Constraints on low-energy effective theories from weak cosmic censorship

We examine the weak cosmic censorship conjecture (WCCC) for the extremal charged black hole in the possible generalizations of the Einstein-Maxwell theory due to the fourth-derivative higher order corrections. Our derivation is based on Wald's gedanken experiment to destroy an extremal black hole. We find that WCCC no longer holds for all the possible generalizations. Thus, WCCC can serve as an additional constraint to the swampland conjecture. However, our constraint is independent of photon's self-interactions so that the precision measurement of quantum electrodynamics cannot constrain WCCC. For the higher-dimension operators induced by the one-loop correction for the minimally coupled spinor and scalar to gravity, our constraint is satisfied.


Introduction and Sumamry
Weak cosmic censorship conjecture (WCCC) [1] has been proved recently in [2] for Einstein-Maxwell theory via Wald's gedanken experiment of throwing matters into the extremal or near-extremal Kerr-Newman black hole 1 . This ensures the naked singularity is always hidden behind the horizon and avoids the philosophical challenge to the validity of Einstein gravity. Especially, this result may also be related to the fact that the trajectory of constant entropy meets tangentially with the boundary of the extremality bound, which for a non-rotating black hole of mass where m ≡ M/4π, q ≡ Q/4π and κ = 8πG N .
However, from the effective theory point of view, the Einstein-Maxwell theory should be subjected to higher order corrections due to quantum effect. To be concrete, we consider the most general fourth-derivative higher order corrections to Einstein-Maxwell theory, namely, where 2 ∆L = c 1 R 2 + c 2 R µν R µν + c 3 R µνρσ R µνρσ + + c 4 RF µν F µν + c 5 R µν F µρ F ν ρ + c 6 R µνρσ F µν F ρσ + + c 7 F µν F µν F ρσ F ρσ + c 8 F µν F νρ F ρσ F σµ .
We will assume c i 's are small and restrict our consideration to O(c i ). In [5] it was shown that the extremality bound is changed from (1) to the following where 1 See also [3,4] for earlier attempts and discussions. 2 We have neglected terms proportional to ∇ µ F µρ ∇ ν F νρ , as it does not affect the black hole metric or our parameter bound. Further note that terms like (∇ µ F νρ )(∇ µ F νρ ) and (∇ µ F νρ )(∇ ν F µρ ) can be recasted (up to some constant factor) into ∇ µ F µρ ∇ ν F νρ plus existing terms in ∆L and an additional boundary term, upon using Bianchi identities, Ricci identities, and integrating by parts [6].
Despite the extremality condition is changed, by some argument 3  This is what we will examine in this paper by adopting the method developed in [2,7]. The key procedure can be sketched as follows. Firstly we obtain the first law of black hole mechanics based on the Noether charge method of [8] for constructing the following first-order variational identity for higher theory (2): where M is the ADM mass of black hole, Φ H ≡ − (ξ a A a ) | H electromagnetic potential on the horizon H, with abcd the spacetime volume form, ξ a the time-like Killing vector, A a the Maxwell gauge field, and δj a and δT e a are respectively the charge current and stress tensor of the matter falling into the horizon. The right-hand side should be positive if the matter obeys the positive energy condition. Secondly, based on the perturbative solutions of extremal black holes up to the first order of c i 's for the higher theory (2), we will evaluate δM, Φ H and δQ ≡ H abcd δj a which can be thought as the variation of black hole's charge caused by the in-falling matter, so that (6) turns into an inequality relating the variations of mass and charge of the extremal black hole due to the falling matters, which explicitly depends on c i 's. Finally, compare this inequality with the condition for the extremal black hole of the higher theory (2), we arrive the main result of this paper, i.e., the condition to preserve the WCCC is Our result implies that WCCC does not hold for all the higher order theories.
This then raises an issue if WCCC should be taken as a criterion for a consistent theory of quantum gravity, to which (2) is considered as a low energy effective theory. If not, we should face the issue of dealing with the unphysical curvature divergence of the naked singularity. If yes, this is quite similar to the swampland conjecture [9,10] introduced with the same purpose here. One concrete example of swampland is based on weak gravity conjecture (WGC) [10] which states that the gauge force must be stronger than gravity. This conjecture is motivated by the assumption of finiteness of the number of stable particles not protected by a symmetry principle, which requires the the minimum of the ratio of mass to charge is less than one, i.e., m < 2 κ |q|. When invoking this conjecture to the extremal black holes, it implies the extremality condition cannot be m = 2 κ |q|. Instead, one shall expect the higher-order quantum corrections should turn it into m < 2 κ |q|, which then gives the following constraint [5]: Recently the same constraint is obtained in [11] by requiring that the higher-order correction to the entropy of extremal black hole must be positive.
It is interesting to see that (7) is quite different from (8), especially in (7) (7) can serve as the constraints on these interactions and as an additional constraint to the swampland conjecture [9] besides (8). Below we will elaborate our derivation of (7) and then check some examples.

