{"id":"a0736301-34dc-48d3-8f7f-5c1999028774","arxiv_id":"2506.11353","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under specific algebraic conditions on the coupling functions, Reissner-Nordström and Schwarzschild-(A)dS solutions persist in higher-order Maxwell-Einstein theories, including stealth solutions and dyonic solutions in a degenerate class.","lead":"Higher-order Maxwell-Einstein theories are a recently constructed class of modified gravity actions with U(1) symmetry. This paper shows that standard Reissner-Nordström and Schwarzschild black holes, including stealth solutions where an electric field leaves the geometry unchanged, can survive in many of these theories under tuned coupling conditions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sec. VI A's dyonic claim is demonstrably wrong: the α0 interaction term vanishes identically on the static spherically symmetric dyonic ansatz, so the RN-(A)dS solution holds for all P,Q, not only P=Q.","rationale":"The reader's flagged weakest assumption, the unproved identity (53) in Sec. VI B, actually survives scrutiny: for any static spherically symmetric metric the Weyl tensor is purely electric, so its contraction with the bivectors built from a dyonic field, which lie in the span of e0∧e1 and e2∧e3, vanishes identically. The Class C1II dyonic solution is therefore sound. The real defect is in Sec. VI A. The interaction α0 F^p G^q (*F)_{cd}∇^cF∇^dG depends only on r through the invariants F and G, so its two gradients are both radial; contracted with the antisymmetric dual tensor, the term becomes (*F)^{rr}F'G'=0. Thus the interaction vanishes identically on the ansatz for arbitrary A0(r), P, and Q, and the dyonic RN-(A)dS metric solves the full theory for any P,Q, including p=0. The paper's P=Q condition and its p=0 exception are false. This is a concrete algebraic error in a section that is part of the paper's abstract claims about degenerate classes. It does not by itself disprove the Secs. III-V existence results, but it undermines confidence in the degenerate-class analysis and justifies keeping the verdict at CONDITIONAL: the paper should be revised to correct Sec. VI A and the other degenerate-class variations should be rechecked. A direct substitution of the exact dyonic solution into the field equations would settle the remaining sufficiency questions.","tokens_in":17653,"tokens_out":30433,"duration_ms":340116,"concrete_test":"Using a symbolic algebra system (e.g., xAct), evaluate the α0 interaction term in (47) on the general ansatz (17) with an arbitrary A0(r) and dyonic magnetic charge P. Verify that the term reduces to α0 F^p G^q (*F)^{rr} F'(r) G'(r), and therefore vanishes identically because (*F)^{rr}=0. Then substitute the exact dyonic RN-(A)dS solution (48) into the full Euler-Lagrange equations of (47); if the α0 term vanishes identically, the equations reduce to those of pure Maxwell-Einstein theory and the solution must solve them for arbitrary P,Q, directly contradicting the claimed P=Q condition.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section VI A considers the totally degenerate action (47), whose nontrivial interaction is α0 F^p G^q (*F)_{cd} ∇^c F ∇^d G. For the static spherically symmetric metric (17) and the dyonic vector ansatz (45), the invariants F and G depend only on the radial coordinate r. Hence ∇^c F = g^{cr} F'(r) and ∇^d G = g^{dr} G'(r). Contracting these two radial vectors with the antisymmetric tensor (*F)_{cd} gives (*F)_{cd} g^{cr} g^{dr} F' G' = (*F)^{rr} F' G' = 0, identically, for arbitrary f(r), h(r), A0(r), and for arbitrary electric and magnetic charges. The α0 interaction therefore contributes nothing to the reduced Lagrangian or to the Euler-Lagrange equations. Consequently the dyonic Reissner-Nordström-(anti-)de Sitter solution (48) is a solution of the theory (47) for all values of P and Q, with no need for the condition P=Q and with no exception for p=0. The paper's statement that the solution exists only for P=Q, and not at all when p=0, is false. This is not a missing proof but a concrete algebraic error, indicating that the variational computation for the degenerate classes should be rechecked. The main Secs. III-V existence claims are not directly disproved by this error, but the paper as written is not reliable in its present form.