Lin, Ning and Chen (Phys. Rev. D 108, 044025 (2023)) showed that for perturbative near-extremal charged black holes in quartic higher-derivative extensions of Einstein-Maxwell theory, requiring the second law of black hole thermodynamics for the Wald entropy yields exactly the weak cosmic censorship condition, while the Sorce-Wald first-law approach fails in these theories. Their demonstration is restricted to perturbative, spherically symmetric, stationary-to-stationary processes and assumes rather than proves the second law. The open problem addressed here is to place this thermodynamic route to weak cosmic censorship on fully general, dynamical footing: prove an entropy increase theorem for the black-hole entropy of generic higher-derivative gravity under physically reasonable energy conditions (resolving the gauge dependence of the horizon chemical potential and the known ambiguities of the Noether-charge construction); construct a gauge-invariant canonical energy and the sourceless second-order first law needed to treat generic non-spherical collapse; extend the analysis to rotating charged black holes with higher-derivative corrections; and rigorously establish the smoothness and nondegeneracy assumptions near extremality on which the second-law-to-WCCC implication rests. Second laws for higher-derivative gravity are established to linearized order and to effective-field-theory order in perturbations, but the general dynamical theorem and the WCCC extensions remain open.

Background

Penrose's weak cosmic censorship conjecture (WCCC) asserts that singularities formed in gravitational collapse remain hidden behind event horizons, so that the exterior spacetime is predictable. The classic test is a gedanken experiment of Wald (1974): throw matter into an extremal (maximally charged or maximally spinning) black hole and ask whether the horizon can be destroyed, i.e. whether the black hole can be 'overcharged' or 'overspun' into a naked singularity. Writing the condition for a horizon to exist as W(m,q)≥0W(m,q)\geq 0 in terms of the black-hole mass mm and charge qq (or angular momentum), the question is whether matter increments δm,δq\delta m,\delta q compatible with physical energy conditions can drive the final state to W<0W<0.

Sorce and Wald (Phys. Rev. D 96, 104014 (2017)) reanalysed this for Kerr-Newman black holes in Einstein-Maxwell theory using the first law of black hole mechanics extended to second order in perturbation theory: the second-order first law relates the second-order variation of the ADM mass (the total energy measured at spatial infinity) to the canonical energy E\mathcal{E} of the induced gravitational and electromagnetic waves — the conserved quadratic form on the space of perturbations constructed from the Iyer-Wald covariant phase space — and an energy condition on the infalling matter then forbids destruction. First-order analyses had suggested overcharging was possible; the second-order terms repair this.

In general relativity such arguments rest on the second law of black hole mechanics, Hawking's area theorem: the horizon area never decreases if the null energy condition holds. In higher-derivative gravity, obtained by adding to the Einstein-Maxwell Lagrangian terms with four derivatives,

L=12κR−14FμνFμν+∑ici Li,∣ci∣≪1,\mathcal{L}=\frac{1}{2\kappa}R-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\sum_i c_i\,\mathcal{L}_i,\qquad |c_i|\ll 1,

where the Li\mathcal{L}_i are quadratic in curvature (such as R2R^2 and RμνRμνR_{\mu\nu}R^{\mu\nu}) or up to quartic in the Maxwell field strength FμνF_{\mu\nu} (the 'quartic' class of Lin-Ning-Chen), the entropy is no longer a quarter of the horizon area but the Wald entropy — the Noether-charge functional S=−8π∫∂L/∂Ruvuv dAS=-8\pi\int\partial\mathcal{L}/\partial R_{uvuv}\,dA obtained by differentiating the Lagrangian with respect to the Riemann tensor, integrated over a horizon cross-section. The area theorem has no direct analogue: a second law for the Wald entropy in dynamical situations is not established, and even the first law becomes subtle because the horizon chemical potential ΦH=−ξμAμ∣r=r+\Phi_H=-\xi^\mu A_\mu|_{r=r_+} depends explicitly on the gauge choice of the Maxwell potential, while the Noether-charge construction itself suffers from known field-redefinition ambiguities.

