{"id":"32e10592-9049-4808-8109-11238eb1b563","arxiv_id":"2505.15838","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"A varying speed of light in FLRW is presented as gauge freedom via the lapse function, but the key variational derivation omits the √-g measure and the claimed Hubble-tension resolution contradicts Eq. (30).","lead":"This paper argues that a varying speed of light in cosmology is a coordinate choice, not real physics. The proof has a faulty variational step, and the Hubble-tension fix it promises is excluded by its own distance formula.","discovery_kind":"unclear","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section VI's central derivation is invalid: Eq. (34) varies only R, but the action in Eq. (33) is ∫R√-g with √-g=c̃a³; including the measure (and the matter term) replaces the claimed constraint a(t)=(c1+c2t)^{1/4} with the Friedmann equation (25).","rationale":"I agree with the reader's weakest_assumption. Eq. (33) contains R√-g, yet Eq. (34) is obtained by varying R alone; √-g=c̃a³ is c̃-dependent. Including the measure converts the c̃-equation into a Friedmann-type constraint, not the advertised ä/a+3(ȧ/a)²=0. Since this equation is the paper's only derivation of the non-dynamical status of c̃, the central claim is unsubstantiated. The Hubble-tension aside in Section V is also contradicted by Eq. (30), which equates the meVSL Hubble radius to the SMC value. The rejection stands.","tokens_in":16684,"tokens_out":15302,"duration_ms":153646,"concrete_test":"Recompute δS/δc̃ for Eq. (31) with √-g=c̃a³ and the matter contribution ρ_i0(1+ω_i)c̃0² c̃ a^{-3ω_i} from Eq. (24), then form the Euler-Lagrange equation. A few lines of hand algebra or a symbolic manipulator will settle whether the result is Eq. (34) or the Friedmann equation (25); no numerics are required. If the result is the Friedmann equation, Eq. (34) and the derived constraint on a(t) are refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Eq. (34). Starting from Eq. (33), S=∫(1/(2κ0))R√-g dtd³x, the paper computes d/dt(∂R/∂ċ̃)-∂R/∂c̃=0, i.e., it varies the Ricci scalar while treating √-g as inactive. This is not the valid variational derivative of the action, because for the metric (4), √-g=c̃a³. The matter term also cannot be dropped: after substituting Eq. (24), ρ_i(1+ω_i)c̃²√-g = ρ_i0(1+ω_i)c̃0² c̃ a^{-3ω_i}, which depends on c̃ unless the fluid sector is absent. The correct δS/δc̃=0 equation is the Friedmann constraint (25) (vacuum limit: ȧ²/a² + k c̃²/a² - Λc̃²/3 = 0), not ä/a + 3(ȧ/a)² = 0. Thus the derived solution a(t)=(c1+c2t)^{1/4} and the conclusion that c̃ has no dynamics are not established by the action principle. The paper's own Eq. (30) also makes the Hubble radius identical to SMC, so the advertised Hubble-tension insight is unsupported by the model's distance observables.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper argues that a time-dependent speed of light c̃(t) in the FLRW metric is not an independent physical degree of freedom but reflects a coordinate choice, specifically the lapse function. The central claim is supported by an action-principle calculation in Section VI, where the Euler-Lagrange equation for c̃ is said to yield a constraint on the scale factor, a(t) = (c1 + c2 t)^(1/4), rather than a dynamical equation for c̃. The paper further proposes that this reframing offers a new interpretation of cosmological observables, including the Hubble tension. It also derives modified Friedmann equations (Eqs. 25-29) and notes that the Hubble radius in this model is identical to the standard model's (Eq. 30).","tokens_in":17012,"tokens_out":4539,"duration_ms":45793,"significance":"If the central claim were correct, it would constitute a clean no-go argument against treating VSL as new physics in FLRW backgrounds and would clarify the gauge nature of c̃. The paper has some positive features: the derivation of the modified Friedmann equations is systematic, and the explicit observation in Eq. (30) that the Hubble radius coincides with the standard model is a useful check. However, the load-bearing variational calculation in Section VI is invalid, and the proposed observational test of a gauge parameter is logically inconsistent. The main conclusion that c̃ has no dynamics is therefore not established by the manuscript, and the advertised connection to the Hubble tension is contradicted by the paper's own equations.","major_comments":[{"comment":"The Euler-Lagrange equation is computed by varying only the Ricci scalar R, while the action in Eq. (33) is ∫ R√-g d⁴x with √-g = c̃ a³ for the metric in Eq. (4). Since the measure depends explicitly on c̃, the correct variation of the action with respect to c̃ must include δ√-g; this yields the Hamiltonian (Friedmann) constraint, not the equation ä/a + 3(ȧ/a)² = 0 reported in Eq. (34). Consequently, the derived