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REVIEW 5 major objections 5 minor 45 references

Nonstatic Reissner-Nordstr\"om metric in the perturbative $f(R)$ theory: Embedding in the background of the FLRW cosmology, uniqueness of solutions, the TOV equation

T0 review · 5 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read In a perturbatively modified gravity, a spherical charged source can emit gravitational waves—something Einstein's equation forbids.

desk verdict The GR section is solid and the embedding derivation is clean, but the central f(R) gravitational-wave claim fails on the unverified (0,1) field equation. read the letter →

arxiv 2502.12160 v2 pith:W4D3VSIN submitted 2025-02-10 gr-qc astro-ph.HEhep-th

Pham Van Ky This is my paper · ORCID
classification gr-qcastro-ph.HEhep-th PACS 04.50.Kd04.70.-s04.30.-w
keywords Reissner-Nordströmmetricf(R)gravityperturbativetheorygravitationalwavesBirkhofftheoremFLRWembeddingTOVequationchargedblackhole
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that in a perturbative f(R) theory—where the gravitational Lagrangian is a small correction to Einstein's, $f(R)=R+\lambda h(R)$—the exterior metric of a charged, spherically symmetric source can depend on time even though Einstein's equation forces such a metric to be static. That time dependence is precisely what gives a spherical source a route to gravitational radiation, a channel the author says general relativity closes. Along the way the paper derives first-order solutions valid inside and outside the source, presents a simple embedding of the charged metric in an FLRW expanding background, argues for uniqueness of f(R) solutions in the presence of matter, and writes an explicit perturbative TOV equation for charged stars. If the claim holds, standard black-hole and neutron-star computations acquire small f(R) corrections, and spherical collapse becomes a potential source of gravitational waves.

What carries the argument

The load-bearing device is the first-order perturbative f(R) field equation (94), obtained by expanding $f(R)=R+\lambda h(R)$ around the Einstein solution and replacing $R$ and $R^\mu_\nu$ by their Einstein values inside the $\lambda$ terms. The central object it produces is the exterior metric (118)–(121), whose time dependence enters through the term $\lambda h''(kT^0_0)\,[\partial_t(M/[R_0(t)]^3)]^2\,\alpha(t)$; in the power-law model this reduces to replacing $M_f$ by $M_{ff}(t)$ in the Reissner–Nordström metric. The auxiliary function $\alpha(t)$ in (115) and the Einstein-limit metric tensors inside the source carry the explicit $R_0(t)$ dependence, which is what breaks staticity.

What would settle it

Substitute the exterior metric (118)–(121) for a concrete model, say $f(R)=R+\lambda R^2$ with a nonconstant $R_0(t)$, into the (0,1) component of the first-order field equation (94). If that component does not vanish identically for generic $R_0(t)$, the claimed time-dependent exterior solution is not a solution of the theory and the gravitational-wave effect is not established.

Watch

Extended reading notes

Core claim

The central discovery is that Birkhoff's theorem fails at first order in perturbative f(R) gravity. Starting from the spherically symmetric line element $ds^2=e^{u(r,t)}dx^{0\,2}-e^{v(r,t)}dr^2-r^2(d\theta^2+\sin^2\theta\,d\varphi^2)$, the author integrates the first-order perturbed field equation (94) to the exterior solution (118)–(121). For models with $f(R)=R+\lambda R^b$ the solution takes the Reissner–Nordström form with the constant total mass $M_f$ replaced by a time-dependent effective mass $M_{ff}(t)$, so $g_{00}$ and $g_{11}$ depend on time whenever the source radius $R_0(t)$ expands or contracts. The author identifies this time dependence with gravitational-wave emission from a spherically symmetric charged source, while the electromagnetic sector remains static and the source still does not radiate electromagnetic waves.

Load-bearing premise

The result depends on the unproven assumption that the time-dependent exterior metric satisfies every component of the first-order f(R) field equations; in particular, the (0,1) (time-radial) component is never solved or checked after the charge is added.