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies static, spherically symmetric black hole solutions of the general U(1)-invariant higher-order Maxwell-Einstein action (3). For constant coupling coefficients and for quartic- and sixth-order couplings, it derives algebraic conditions (24), (41), (43) under which the Reissner-Nordström-(A)dS solution of Einstein-Maxwell theory remains an exact solution, and, when the ordinary Maxwell kinetic term is absent, a Schwarzschild-(A)dS metric with a nonzero electric field, i.e., a stealth solution. The second part examines degenerate classes: a totally degenerate alpha0 interaction and Class C1II with a beta(F,G) coupling to the Weyl tensor, claiming dyonic Reissner-Nordström-(A)dS solutions exist, with P=Q in the former case and for arbitrary charges in the latter via the identity (53).","tokens_in":18032,"tokens_out":11305,"duration_ms":129778,"significance":"If correct, the paper provides a useful catalogue of embeddings of Einstein-Maxwell black holes into a broad family of higher-derivative U(1)-invariant vector-tensor theories, extending the stealth-solution programme from scalar-tensor to Maxwell-Einstein settings. The existence conditions are concrete algebraic relations among the action coefficients, and the results are in principle checkable by direct substitution. However, the degenerate-class results, which are the most novel part of the paper, rest on a concrete algebraic error in Sec. VI A and on an unproved Weyl-tensor identity in Sec. VI B, so the overall significance is conditional on those issues being corrected and on the leading-order verifications in Secs. III-V being upgraded to complete substitutions.","major_comments":[{"comment":"The claim that the dyonic Reissner-Nordström-(A)dS solution exists only for P=Q, with no such solution when p=0, is contradicted by direct computation. On the static spherically symmetric metric (17) and the dyonic ansatz (45), F and G are functions of r alone, so ∇^c F = g^{cr} F'(r) and ∇^d G = g^{dr} G'(r); the alpha0 interaction in (47) therefore contains the factor (*F)_{cd} g^{cr} g^{dr} = (*F)^{rr}, which vanishes identically by antisymmetry for arbitrary f(r), h(r), A0(r), P, Q, p, and q. Hence the interaction term contributes nothing on the whole ansatz, and the dyonic solution (48) is a solution for all P and Q, with no exception at p=0. The stated condition (49) and the accompanying restriction are incorrect, and the variational computations for the degenerate classes should be rechecked.","section":"Sec. VI A, around Eq. (47) and the statement following Eq. (48)"},{"comment":"The identity asserted for the dyonic ansatz is not derived or checked anywhere in the manuscript, and it is the only reason the beta(F,G) interaction does not affect the dyonic Reissner-Nordström-(A)dS solution (48). Because the contraction is taken in the index order C_{acbd} with a tensor built from F and *F, the vanishing is not an immediate consequence of the standard symmetries of the Weyl tensor. The authors should present an explicit derivation or a component-level verification; as written, the central claim of Sec. VI B that the solution exists for arbitrary beta(F,G) is unproven.","section":"Sec. VI B, Eq. (53)"},{"comment":"The existence conditions are introduced as 'we find' and are supported only by the leading terms of a 1/r expansion, e.g., Eqs. (26)-(28) for Sec. III A. The manuscript does not show the reduced Lagrangian or the full Euler-Lagrange equations for the ansatz, nor does it provide a recurrence showing that all higher-order coefficients vanish after imposing the stated conditions. Since these conditions carry the main existence claims of the paper, the authors should supply a complete substitution check, either analytically or with a documented symbolic computation.","section":"Secs. III-V, Eqs. (24), (41), (43)"}],"minor_comments":[{"comment":"In the definition of E_h, the derivative is taken with respect to f'(r) rather than h'(r), and the possible second-derivative term in h is omitted; the displayed variational equations should be corrected.","section":"Eq. (19)"},{"comment":"The text says the Schwarzschild-(A)dS solution is obtained 'in the higher-order Maxwell-Einstein theories (3) with Eq. (42)', but Eq. (42) defines the sixth-order couplings of Sec. V; the reference should be to Eq. (40).","section":"Sec. IV D"},{"comment":"The solution (48) is called the 'Reissiner-Nordstr om solution-(anti-)de Sitter solution' and reference is made to Eq. (23) as if it contained the cosmological constant; the typo and the reference should be corrected.","section":"Sec. VI A"},{"comment":"The sentence 'Since the electric