solution a(t) = (c1 + c2 t)^(1/4) and the conclusion that c̃ is non-dynamical are not established.","section":"Section VI, Eq. (34)"},{"comment":"The claim that the perfect-fluid Lagrangian is independent of c̃ is based on Eq. (24), but the matter action is ∫ ρ_i(1+ω_i)c̃² √-g d⁴x = ∫ ρ_i0(1+ω_i)c̃0² a^(-3(1+ω_i)) c̃ a³ d⁴x, which depends on c̃ through the volume element. Dropping this c̃-dependence changes the equation of motion and invalidates the subsequent constraint derivation.","section":"Section VI, Eqs. (31)-(32)"},{"comment":"Equation (30) shows that the Hubble radius in the meVSL model is identical to the standard model value, and the text correctly states that luminosity and angular diameter distances, being integrals over the Hubble radius, are unchanged. This directly contradicts the abstract's and conclusion's claim that the framework offers a new interpretation of observational tensions such as the Hubble tension: if the background distance observables are unchanged, the model cannot resolve the tension by modifying the expansion history.","section":"Section V.C, Eq. (30)"},{"comment":"The identification of c̃ with the lapse function via N ≡ c̃/c in Eq. (35) is a definition, not a derived result. Moreover, the conclusion proposes to test the gauge parameter b by cosmological observations, while simultaneously asserting that c̃ is a coordinate/gauge choice with no physical content. If c̃ is pure gauge, b is unobservable; if b is observable, c̃ is not pure gauge. The manuscript does not resolve this contradiction and instead relies on it for its advertised testability.","section":"Section VI, Eq. (35) and Conclusion"}],"minor_comments":[{"comment":"The text says 'we conclude with a discussion of our findings and their broader implications in Section 6', but the conclusions appear in Section VII, not Section VI; the cross-reference should be corrected.","section":"Introduction, last paragraph"},{"comment":"The notation alternates between c̃ and c without always making clear whether c denotes the constant present-day value or the speed of light in a given frame; a consistent definition at first use would improve readability.","section":"Section III, Eq. (4)"},{"comment":"The meVSL relation c̃1 = (a1/a2)^(b/4) c̃2 is stated as a special case f(a)=a^(1-b/4), but the intermediate steps connecting f(a) to the exponent b/4 are not shown, making the equation hard to follow.","section":"Section III, Eq. (5)"},{"comment":"The terms 'dynamical', 'independent degree of freedom', and 'gauge' are used without precise definitions; the paper would benefit from stating explicitly, e.g., whether the lapse is a Lagrange multiplier in the ADM formalism and what exactly counts as a dynamical field.","section":"Abstract and Section I"},{"comment":"A large fraction of the references are to the author's own previous work; while self-citations are legitimate, the paper would be stronger if it cited independent derivations or critical discussions of the meVSL framework.","section":"References"}],"recommendation":"reject","confidential_remarks":"The paper essentially reformulates the standard result that the lapse function in the ADM decomposition is a Lagrange multiplier that enforces the Hamiltonian constraint. The central 'proof' that c̃ is non-dynamical rests on a variational error (dropping the c̃-dependence of √-g and the matter action), and the advertised Hubble-tension insight is contradicted by Eq. (30). The proposal to observationally test the gauge parameter b is internally inconsistent. The heavy reliance on the author's own prior work and the failure to engage with the literature on lapse/non-dynamical fields further weaken the novelty. I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the gauge/lapse interpretation of VSL in FLRW is correct and standard — g00 can always be rescaled by a time redefinition — and the author already made this point in Ref. [5]. Second, the paper's claimed new derivation of that point, Section VI, is wrong. Equation (34) varies the Ricci scalar while holding √-g = c̃ a³ fixed, but the action is ∫R√-g dtd³x, so the c̃-dependence of the measure cannot be dropped. Including it gives the Friedmann constraint, not ä/a + 3(ȧ/a)² = 0. The matter term also depends on c̃ through √-g, contrary to the claim around Eq. (32). So the central conclusion that c̃ imposes a(t) = (c1+c2t)^{1/4} is unsupported.\n\nWhat is genuinely useful: the survey of VSL models in Section II is careful, and Eq. (30) is correct — the Hubble radius in meVSL is identical to SMC, so luminosity and angular distances are unchanged. That result is worth knowing, but it undercuts the paper's own Hubble-tension claim: if no distance observable changes, there is nothing to reconcile. The paper also proposes to observationally test the gauge parameter b, which is self-contradictory. If c̃(a) is a coordinate choice, all values of b are physically equivalent; no experiment can favor one. This is a logical inconsistency in the paper's central narrative, not a minor slip.