Editorial extensions

If this is right

  • Birkhoff's theorem breaks in perturbative f(R) gravity: a spherical source whose radius changes gives a time-dependent exterior metric, whereas general relativity forces that metric to be static.
  • A spherically symmetric charged source can emit gravitational waves but no electromagnetic waves, because the Maxwell equations keep their spherical static form.
  • Black-hole masses and horizon radii acquire f(R) corrections: for Sgr A* with $f(R)=R+\lambda R^2$, the event horizon is about $0.999187$ of the Einstein value for the chosen $\lambda$.
  • The perturbative TOV equation (198) gives an explicit hydrostatic equilibrium condition for charged stars with effective density and pressure built from the f(R) corrections.
  • The Reissner–Nordström metric embeds in the flat FLRW background without imposing the $G_{01}=0$ condition used by earlier embedding methods.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper's claims, the time-dependent effective mass $M_{ff}(t)$ suggests the leading observable signal would be a zero-angular-momentum (monopole) radiative term in $g_{00}$; since standard gravitational-wave templates are quadrupole-based, testing this prediction would require new template families for spherical collapse.
  • The paper leaves the exact wave zone unspecified; one concrete extension would be to compute the radiated power or the scalar curvature invariants for the metric (131)–(134) to see whether the time dependence carries energy to infinity rather than being a coordinate artifact.
  • The uniqueness discussion in Section 6 is explicitly non-rigorous; a natural strengthening would be to pose the interior-exterior matching with nonzero $T_{\mu\nu}$ as an initial-value problem and prove existence and uniqueness for the first-order system.
  • The embedding method, applied to the time-dependent metric, yields a McVittie-like spacetime with mass $M_{ff}(t)$; the paper does not analyze its apparent horizons or geodesic motion, which are the standard tests for whether such an embedded metric is physically well behaved.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. This manuscript develops a nonstatic Reissner-Nordström metric in general relativity and in a perturbative f(R) gravity framework. In the GR part (Sections 2–3), the author derives the exterior charged metric, discusses the time independence of the exterior field via the (0,1) Einstein equation, defines the total effective mass M_f, and analyzes the gravitational acceleration using proper distance and proper time, concluding that the gravitational field is always attractive. In the f(R) part (Section 4), the author writes down a first-order perturbative field equation and presents exterior solutions, claiming that, for models such as f(R)=R+λR^b, the exterior metric can depend on time when the source radius changes, thereby allowing a spherically symmetric source to emit gravitational waves. Section 5 proposes an embedding of the static and nonstatic Reissner-Nordström metric into an FLRW background by promoting a constant scale factor to a(t). Section 6 contains a discussion of uniqueness of f(R) solutions based on a toy model and presents a perturbative TOV equation for charged sources.

Significance. The GR portion of the paper is largely coherent and contains a useful, explicit treatment of the Reissner-Nordström gravitational acceleration, including a correct resolution of the historical repulsion debate by using the proper acceleration. The perturbative f(R) framework is clearly defined, and the central claim—that a spherically symmetric charged source can emit gravitational waves in this framework—would be a striking, observable consequence if correct. However, the central time-dependent exterior solution does not satisfy the full set of field equations, as detailed below, so the main physical effect is not established. The embedding, uniqueness, and TOV sections are also insufficiently supported. The strength of the paper is its explicit, self-contained GR derivation; its weakness is the unsupported f(R) generalization.

major comments (5)
  1. [Section 4.1, Eqs. (94), (131)-(134)] The manuscript never solves or checks the (0,1) component of the first-order f(R) field equation. For model II (f(R)=R+λR^b with b>0), outside the source T^(m)=0 and T^(e)=0 so the total trace T=0; consequently h(kT)=h(0)=0 and h'(kT)=h'(0)=0 for b>1 (and for b=1, h' is constant, so all derivatives of h' vanish). All λ-dependent terms in Eq. (94) therefore vanish in the exterior, and the field equations reduce to the Einstein-Maxwell equations with T^0_1=0, which force R^0_1=0. For the metric (131)-(134), which has u=-v and a time-dependent M_ff(t), a direct computation gives R_{01} = (1/r) ∂_t v = -(1/r) ∂_t u, whose leading term is proportional to \dot{M}_ff/r^2. This is nonvanishing unless \dot{M}_ff=0. Thus the proposed time-dependent exterior metric is not a solution of Eq. (94), and the claimed monopole gravitational-wave emission from a spherically symmetric source is not established.
  2. [Section 4.1, Eq. (103)] The relation u(r,t)=-v(r,t) outside the source is asserted on the basis of T^0_0=T^1_1, but the perturbative equation (94) contains λ-dependent terms such as ∇^0∇^E_0 h'(kT) and ∇^1∇^E_1 h'(kT) that are not equal in general. The manuscript does not show that these terms preserve the equality, nor does it solve the resulting equations for u and v independently. For a general f(R) model, the metric forms in Eqs. (109)-(112) and (118)-(121) therefore do not follow from the diagonal components of (94).
  3. [Section 5, Eq. (180)] The FLRW embedding is performed by replacing the constant a in the isotropic metric with a(t) and by ignoring dqdt mixing terms, without verifying that the resulting metric satisfies the field equations. For a time-dependent scale factor, the metric (180) generally has nonvanishing G_{01} and other components that are not accounted for by the assumed matter content; the McVittie limit (Q=0) is a solution of Einstein's equations only with a specific energy-momentum tensor or under restrictive conditions. The claim that the embedding appears 'without any conditions being imposed' is therefore not supported by the calculations shown.
  4. [Section 6, Eq. (198)] The perturbative TOV equation is stated without derivation. The effective energy-momentum tensor in Eq. (195) includes the electromagnetic contribution, which is anisotropic (T^0_0=T^1_1=-T^2_2=-T^3_3), whereas the standard TOV equation (198) applies to an isotropic perfect fluid. The manuscript does not show how the anisotropic stresses are handled, nor does it derive (198) from (194). Thus the claim that this is the TOV equation for a charged source in perturbative f(R) is not substantiated.
  5. [Section 6, Eqs. (181)-(193)] The uniqueness argument is based on a toy system of two ordinary differential equations that is not derived from the f(R) field equations. The conclusion that each f(R) model has a unique spherically symmetric solution is not proven; the manuscript itself concedes that the proof is not rigorous. Since uniqueness is listed as a contribution in the abstract and conclusions, this claim needs either a proper proof or an explicit statement that it is a conjecture.
minor comments (5)
  1. [Title and abstract] The title contains 'FLR W' with a space; this should read 'FLRW' throughout.
  2. [Eq. (19)] The delta-function identity in Eq. (19) writes δ(y) twice; the intended expression should have δ(x)δ(y)δ(z).
  3. [General notation] The notation for the source radius is inconsistent: R0(t) and Ro(t) are both used in Eqs. (115)-(117), (127)-(128), (A.13)-(A.20), and elsewhere.
  4. [Section 4.1.2, Eqs. (135)-(143)] The numerical value of λ is fitted to Mercury's perihelion precession in Ref. [14]; the Sgr A* horizon correction in Eq. (143) is therefore an illustration of a fitted parameter applied to a different system, not an independent prediction. The text should state this clearly.
  5. [Appendix A, Eq. (A.10)] The integral expression in Eq. (A.10) is displayed with a square-root symbol separated into '/radicaltp/radicalvertex/radicalvertex√'; this appears to be a typesetting artifact and should be rendered as a single radical.