charge in the sector of the vector field does affect the spacetime geometry' is the opposite of what is meant; it should read 'does not affect', since that is the defining property of the stealth solutions.","section":"Sec. VII, Conclusions"}],"recommendation":"major_revision","confidential_remarks":"The error in Sec. VI A is concrete and easily demonstrated, so the manuscript cannot be accepted before that section is corrected. My main concern for the editor is the reliability of the 'we find' statements in the degenerate sections: they are not accompanied by derivations or by a documented computation, and the Sec. VI A mistake suggests that the unshown variational work should be independently checked. The Secs. III-V results are not directly invalidated by that error, but they would benefit from the same full-substitution documentation requested in the report."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is the first to look for standard black hole solutions in the higher-order U(1)-invariant Maxwell-Einstein theories recently built in Refs. [71-73]. For the constant-coefficient, quartic, and sixth-order coupling cases it derives algebraic conditions under which RN-(A)dS and stealth Schwarzschild-(A)dS solutions survive. That is a useful catalog, and the organization is clear. The stealth solutions in the absence of the Maxwell term are a legitimate and interesting feature.\n\nThe main soft spot is verification: the conditions (24), (41), (43) are reported as \"we find\" and checked only via the leading terms of a 1/r expansion. No full substitution into the Euler-Lagrange equations is shown. That is common in this literature and not fatal, but it leaves room for exactly the kind of error that shows up in Sec. VI A.\n\nThe stress-test note is right. For the totally degenerate action (47), the interaction term α0 F^p G^q (*F)_{cd} ∇^c F ∇^d G vanishes identically on the static spherically symmetric dyonic ansatz. Since F and G depend only on r, their gradients are radial, and the antisymmetric (*F) has no (r,r) component. So the term contributes nothing to the equations of motion for any electric and magnetic charges. The paper's claim that the dyonic RN-(A)dS solution exists only for P=Q, and not at all for p=0, is simply wrong. This is not a missing proof; it is an algebraic slip. The main sections III-V are not directly affected, but the paper needs a corrected Sec. VI A and the derivative terms in the degenerate classes should be rechecked.\n\nAlso, the \"identical relation\" (53) for the Class C1II coupling is asserted without proof. It might be true, but a referee should ask for a derivation or a symbolic verification.\n\nOverall, this is a serious paper with a concrete local error. It deserves peer review, with the referee asked to check the variational computations, especially in Sec. VI. A revised version that fixes the dyonic claim and either proves or replaces (53) would be a solid contribution.\n\nRecommendation: send to peer review; the error is significant but localized, not a desk-reject.","headline":"Useful first black-hole solution catalog for the new U(1)-invariant higher-order Maxwell-Einstein theories, but a concrete algebraic error in the dyonic degenerate-class section (Sec. VI A) needs correction before the paper is reliable.","tokens_in":18464,"tokens_out":3191,"would_cite":false,"duration_ms":31431,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that, for specific tunings of the higher-order couplings, the Reissner–Nordström and Reissner–Nordström-(anti-)de Sitter solutions of pure Einstein–Maxwell theory remain exact solutions of the higher-order U(1)-invariant…","keywords":["higher-order Maxwell-Einstein theories","stealth black holes","Reissner-Nordström solution","Schwarzschild solution","dyonic black holes","degenerate theories","U(1) symmetry"],"falsifier":"A direct computer-algebra check of the left-hand side of Eq. (53) on the metric ansatz (17) with the dyonic vector (45), for generic unequally charged $P \\neq Q$, would settle it: any nonzero component means the claimed $eta$-independence fails. A second check is to insert the dyonic Reissner–Nordström-(anti-)de Sitter metric into the full Euler–Lagrange equations while varying $eta(F,G)$; the $eta$ terms must cancel identically for the claim to hold.","tokens_in":17451,"feed_emoji":"🕳️","tokens_out":8894,"duration_ms":99426,"temperature":0.7,"pith_summary":"Higher-order Maxwell-Einstein theories are the most general U(1)-symmetric extensions of