Charged black hole in higher theory
The field equations obtained by the variation of the action (2) with respect to A µ and g µν are given respectively by and In the aboveT µν = F µ ρ F νρ − 1 4 g µν F ρσ F ρσ is the stress tensor of the Maxwell theory, and ∆T µν and S µν are the corrections respectively to the stress tensor and Maxwell source field from the higher-dimension operators, and the details are given in Appendix A.
Next, we follow Ref. [5] to solve (9) and (10) for the O(c i ) corrections to the Reissner-Nordström black hole utilizing the spherical symmetry. A general static metric with spherical symmetry can be written as ds 2 = −e ν(r) dt 2 + e λ(r) dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) (11) and also assuming the boundary condition that this metric approaches the Schwarzschild metric at r → ∞. Then, after some manipulations the functions λ(r) and ν(r) sat- We take the background to be the Reissner-Nordström black hole of Einstein-Maxwell theory, The condition for the extremal black hole of the metric (13) is to saturate the extremality bound of (1), i.e., which is also the location of the degenerate horizon.
Using the Maxwell source field S µν and stress tensor with the higher corrections ∆T µν listed in Appendix A, we can solve the O(c i ) corrections to the Reissner-Nordström background. We have elaborated the solving procedure in Appendix B, and the corrected Maxwell gauge field is The corrected metric is also obtained as listed in (50) of Appendix B, here we just show the tt-component, Note that in (16) there is no O(c 1 ) correction. From (16) we can solve the position of the degenerate horizon, and derive the condition of extremal black hole in the higher theory. Moreover, from the corrected metric one can derive the new mass-charge relation for extremal black holes up to O(c i ) correction by finding the linearized solution to the double-root condition of e ν = 0. This then yields the condition by saturating the extremality bound of (4) [5], i.e., with c 0 given by (5).

Gedanken experiments and weak cosmic censorship
We now consider Wald's gedanken experiments to destroy the quantum gravity corrected extremal black hole that we obtained in the last section. We throw charged matter into the black hole and test whether or not it will be overcharged. For simplicity, we assume that the in-falling matter finally turns the original extremal black hole (16) of the higher theory into a one-parameter family solutions of the same theory but with mass m(τ ) and charge q(τ ). This then turns the extremality condition (18) into a discriminant function When f (τ ) ≥ 0, the spacetime is a black hole, otherwise it becomes a naked singularity.
Assuming small perturbation due to in-falling matter, i.e., m(τ ) = m + τ δm + O(τ 2 ) and q(τ ) = q + τ δq + O(τ 2 ) with τ << 1, where the mass and charge of the unperturbed extremal black hole of higher theory are denoted by m and q, respectively. Then, to the first order in τ , the discriminant function can be . Therefore, the perturbed spacetime is still a black hole, thus WCCC holds only if the first order perturbations satisfy

The linear variational identity
Next we need to study the first law of black hole mechanics, based on which we can see if the superextremality condition (20) can be satisfied or not. To proceed we follow the Noether charge construction developed by Iyer and Wald [8] to derive the linear variation identity [2,7] which takes the following form Here δM is the variation of the ADM mass, ξ a is the time-like Killing vector, and δj e and δT e a are the associated current and stress tensor of the in-falling matter passing through the hypersurface Σ. In Wald's gedanken experiment, Σ is chosen to be H ∪ Σ 1 where H is the event horizon of the black hole and Σ 1 denotes a space-like hypersurface connecting to H at the late time and then extending to infinity.
We briefly sketch how (21) is obtained in [2,8]. Introduce the Lagrangian 4-form L = L associated with a Lagrangian L(φ), where φ denotes (g ab , A a ) and is the volume form associated with the metric. Then, the variation of L yields where E(φ) = 0 is the Euler-Lagrangian equation, and Θ(φ, δφ) is the symplectic 3-form potential. For an arbitrary vector ξ a , one can construct the associated can be rewritten as J ξ = dQ ξ + ξ d C d with the 3-form constraint C d = 0 when the equations of motion are satisfied. Assuming E(φ) = 0 and ξ a is a Killing vector, i.e., L ξ φ = 0, it is easy to show that δJ ξ = di ξ Θ(φ, δφ) which is then combined where ∂Σ = ∞ denotes the spatial infinity. If we assume ξ a is the time-like Killing vector t a for the non-rotating black hole, then and furthermore identify we can then arrive the the linear variation identity (21). For simplicity, we assume all the matters fall into the black hole far earlier than the joint moment of H and Σ 1 , thus we can replace Σ in (21) by H , then (21) turns into (6).
To really utilize the power of the linear variational identity (6) to check the WCCC for the extremal black hole in the higher theory, we need to derive the explicit forms of Q ξ and C a and their variations. A canonical procedure to derive them is developed in [8], following which we derive the explicit results in Appendix C.