Circularity Check

2 steps flagged · score 6.0 of 10

Central f(R) gravitational-wave claim is imported from the authors' own [14]-[18]; the Sgr A* horizon ratio is an evaluation of the Mercury-fitted λ, so the advertised predictions are not independently derived.

  1. self citation load bearing [Section 4, Eq. (97)-(103); Section 4.1.2, Eqs. (131)-(134); Abstract]
    "Notice that in the f(R) theory, the equation (48) is incorrect, so the metric of a spherically symmetric field outside the gravitational source can still depend on time (see [14]). Outside the gravitational source, the metric tensors can depend on time, this makes it possible for a spherically symmetric gravitational source to emit gravitational waves (Einstein's equation cannot give this effect)."

    The paper's advertised time-dependent exterior metric (131)-(134) rests on the g11 formula (97), which is not derived here but imported from the same authors' Ref. [14] ('see details in [14]'). The claim that Birkhoff's theorem is broken, i.e. that Eq. (48) is 'incorrect' in f(R), is likewise justified only by [14]. No component of the first-order field equation (94) is solved or checked for the charged, time-dependent generalization, so the gravitational-wave effect is carried by an unverified self-citation chain rather than by the field equations displayed in this paper.

  2. fitted input called prediction [Section 4.1.2, Eqs. (139)-(143)]
    "To correct the GR theory for Mercury's precession to be more consistent with the observations, we take λ = 0.1511677×10^18 m^2 (see details in [14]). ... The event horizon of the black hole Sgr A* would be rsf (R)/rsE = Mff/M = M−λM1/M = 0.999187."

    The numerical value of λ is a prior fit to Mercury's perihelion precession in the authors' [14]. The Sgr A* horizon shift (143) is then obtained by direct substitution of that fitted λ into M1; the quoted ratio 0.999187 is a one-line evaluation of the fitted parameter, not an independent prediction. Presenting it as the f(R) correction to GR makes the output follow from the input fit by construction, with no new observational or theoretical content added by this paper.