\n\nThe citation pattern is also heavy: Refs. [1]–[5] and [87], [89], [103]–[106] are the author's own work, and the key idea appears in Ref. [5]. That does not by itself disqualify the paper, but combined with the invalid derivation, the incremental contribution is very small.\n\nWho is this for? A reader interested in the meVSL program might find Eq. (30) and the review useful, but this paper does not advance the program. The central new argument fails, and the advertised observational implication is logically flawed. I would not cite it, and I would not bring it to reading group unless the discussion is about common variational errors in cosmology.\n\nRecommendation: reject. If the editor wants to be generous, the author could correct Eq. (34) by varying R√-g, but the corrected equation will not yield the claimed constraint, and the b-test claim would still need to be withdrawn. A desk reject is defensible; a full peer review is only warranted if the editor believes the author can fix the derivation, which I doubt.","headline":"The paper's only new derivation is invalid because it drops the √-g measure in the variation; the gauge interpretation is standard and was already in the author's earlier work.","tokens_in":17559,"tokens_out":5481,"would_cite":false,"duration_ms":49923,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper argues that a varying speed of light in the FLRW metric is not a dynamical field but a coordinate artifact carried by the lapse function, which imposes a constraint on the scale factor instead of having its own equation of motion.","keywords":["varying speed of light","FLRW metric","lapse function","gauge freedom","cosmological time dilation","Hubble tension","Einstein-Hilbert action"],"falsifier":"Recompute the Euler-Lagrange equation for $\\tilde c$ from Eq. (31) with $\\sqrt{-g} = \\tilde c a^3$ and with the $\\tilde c$-dependent measure kept; if the result is the vacuum Friedmann constraint $\\dot a^2/a^2 + k\\tilde c^2/a^2 = \\Lambda \\tilde c^2/3$ rather than $\\ddot a/a + 3(\\dot a/a)^2 = 0$, then the conclusion that $\\tilde c$ is non-dynamical does not follow from this action.","tokens_in":16410,"feed_emoji":"🕰️","tokens_out":5361,"duration_ms":53633,"temperature":0.7,"pith_summary":"The paper tries to show that a varying speed of light in the Friedmann-Lemaître-Robertson-Walker metric is not a new physical field: it is a coordinate effect, exactly the lapse function that sets the clock rate between spacelike hypersurfaces. Because the Weyl postulate makes cosmic time the proper time of comoving observers, the apparent time-dependence of $\\tilde c$ is a choice of temporal gauge rather than an evolution of a fundamental constant. Varying the Einstein-Hilbert action with respect to $\\tilde c$ gives no dynamical equation for $\\tilde c$; it instead gives a constraint on the scale factor, which the paper reads as evidence that $\\tilde c$ is fixed by the time parametrization. A sympathetic reader would care because this recasts VSL models, and their proposed resolutions of the Hubble tension, as gauge freedom within general relativity rather than new physics.","feed_headline":"Varying speed of light may be a gauge choice, not new physics","feed_subtitle":"In the FLRW metric, a changing speed of light acts as the lapse function, recasting VSL as a clock-rate choice.","key_machinery":"The central machinery is the lapse function, defined in the paper as $N \\equiv \\tilde c/c$, which sets the rate at which proper time advances relative to coordinate time across spatial hypersurfaces. The argument runs by writing the Einstein-Hilbert action for the VSL metric, using the fixed-$\\tilde\\kappa$ relation Eq. (9) and the conservation equation Eq. (24) to eliminate apparent $\\tilde c$ dependence from the matter sector, then varying the action with respect to $\\tilde c$. The resulting Euler-Lagrange equation produces the scale-factor constraint Eq. (34), which is what carries the claim that $\\tilde c$ is non-dynamical and should be interpreted through the lapse function rather than as a new field.","core_discovery":"The paper claims that in the line element $ds^2 = -\\tilde c(t)^2 dt^2 + a(t)^2 dl^2_{3D}$, the time-dependent speed of light $\\tilde c(t)$ plays exactly the role of the lapse function $N(t)$ in the ADM foliation, rescaling coordinate time relative to proper time. Varying the Einstein-Hilbert action, while holding the Einstein gravitational constant $\\tilde \\kappa = 8\\pi \\tilde G/\\tilde c^4$ fixed, leads to an Euler-Lagrange equation for $\\tilde c$ that does not determine $\\tilde c$; instead it yields the constraint $\\ddot a/a + 3(\\dot a/a)^2 = 0$, whose solution