation, we have used perturbative techniques [19,22] applicable to spherically-symmetric black holes to solve for the metric itself at leading order in the d i , making use of the invertibility of the Ricci tensor for spherical geometries; see the appendix for details. Inserting the metric into the Wald formula, we obtain ∆S for the ν = 0 case, and we find that it agrees with Eq. (6). As a further consistency check, we have verified that the surface gravity at the horizon in the perturbed metric agrees with the thermodynamic temperature extracted from differentiating the Wald entropy with respect to the mass.
EFT Bounds.-We now show how ∆S > 0 constrains the EFT. As discussed earlier, this inequality holds for black holes that are thermodynamically stable. For a general KN black hole, the requirement of positive heat capacity at constant J is not enough to guarantee thermodynamic stability. We must also require positive isothermal mo-ment of inertia [23]. This enhanced stability condition restricts our consideration to In the ν = 0 nonrotating case, this window reduces to ξ ∈ (0, 1 2 ), i.e., q 2 + q 2 > 3m 2 /4. The stability condition in Eq. (7) cannot be satisfied in the neutral case, where ν = (1 − ξ)/(1 + ξ). Thus, we cannot use neutral Kerr black holes to place a bound because they are thermodynamically unstable. In the ξ → 0 extremal limit, Eq. (7) reduces to ν ∈ [0, 1/ √ 3), i.e., q 2 + q 2 > 2a 2 . As discussed earlier, the entropy shift satisfies ∆S(ξ, µ, ν) > 0 for all (ξ, ν) satisfying Eq. (7) and all µ ∈ [−1, 1], which is the main result of this work. This condition implies new bounds on the EFT of gravitons and photons mandated by the entropy of rotating dyonic black holes.
Numerical studies indicate that the parameter space excluded by Eq. (8) is already forbidden by the special case ν = 0, i.e., the bounds from rotating black holes are already accounted for by the nonrotating case. See Fig. 1 for examples of excluded regions, which we now discuss.
Furthermore, d 3 is bounded from below by d 0 and d 9 .  Table I. Higher-dimension operator coefficients di and black hole entropy shift ∆S induced at low energies in various field theory UV completions defined by a massive particle φ that can be either a scalar, pseudoscalar, or tensor field, as well as the heterotic string. In the former, the Lagrangian is L UV = L + L kin + Lint where L kin is the canonical kinetic term for φ.
The tensor field has a Fierz-Pauli mass term and couples to the energy-momentum tensor, T µν = FµρF ρ ν − 1 4 gµν FρσF ρσ . All examples produce positive ∆S for any choice of parameters. lower bound on d 3 is expected in light of the unitarity argument for positivity of d 3 given in Ref. [11].
As illustrated in the last panel of Fig. 1, regions outside of the allowed window in d 6 are excluded since there exist values of ξ and µ such that ∆S(ξ, µ, 0) < 0 for any value of d 3 . We can obtain this bound analytically by taking ∆S(0, µ, 0) and marginalizing over µ, yielding assuming both d 0 and d 9 are nonzero. If d 0 and d 9 are both generated at Λ, |d 6 | larger than ∼ 1/Λ 2 is forbidden. A similar conclusion, disallowing d 6 without the appearance of heavy states at the scale |d 6 | −1/2 , was reached using causality in Ref. [24].
UV Examples.-As a consistency check, we calculate the operator coefficients d i for various tree-level UV completions and verify that they obey the inequalities in Eq. (8). These include UV completions with a massive scalar or pseudoscalar coupled to the electromagnetic field strength and gravitational curvature tensor in all ways consistent with parity, as shown in Table I. We also consider a UV completion with a massive tensor field with a ghost-free Fierz-Pauli mass term and minimal coupling to the energy-momentum tensor. It is known that such a theory can have a cutoff parametrically higher than the mass of the tensor field [25]. In principle, other interactions with the tensor field could be possible, but such theories have not been well studied and we therefore have no reason to expect a consistent EFT, so such a generalization would not constitute a well defined check of our entropy bounds. For example, a coupling such as φ µ µ F νρ F νρ would violate bounds coming from amplitude analyticity as well as entropy.
As shown in Table I, the entropy shift is manifestly positive in our example scalar, pseudoscalar, and tensor completions, providing a check of our bounds. In these examples, only d 0 and d 9 are generated. See the first panel of Fig. 1 for the parameter space spanned by these field theory UV completions.
Finally, we considered the low-energy effective theory of the heterotic string [19,26], whose higher-dimension operators are not generated by quantum fields but nonetheless satisfy our bounds, as shown in Table I. Here we have ignored the presence of the dilaton, but if this state is stabilized below the string scale, then its dynamics will be encoded in the massive scalar UV completion.