Parameter bounds from WCCC
The key ingredient for Wald's gedanken experiment to examine WCCC is to throw Killing vector ξ a become null, thus ξ a ∝ n a , so that the null energy condition of the in-falling matter, δT ab n a n b ≥ 0 ensure the RHS of (6) is nonnegative. Namely, on the horizon one may write where˜ bcd is the volume element on the horizon, then the RHS of (6) becomes H˜ bcd δT ea ξ a n e ≥ 0. Thus, (6) turns into an inequality relating δM and δQ, Here the electromagnetic potential Φ H := − (ξ a A a ) | H can be readily calculated by using the background A a given in (15) as well as the location of horizon given in (17), the result is where However, the variational quantities δM and δQ for the black hole in the higher theory may not be the same as the ones for the Reissner-Nordström black hole, i.e., δM and δQ. Given the explicit forms in Appendix C, the corrections due to the higher-dimension operators asymptotically approach infinity as r −h with h ≥ 4, thus they fall off quickly and have no contributions to the ADM mass M, which is evaluated at the infinity. As a result, we find that Similarly, using C a in Appendix C, after straightforward calculations we arrive 4 Combine all the above, we can conclude that (27) gives where we have expressed in terms of reduced quantities. This is the relation between δm and δq from black hole mechanics provided null energy condition is satisfied.
Now if we require the condition (20) for the WCCC to hold so that an extremal black hole in the higher theory cannot be overcharged, then it is not hard to argue that we must have c 0 ≥ c 0 . Explicitly we have This is the key result of this paper, which gives the parameter bounds on the low-energy effective theory of quantum gravity by demanding that the low-energy theory preserves the weak cosmic censorship.
In this work we only consider the extremal black hole. It is then interesting to explore the WCCC for the near-extremal black hole under the condition that (33) is satisfied by following the same procedure in [2]. However, it should be technically far more involved as the the second order variational identities will involve the backreaction of gravitational and electromagnetic fluxes and need the generalization of the work done in [7] to higher derivative theories. Moreover, for the extremal black hole, the surface gravity vanishes, it is not clear how the change of Wald entropy plays the roles in the first law of black hole mechanics, which manifests as the variational identity. This should also be answered when considering the WCCC for the near-extremal case. 4 This can also be seen as follows. By the construction of source theory, j a = ∇ b (F ab − S ab ) in which S ab is O(c i ), and using (9) F ab = F With our new bound from weak cosmic censorship, it is then natural to ask how this bound works in the real world. At the low energy level, the leading order correction to the Einstein-Maxwell background is given by the graviton-photonphoton amplitudes with a scalar or spinor loop. For the minimally-coupled case, the one-loop effective actions for the Einstein-Maxwell background induced by spinor and scalar are given by [12,13] where again we have neglected terms proportional to ∇ µ F µρ ∇ ν F νρ , as it does not affect the black hole metric or our parameter bound. Simply plugging the values of c 4 , c 5 and c 6 into the bound (33), we find the inequality hold for both theories.
An important implication is then, the weak cosmic censorship conjecture not only holds for the Einstein-Maxwell theory, but may also hold at one-loop level. This could possibly mean the correctness of the conjecture in the real world! When the non-minimal coupling between matter and gravity is present, however, the bound (33) may subject to change under different situations. This is consistent with the fact that the combination of c-coefficients in our bound is not invariant under the field redefinition g µν → g µν + δg µν [11], where δg µν = r 1 R µν + r 2 g µν R + r 3 F µρ F ρ ν + r 4 g µν F ρσ F ρσ .
With a proper choice of the matter-gravity coupling, it is even possible that there yields no bound for the Wilson coefficients, and that the WCCC always holds. A further discussion is beyond the scope of this paper and we would like to explore it in the future.

A Corrections to the Maxwell source and stress tensor
Here we list the details of the corrections to the Maxwell source field and stress tensor and , i.e., S µν in (9) and ∆T µν in (10): and Note that F 2 = F ρσ F ρσ and = ∇ a ∇ a .