is $a(t) = (c_1 + c_2 t)^{1/4}$. The paper concludes that a varying speed of light is not an independent degree of freedom but a manifestation of the freedom to choose the temporal coordinate, and that VSL models should be understood as a coordinate-dependent feature of cosmic time rather than a modification of physical laws.","pith_inferences":["A testable consequence the paper leaves implicit: if $\\tilde c$ is pure gauge, any VSL model can be rewritten with a constant $c$ by a time reparametrization, so physical observables must be gauge-invariant and distance-redshift data alone cannot prefer VSL over standard cosmology.","The derivation's handling of $\\sqrt{-g}$ is the natural place to probe the claim; carrying the full variation with $\\sqrt{-g} = \\tilde c a^3$ kept intact would either confirm the constraint or turn it into the first Friedmann equation, which would decide whether the non-dynamical conclusion survives.","The same lapse logic could be applied to other seeming drifts of constants in cosmological fits, since the model already requires $\\tilde G \\propto \\tilde c^4$ to keep the Einstein gravitational constant fixed."],"forward_implications":["The Hubble radius $\\tilde c(a)/H(a)$ in the model equals the standard-model value $c_0/H^{(GR)}(a)$, so luminosity distances and angular diameter distances built from it are unchanged when the redshift is defined in the same way.","The past evolution of the Hubble parameter differs from standard cosmology by a factor $\\tilde c^2/\\tilde c_0^2$, which the paper says could reconcile local and distance-inferred values of $H_0$ without introducing a new field.","The Hubble tension and time-dilation measurements become probes of the temporal gauge: if observations favor a nonzero parameter $b$ in $\\tilde c = \\tilde c_0 a^{b/4}$, the standard gauge choice $b = 0$ would be disfavored.","Because no new dynamical field appears, the framework remains general relativity with a nontrivial temporal coordinate choice, avoiding the stability and symmetry issues that plague scalar-field VSL models."],"supporting_citations":[{"why":"Supplies the minimal-extension VSL model and the scaling relations for $\\tilde c$ and $\\tilde G$ that keep the Einstein gravitational constant fixed in the action.","marker":"[1]"},{"why":"Defines the lapse-function and clock-rate interpretation of $\\tilde c$ that the paper extends into the non-dynamical argument.","marker":"[5]"},{"why":"Provides the general isotropic and homogeneous metric and lapse-function notation adopted for the FLRW line element.","marker":"[81]"},{"why":"Gives the FLRW curvature tensors for the VSL metric used to write the action and the modified Friedmann equations.","marker":"[102]"},{"why":"Is cited for the Hubble-tension application that the gauge reinterpretation of $\\tilde c$ is said to address.","marker":"[105]"}],"fun_headline_variants":["Varying speed of light is just a time-coordinate choice","VSL as gauge freedom: no new physics needed","Variable c is a clock-rate choice, not a new field","Cosmic speed of light: just a gauge, not physics","Varying c reduces to lapse function, no new fields"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation's load-bearing step is treating $\\sqrt{-g}$ as independent of $\\tilde c$ when varying the action, even though the metric gives $\\sqrt{-g} = \\tilde c a^3$; if that dependence is included, the constraint equation for $\\tilde c$ changes.","fun_headline_variants_meta":{"raw":{"variants":["Varying speed of light is just a time-coordinate choice","VSL as gauge freedom: no new physics needed","Variable c is a clock-rate choice, not a new field","Cosmic speed of light: just a gauge, not physics","Varying c reduces to lapse function, no new fields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000255,"raw_usage":{"total_tokens":1601,"prompt_tokens":1005,"completion_tokens":596,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":621,"completion_tokens_details":{"reasoning_tokens":513}},"tokens_in":621,"tokens_out":596,"duration_ms":6241,"temperature":1.0,"reasoning_tokens":513,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:48:22.647142+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the Euler-Lagrange equation for $\\tilde c$ from Eq. (31) with $\\sqrt{-g} = \\tilde c a^3$ and with the $\\tilde c$-dependent measure kept; if the result is the vacuum Friedmann constraint $\\dot a^2/a^2 + k\\tilde c^2/a^2 = \\Lambda \\tilde c^2/3$ rather than $\\ddot a/a + 3(\\dot a/a)^2 = 0$, then the conclusion that $\\tilde c$ is non-dynamical does not follow from this action.","supporting_citations":[{"cited_title":"Lee, [arXiv:2504.07975 [astro-ph.CO]]","cited_arxiv_id":null,"evidence_quote":"Gives the FLRW curvature tensors for the VSL metric used to write the action and the modified Friedmann equations."}],"review_version":1}