Weak Gravity Conjecture.-Our entropy bounds are intimately connected to the WGC. We define the extremality parameter ζ = a 2 + q 2 + q 2 /m = 1 − ξ 2 . In pure EM theory, black holes are described by the KN metric, which is free from naked singularities provided ζ ∈ [0, 1]. The higher-dimension operators in Eq. (1) correct the equations of motion, changing the allowed range of physical values to ζ ∈ [0, 1+∆ζ], as shown for the case of nonrotating electric black holes in Ref. [19]. As in Ref. [18], we calculate the extremality shift by writing g rr in terms of g rr and the shifts in various parameters. Since ∂ θ g rr = ∂ µ g rr = ∂ ν g rr on the KN horizon and ∂ r g rr = 0 for the extremal horizon, we find that the extremality shift is ∆ζ = −∆g rr /∂ ζ g rr | r H , evaluated on r H = m(1 + ξ). Similarly, for a black hole of fixed (m, a, q, q), the shift in the horizon is ∆r H = −∆g rr /∂ r g rr | r H .
In the extremal limit, the ∆r H term in the entropy shift dominates, since ∂ r g rr → 0, so we have ∆S = (2π/κ 2 )∆r H ∂ rĀ , whereĀ = 4π(r 2 H + a 2 ) is the area of the KN black hole. Using the fact that ∂ ζ g rr /∂ r g rr | KN,r=r H = mζ/ξ, we directly relate the shift ∆ζ in the extremality parameter to the entropy shift, Sans rotation, we find that ∆ζ ∝ +d 0 for the electric black hole [18], ∆ζ ∝ +d 0 + 2d 6 for the magnetic black hole, and ∆ζ ∝ +d 9 for q = q.
Let us now consider the decay of a black hole to lighter states with masses and charges m i , q i , q i in natural units and angular momenta j i = 8πm i a i /κ 2 including both intrinsic spin and orbital components. Charge conservation implies that i (q i , q i ) = (q, q), while conservation of energy requires i m i < m, where the inequality is needed if there is to be nonzero phase space for the decay products. Angular momentum conservation requires J = aM ≤ i j i , where the inequality is due to the possibility of misaligned angular momenta. Defining unitless ratios z = q/m, z = q/m, and α = amκ −2 = J/8π, the set of KN black holes free of naked singularities forms a spheroid, ζ 2 = z 2 + z 2 + (α 2 κ 4 /m 4 ) ≤ 1, depicted in Fig. 2. Defining the analogous ratios for the decay products, Putting this all together to define w = (z, z, α), we conclude that a state with vector w must decay to states w i for which w is either in the convex hull of the w i or strictly between that convex hull and the α = 0 plane.
The bound in Eq. (8) implies by Eq. (10) that ∆ζ(µ, ν) > 0 for all ν ∈ [0, 1/ √ 3). Thus, extremal black holes located in the barrel-shaped region with |a|/m = |α|κ 2 /m 2 < 1/ √ 3 can decay directly to other rotating black holes. The lighter rotating dyonic black holes form a closed surface everywhere outside the barrel-shaped region, so the tower of stable large black holes collapses to the scale at which the EFT breaks down. In this sense, the WGC bounds implied by extremality corrections point to a principle of black hole self-sufficiency in providing their own decay channel. While our ∆S > 0 bounds do not apply to black holes with ν > 1/ √ 3, this is due to the fact that such states are known to be already thermodynamically unstable, even in the extremal limit [23]. In particular, these objects can shed their spin by emitting gravitons in states with nonzero angular momentum, decaying to the α = 0 plane. For nonrotating black holes, our entropy bound implies a dyonic version of the WGC: the set of near-extremal dyonic black holes can decay because they form a closed curve slightly outside the unit circle in the α = 0 plane.
If the leading higher-dimension operators in Eq. (1) have vanishing coefficients then the entropy will be corrected by yet-higher-derivative operators, which are still constrained by our reasoning since Eq. (10) implies that ∆S ∝ ∆ζ in the extremal limit. We leave such an analysis to future work [27]. On the other hand, if ∆S strictly vanishes to all orders then by definition there is a flat direction in the Euclidean action, indicating a symmetry protecting the states. While such a symmetry is not present for the rotating dyonic black holes considered here in EM theory, it would be interesting to see how supersymmetry or massless moduli alter this picture. z e z ↵ Figure 2. The cyan spheroid z 2 + z 2 + α 2 κ 4 /m 4 ≤ 1 denotes the space of all possible black holes without naked singularities. The blue barrel-shaped region incorporates the additional criterion |α| < m 2 / √ 3κ 2 , which defines the subspace of black holes that, due entropy bounds, can decay directly to lighter, extremality-corrected black holes. The gray disk represents all nonrotating dynoic black holes.
Appendix: Dyonic Metric Perturbation.-In the nonrotating case, we can use a spherically-symmetric ansatz for the black hole metric, Following Refs. [19,22], we can write f (r) and g(r) as integrals over the perturbed energy-momentum tensor T µν + ∆T µν that appears on the right-hand side of the perturbed Einstein equation, R µν − 1 2 Rg µν = κ 2 (T µν + ∆T µν ). The shift ∆T µν can be written as the sum of two contributions, one coming from the metric variation of ∆L and one coming from the change to the field strength and metric in the original Maxwell stress tensor.