Einstein–Maxwell theory whose Lagrangians mix curvature with powers of the field strength and its derivatives. This paper asks whether the familiar static black holes of the plain theory survive these additions. It finds that they do, provided the higher-order couplings satisfy algebraic tuning conditions: with the standard Maxwell term present, the Reissner–Nordström-(anti-)de Sitter metric remains exact, and without it the Schwarzschild-(anti-)de Sitter metric with a nonzero electric field is exact, a stealth black hole because the field leaves the geometry unchanged. In one degenerate class, the dyonic Reissner–Nordström-(anti-)de Sitter solution is exact for an arbitrary coupling function, while in another degenerate class it is exact only when the electric and magnetic charges are equal. These results matter because they delimit when higher-derivative electrodynamic corrections can hide from black hole observations while still changing the theory.","feed_headline":"Exact Einstein–Maxwell black holes persist in higher-derivative gravity","feed_subtitle":"With tuned couplings the standard charged solution stays exact; drop the Maxwell term and the charge turns invisible to spacetime.","key_machinery":"The machinery is the general action (3) built from 21 U(1)-invariant higher-derivative building blocks, namely the curvature couplings $R_1$, $R_2$, $R_3$ and the derivative couplings $F_1,\\dots,F_{18}$, together with the static spherically symmetric metric ansatz (17), the purely electric vector ansatz (18), and the dyonic vector ansatz (45). Under these ansätze the field equations reduce to ordinary differential equations; the analysis uses large-distance asymptotic expansions whose leading coefficients, displayed in Eqs. (26)-(28) and (32)-(34), determine the algebraic coefficient conditions (24), (41), and (43) under which the standard solutions survive. The decisive identity for the degenerate-class result is Eq. (53): for the dyonic ansatz, the combination $2F(*F)_{ab}F^{cd} + G(F^{ab}F^{cd} - (*F)^{ab}(*F)^{cd})$ contracted with the Weyl tensor vanishes identically, so the entire $eta(F,G)$ coupling is invisible on the background and the dyonic Reissner–Nordström-(anti-)de Sitter solution exists for arbitrary $eta$.","core_discovery":"The paper's central claim is that the standard Reissner–Nordström and Schwarzschild black holes can be embedded as exact solutions of the higher-order Maxwell-Einstein action (3), not by setting the higher-order couplings to zero, but by choosing them to satisfy specific algebraic relations. For constant coefficients, the required relations are (24): the curvature couplings $a_2$ and $a_3$ must vanish, while the derivative-interaction coefficients $b_7$, $b_8$, $b_{13}$, $b_{15}$, and $b_{18}$ are fixed in terms of the others. For quartic and sixth-order couplings, the tuning changes: curvature couplings may survive provided $a_3 = 2a_2$, with further conditions (41) and (43). In every case, turning off the ordinary Maxwell kinetic term ($c_1 = 0$) turns the Reissner–Nordström solution into a Schwarzschild solution carrying a nonzero electric field that does not back-react on the metric, i.e., a stealth black hole. In the degenerate Class C1II theories, the dyonic Reissner–Nordström-(anti-)de Sitter solution is claimed to be exact for any $eta(F,G)$, thanks to the Weyl-contraction identity (53), and with $c_1 = 0$ it becomes a dyonic Schwarzschild-(anti-)de Sitter stealth solution. The paper also reports that the totally degenerate class admits the dyonic solution only for equal electric and magnetic charges, and that the Class C3 degenerate theories do not admit it.","pith_inferences":["Because the dyonic background is the same for arbitrary $eta$, the theory-dependent physics must show up at the level of perturbations; computing quasinormal modes or tidal Love numbers around this background would separate the degenerate theories even though their static solutions coincide.","The tuning conditions (24), (41), and (43) are derived through asymptotic expansion and are sufficient as stated; a recurrence analysis could settle whether they are also necessary, which would give a sharp classification of which higher-order actions keep the standard electrovacuum solutions.","The $P = Q$ requirement in the totally degenerate class suggests a charge-duality selection rule; extending the analysis to rotating or non-spherically-symmetric ansätze would show whether the stealth property is an artifact of spherical