B Corrections to the Reissner-Nordström black hole
The functions λ(r) and ν(r) of (11) are related to the components of Ricci curvature To solve for λ and ν explicitly, we need an additional boundary condition. Assuming that at r → ∞ the metric approaches the Schwarzschld solution, the results are then given by We further take the trace-reverse of Eq. (10) and obtain that where T is the trace of the total energy-momentum tensor T µν , and is given by Plugging the trace-reversed Einstein field equation into the integral expression (39), we get Once we know the diagonal components of the energy-momentum tensor, it will be straightforward to compute the corrections to the spherically symmetric static spacetime as induced by T µν .
We now take our background spacetime to be Reissner-Nordström black hole in four-dimension. That is, Here ν (0) (r) and λ (0) (r) refer to the metric components in the unperturbed black hole spacetime, and F (0) µν is the background electromagnetic energy-momentum tensor. Considering the action in Eq. (2), we treat the corrections from higherdimension operators as perturbations. For convenience, we also introduce a power counting parameter ε, and consider a one-parameter family of actions I ε , which is given by The original action will be recovered after setting ε = 1. We then expand everthing into powers series in ε. For instance, At order ε 1 , the stress energy tensor is given by Noting that in order to compute the corrections to the metric, we need to calculate T µ ν instead of T µν . At order ε 1 , T µ ν (1) is given by From Eq.(9) we solve for the corrections to Maxwell equations, and obtain that the This corresponds to the gauge field A a given by With the corrections to F µν , we can solve for the corrected energy-momentum tensor T µ ν (1) . We then find the corrected metric tensor component to be In the above we have defined the reduced quantities m = M/4π and q = Q/4π. Note that the R 2 -term in the action has no contributions to the equation of motion at leading order in ε. The contributions from R µν R µν and R µνρθ R µνρθ can be canceled out by choosing c 2 = −4c 3 . This directly confirms that the Gauss-Bonnet term is a topological invariant and does not influence the equation of motion. Due to the fact that only the tr-and rt−component of F µν are nonzero, the term F µν F µν F ρσ F ρσ always have twice the contributions from F µν F νρ F ρσ F σµ towards the equation of motion [5].
C Explicit forms of Q ξ and C a for the higher theory The Lagrangian 4-form L for the higher theory can be written as In this appendix, by following the canonical method developed in [8] we derive and present the Noether charge and constraint associated with each term in L.
Variation of the Lagrangian 4-form L 0 yields where G ab = R ab − 1 2 g ab R is the Einstein tensor, and T EM ab is the electro-magnetic stress-energy tensor, which is defined by The symplectic potential can be written as where Θ GR abc (φ, δφ) = 1 2κ dabc g de g f g (∇ g δg ef − ∇ e δg f g ) , Let ξ a be any smooth vector field on the spacetime. We find that the Noether charges associated with the vector field are respectively, The equations of motion and constraints are given by where we have defined T ab = 1 κ G ab − κT EM ab as the non-electromagnetic stress energy tensor, and j a = ∇ b F ab is the charge-current of the Maxwell sources.
We similarly obtain the Noether charges and constraints for all higher-derivative terms. The results are presented below.
(1) L 1 Variation of L 1 yields where we have defined The Noether charge associated with the vector field ξ a is The constraints are given by (2) L 2 Variation of L 2 yields where we have defined The Noether charge associated with the vector field ξ a is The constraints are given by where we have defined The Noether charge associated with the vector field ξ a is The constraints are given by (4) L 4 Variation of L 4 yields where we have defined the equation of motions for g ab and A a respectively as The Noether charge associated with the vector field ξ a is The constraints are given by (5) L 5 Variation of L 5 yields where we have defined the equation of motions for g ab and A a respectively as The Noether charge associated with the vector field ξ a is The constraints are given by (6) L 6 Variation of L 6 yields where we have defined the equation of motions for g ab and A a respectively as The Noether charge associated with the vector field ξ a is The constraints are given by where we have defined the equation of motions for g ab and A a respectively as The Noether charge associated with the vector field ξ a is The constraints are given by where we have defined the equation of motions for g ab and A a respectively as The Noether charge associated with the vector field ξ a is The constraints are given by Finally, the above results can be summarized in the following compact form: where  This is because the two roots of 1/g rr coincide at this location.
Extremal black holes is a one-parameter family, with Q ext (M ), A ext (M ) determined jointly by Eqs. (101) and (102). In practice, when Q < Q ext (M ), we will have contours of constant A (as shown in Fig. 1), determined by On the other hand, we can find out the direction of the extremality curve in the (M, Q, A) space. The tangent vector satisfies However, because we have ∂ A F on that curve, we have ∂ A F = 0 and also This means, on the extremality contour, the direction at which area remains constant is the same as the contour itself. This does not mean that the contour all has the same area -instead, constant area contours reach the extremality contour in a tangential way, as shown in the figure.