Lin, Ning and Chen (Phys. Rev. D 108, 044025 (2023)) constructed the charged black-hole solutions of the quartic class to order O(cicj)\mathcal{O}(c_ic_j) and showed for near-extremal members that requiring the second law (δS≥0\delta S\geq 0 to second order in the infall process) yields exactly the WCCC condition, whereas the Sorce-Wald first-law approach fails: the linear first law is apparently violated at O(ci2)\mathcal{O}(c_i^2), plausibly due to the gauge ambiguity. They also gave a general argument that the second law implies WCCC whenever the horizon radius rhr_h and entropy SS depend smoothly and nondegenerately on the mass excess from extremality, μ=m−mex(qj)\mu=m-m_{\rm ex}(q_j), via rh(μ,qj)=R(qj)+μ ρ(qj,μ)r_h(\mu,q_j)=R(q_j)+\sqrt{\mu}\,\rho(q_j,\sqrt{\mu}) with ∂S/∂rh≠0\partial S/\partial r_h\neq 0. The demonstration is, however, restricted to perturbative, spherically symmetric, stationary-to-stationary processes, assumes rather than proves the second law, and leaves spinning black holes and the generic-collapse machinery untreated. Related earlier work by three of the same authors (Chen, Lin, Ning and Chen, Phys. Rev. Lett. 126, 031102 (2021)) showed that WCCC can constrain the signs of higher-derivative couplings, so the status of censorship in these theories is a live consistency question for low-energy effective theories of quantum gravity.

Problem Statement

Place the second-law route to the weak cosmic censorship conjecture in higher-derivative gravity on fully general, dynamical footing, as left open by Lin, Ning and Chen (Phys. Rev. D 108, 044025 (2023)). The problem has four jointly supporting components, all serving this single objective:

(1) Dynamical second law. Prove an entropy increase theorem for the black-hole entropy of generic higher-derivative gravity — including the quartic Einstein-Maxwell extensions L=12κR−14FμνFμν+∑iciLi\mathcal{L}=\frac{1}{2\kappa}R-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\sum_i c_i\mathcal{L}_i — under physically reasonable energy conditions on the collapsing matter, beyond perturbative stationary-to-stationary processes. This requires a dynamical entropy functional that is gauge-invariant (resolving the explicit gauge dependence of the horizon chemical potential ΦH\Phi_H and the known ambiguities of the Noether-charge construction) and reduces to the Wald entropy in stationary eras.

(2) Canonical energy and the second-order first law. Construct a gauge-invariant, unambiguous canonical energy for gravitational perturbations in generic higher-derivative gravity, and use it to establish and verify the sourceless second-order (n≥2n\geq 2) first law, thereby extending the Sorce-Wald analysis to generic (non-spherical) collapse and clarifying the apparent O(ci2)\mathcal{O}(c_i^2) violation of the linear first law found in the source.

(3) Spinning black holes. Construct rotating charged (Kerr-Newman-like) black-hole solutions with higher-derivative corrections and check WCCC for spinning black holes in the quartic theories; the source verifies only the Einstein-Maxwell Kerr-Newman case, reproducing Sorce-Wald.

(4) Foundations of the implication. Rigorously establish — or refute by counterexample — the smoothness and nondegeneracy assumptions on the horizon radius rh(μ,qj)=R(qj)+μ ρ(qj,μ)r_h(\mu,q_j)=R(q_j)+\sqrt{\mu}\,\rho(q_j,\sqrt{\mu}) (with μ=m−mex(qj)\mu=m-m_{\rm ex}(q_j) the distance from extremality) and on the entropy S(rh,μ,qj)S(r_h,\mu,q_j) with ∂S/∂rh≠0\partial S/\partial r_h\neq 0 near extremality, on which the general proof that the second law implies WCCC rests.

An admissible answer either establishes the general results above for the stated classes of theories — generic higher-derivative gravity and, concretely, the quartic Einstein-Maxwell extensions L=12κR−14FμνFμν+∑iciLi\mathcal{L}=\frac{1}{2\kappa}R-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\sum_i c_i\mathcal{L}_i with ∣ci∣≪1|c_i|\ll1, together with their rotating charged solutions — or refutes one of the four components by an explicit theory and process within those classes. The four components are independently refutable: a negative answer to component (1) (no entropy increase along a dynamical process) or to the smoothness and nondegeneracy assumptions of component (4) resolves the objective negatively, whereas a negative answer confined to component (2) or (3) settles only that component and remains a partial answer to the objective.

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