symmetry."],"forward_implications":["In constant-coefficient higher-order Maxwell-Einstein theories, the standard Reissner–Nordström-(A)dS metric remains an exact solution exactly when the curvature couplings vanish and the derivative couplings obey (24), so the familiar background survives a whole algebraic family of higher-derivative corrections.","Setting $c_1 = 0$ in the same tuned theories yields a Schwarzschild-(A)dS metric with a nonzero electric field, a stealth black hole whose external geometry is indistinguishable from vacuum Schwarzschild-(A)dS even though a U(1) charge is present.","For quartic- and sixth-order interactions, the curvature couplings need not vanish; the tuning $a_3 = 2a_2$ plus conditions (41) or (43) lets the same standard solutions survive, so the standard electrovacuum backgrounds persist despite nontrivial curvature couplings.","In the degenerate Class C1II theories, the dyonic Reissner–Nordström-(A)dS solution is independent of the arbitrary function $eta(F,G)$ and of the ratio of electric to magnetic charge, and with $c_1 = 0$ it becomes a dyonic Schwarzschild-(A)dS stealth solution.","In the totally degenerate class, the dyonic solution exists only for equal electric and magnetic charges ($P = Q$, with $p > 0$), and in Class C3 it does not exist, so the degenerate classification sharply separates which higher-order theories retain the standard electrovacuum black holes."],"supporting_citations":[{"why":"Supplies the general U(1)-invariant higher-order action (3) whose couplings are then tuned to admit the black hole solutions.","marker":"[71]"},{"why":"Classifies degenerate higher-order Maxwell-Einstein theories and specifies Class C1II, the family in which the dyonic Reissner–Nordström-(anti-)de Sitter solution is found.","marker":"[73]"},{"why":"Introduces the stealth black hole concept that frames the Schwarzschild solutions with a nonzero electric field.","marker":"[48]"},{"why":"Provides prior stealth Schwarzschild solutions in quadratic extended vector-tensor theories, the pattern the present paper extends to higher-order Maxwell-Einstein theories.","marker":"[69]"},{"why":"Shows that degeneracy of the kinetic matrix alone is not enough to remove Ostrogradsky ghosts, motivating the degenerate classes used in Sec. VI.","marker":"[72]"}],"fun_headline_variants":["Tuned couplings keep black holes exact in higher-derivative gravity","Charge invisible to gravity: exact stealth black holes in higher-order Maxwell-Einstein","Stealth black holes: charge doesn't warp spacetime in higher-derivative theories"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The dyonic result rests on the unproved algebraic identity (53) asserting that the Weyl-contracted combination of $F$ and its dual vanishes identically on the static spherically symmetric dyonic ansatz; if that identity is wrong, the dyonic Reissner–Nordström-(anti-)de Sitter solution would not be guaranteed for arbitrary $eta(F,G)$.","fun_headline_variants_meta":{"raw":{"variants":["Tuned couplings keep black holes exact in higher-derivative gravity","Charge invisible to gravity: exact stealth black holes in higher-order Maxwell-Einstein","Stealth black holes: charge doesn't warp spacetime in higher-derivative theories"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001336,"raw_usage":{"total_tokens":5475,"prompt_tokens":1033,"completion_tokens":4442,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":649,"completion_tokens_details":{"reasoning_tokens":4379}},"tokens_in":649,"tokens_out":4442,"duration_ms":34738,"temperature":1.0,"reasoning_tokens":4379,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:09:59.440082+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct computer-algebra check of the left-hand side of Eq. (53) on the metric ansatz (17) with the dyonic vector (45), for generic unequally charged $P \\neq Q$, would settle it: any nonzero component means the claimed $eta$-independence fails. A second check is to insert the dyonic Reissner–Nordström-(anti-)de Sitter metric into the full Euler–Lagrange equations while varying $eta(F,G)$; the $eta$ terms must cancel identically for the claim to hold.","supporting_citations":[{"cited_title":"Black holes in the quadratic-order extended vector-tensor theories","cited_arxiv_id":"2105.08936","evidence_quote":"Provides prior stealth Schwarzschild solutions in quadratic extended vector-tensor theories, the pattern the present paper extends to higher-order Maxwell-Einstein theories."}],"review_version":1}