full rationale

The GR part of the paper (Sections 2-3) is self-contained: the exterior Reissner-Nordström metric, the gravitational acceleration, and Birkhoff's theorem are derived from the Einstein equations. The f(R) part is not self-contained. Its central input is Eq. (97), which is obtained by citing the authors' own Ref. [14], and the statement that Birkhoff's theorem fails is likewise imported from [14]. The advertised monopole gravitational-wave effect follows from this imported formula; the first-order f(R) field equations are not fully verified for the charged, time-dependent metric (131)-(134), and no off-diagonal component is written or checked. Under the review rules this is a load-bearing self-citation chain, and the Sgr A* horizon 'correction' (143) is only an evaluation of the Mercury-fitted λ. The skeptical (0,1)-component objection is a serious correctness risk, but it is not itself a circularity: it alleges that the proposed metric fails the field equations, not that the derivation is equivalent to its inputs. Section 6's uniqueness conclusion is explicitly admitted to be non-rigorous ('A more rigorous proof is left for future work'), so it is flagged as missing support rather than counted as a circular step. Section 5's embedding metric coincides with the known Gao-Zhang/McVittie result and is acknowledged as such. The perturbative TOV equation is the standard TOV equation written with an effective stress-energy tensor, a useful re-labeling rather than a circular derivation. Score 6 reflects partial circularity: the central effect is forced by the self-citation chain plus a fitted parameter, while the charged generalization and embedding calculations retain some independent content.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particle or force. Its load-bearing assumptions are the perturbative replacement R≈kT, the imported g11 formula, a homogeneous interior density for the time-dependent integrals, the validity of the FLRW embedding by coordinate promotion, and the use of an isotropic TOV equation for a charged source.

free parameters (2)
  • λ (coupling in f(R)=R+λ R^b) = 0.1511677×10^18 m² for b=2
    Chosen to match Mercury's precession in the author's prior work [14]; used to compute the numerical corrected horizon for Sgr A* (Eq. 143) and the effective mass Mff(t). Not derived in this paper.
  • R0(t), source radius function = unspecified
    The time dependence of the exterior metric is driven entirely by the free function R0(t); no equation of motion or collapse model is provided.
assumptions (6)
  • ad hoc to paper Perturbative f(R) field equation (94) with replacement R≈kT in order-λ terms
    Core approximation of the paper; it assumes corrections to the Ricci scalar in perturbation terms can be evaluated using the Einstein-limit trace T, which is not generally valid for propagating scalar modes.
  • domain assumption g11 solution (97) imported from Ref. [14]
    Imported without re-derivation; the present paper does not check the off-diagonal components of the perturbed field equations.
  • domain assumption Homogeneous matter density T0^(m)_0(t)=M c²/(4/3 π R0(t)^3) inside the source
    Used to evaluate M1, M2, M3 and the time-dependent integrals; ignores density gradients inside the star.
  • ad hoc to paper FLRW embedding: isotropic RN metric with constant a promoted to a(t) solves the field equations
    No field-equation check is provided; for the non-static case, dqdt mixing terms are dropped without justification.
  • domain assumption TOV equation (198) assumes isotropic pressure for a charged source
    The electromagnetic stress in (54) is anisotropic (T^1_1 = -T^2_2 for the exterior), so a single scalar Peff is insufficient unless the EM stress is treated separately.
  • ad hoc to paper Uniqueness in f(R) with a source can be inferred from a toy model
    The author admits the argument 'is not rigorous' and leaves a more rigorous proof for future work.

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Cite this review

Pith. "Pith review of Nonstatic Reissner-Nordstr\"om metric in the perturbative $f(R)$ theory: Embedding in the background of the FLRW cosmology, uniqueness of solutions, the TOV equation." pith.science (2026). https://pith.science/paper/W4D3VSIN

@misc{pith2026250212160,
  author       = {Pith},
  title        = {Pith review of: Nonstatic Reissner-Nordstr\"om metric in the perturbative $f(R)$ theory: Embedding in the background of the FLRW cosmology, uniqueness of solutions, the TOV equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W4D3VSIN}},
  note         = {Machine review of arXiv:2502.12160}
}
abstract

This article introduces a nonstatic Reissner-Nordstr\"om metric, a metric that does not emit electromagnetic waves but can emit gravitational waves. We first use the GR theory to study a charged spherically symmetric gravitational source (CSSGS), the obtained results are further improved in comparison with the previous studies. In particular, this article considers that the field is not necessarily static. The metric tensors $ g_{\mu\nu} $ are considered both outside and inside the gravitational source (the results show that in the first case $ g_{\mu\nu} $ are time independent, in the latter case they are time dependent). The gravitational acceleration and the event horizon of a charged black hole are investigated. The results prove that the gravitational field is always attractive. We then use the perturbative $ f(R) $ theory to consider CSSGS. The obtained results not only correct the solution of Einstein's equation in magnitude (this will describe astronomical and cosmological quantities more accurately than Einstein's equation), but also reveal new effects. Outside the gravitational source, the metric tensors can depend on time, this makes it possible for a spherically symmetric gravitational source to emit gravitational waves (Einstein's equation cannot give this effect). However, a spherically symmetric field still does not emit electromagnetic waves. Next we present a new method for embedding the spherically symmetric metrics of a star (or a black hole) in the background of the FLRW cosmological. Finally, we discuss the uniqueness of the solutions of the f(R) theory. The perturbative TOV equation is also found.

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