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3+1 formalism of the minimally extended varying speed of light model

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Applying the 3+1 formalism to the minimally extended varying-speed-of-light model gives a time-dependent lapse function N = 1 + (b/4)Ht and reproduces the model's Friedmann equations.

desk verdict The consistency claim at the heart of this paper is built on an unjustified switch from c to tilde c in the metric, so the advertised 3+1 check does not actually reproduce the model's Friedmann equations. read the letter →

arxiv 2412.19049 v1 pith:3LJEJVYT submitted 2024-12-26 gr-qc astro-ph.CO

Seokcheon Lee This is my paper · ORCID
classification gr-qcastro-ph.CO MSC 83F0583C05
keywords 3+1formalismADMvaryingspeedoflightmeVSLmodellapsefunctioncosmologicaltimedilationFriedmannequationsRobertson-Walkermetric
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

This paper works out the 3+1 (ADM) decomposition of the Robertson-Walker metric in the minimally extended varying-speed-of-light model, where the speed of light is $c = c_0 a^{b/4}$. It claims that in this decomposition the lapse function becomes $N = 1 + (b/4)Ht$ while the shift vector remains zero. Projecting the Einstein equations onto and along the spatial hypersurfaces then gives a Hamiltonian constraint and an evolution equation that the paper claims are identical to the meVSL Friedmann equations. If true, this makes the model's changing speed of light explicit as a time-slicing effect, namely cosmological time dilation encoded in the lapse, and provides a 3+1 formulation suitable for perturbation theory and numerical relativity.

What carries the argument

The load-bearing mechanism is the 3+1 (ADM) decomposition of the metric with foliation time $T=ct$. The central object is the lapse function $N=\tilde c/c$ obtained from $dT=d(ct)=\tilde c\,dt$; with $c=c_0 a^{b/4}$ this becomes $N=1+\tfrac{b}{4}Ht$. The shift vector $N^i=0$ follows from homogeneity and isotropy. The argument then flows through the induced three-metric $\gamma_{ij}=a^2\sigma_{ij}$ and its extrinsic curvature $K_{ij}=-(H/\tilde c)g_{ij}$, whose projections, organized by the Gauss-Codazzi relations, convert the Einstein equations into the Hamiltonian constraint and the evolution equation.

What would settle it

Re-run the projections of Section 4 using the original meVSL line element $ds^2 = -c(t)^2 dt^2 + a^2\sigma_{ij}dx^i dx^j$, without substituting $d(ct)=\tilde c\,dt$, and compare the Hamiltonian constraint with Eq. (14); if the constraint does not reduce exactly to the paper's Eq. (90) with $\tilde c$ replaced by $c$, the claimed consistency fails. A second check is to search for a coordinate transformation connecting the two line elements; if none exists, they describe different spacetimes.

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Extended reading notes

Core claim

The paper's central claim is that the minimally extended varying-speed-of-light (meVSL) Robertson-Walker metric, with $c(t)=c_0 a^{b/4}$, has a well-defined 3+1 decomposition in which the lapse function is $N=\tilde c/c=1+\tfrac{b}{4}Ht$ and the shift vector vanishes, where $\tilde c=d(ct)/dt$. Projecting the Einstein equations with the Gauss-Codazzi relations yields a Hamiltonian constraint $$\frac{\dot $a^{2}$}{$a^{2}$}+\frac{k\tilde $c^{2}$}{$a^{2}$}=\frac{8\pi G}{3}\rho$$ and an acceleration equation $$\frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{P}{\tilde $c^{2}$}\right)+$H^{2}$\frac{d\ln\tilde c}{d\ln a},$$ which the paper presents as identical to the Friedmann equations of the meVSL model derived earlier. On this basis the paper concludes that the 3+1 formalism reproduces the model's Einstein equations and that the variation of the speed of light is physically explicit as cosmological time dilation encoded in the lapse function.

Load-bearing premise

The argument assumes that the meVSL metric with varying $c$ can be written as $ds^2 = -\tilde c^2 dt^2 + a^2\sigma_{ij}dx^i dx^j$, where $\tilde c = d(ct)/dt$, even though the original metric is $ds^2 = -c^2 dt^2 + a^2\sigma_{ij}dx^i dx^j$; if $\tilde c$ and $c$ differ, these are different spacetimes and the derived equations need not match the original Friedmann equations.

Editorial extensions

If this is right

  • In the meVSL model the lapse function is $N = 1 + (b/4)Ht$, so the proper-time interval between neighboring spatial hypersurfaces grows with cosmic time even for comoving observers.
  • The momentum constraint vanishes and the shift vector is zero, so homogeneity and isotropy are preserved in the 3+1 description.
  • The Hamiltonian constraint and the acceleration equation obtained by projection coincide with the meVSL Friedmann equations, so the model's background dynamics can be treated as a standard initial-value problem.
  • Variation of the speed of light is thereby reinterpreted as cosmological time dilation, with the lapse function playing the role that $(1+z)^{1+\beta}$ plays in observations.

Reading between the lines

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

  • Read as an observational program, the 3+1 formulation suggests that high-redshift time-dilation measurements from supernovae, gamma-ray bursts, and quasars directly constrain the lapse function and hence the parameter $b$.
  • If the identification $\tilde c=d(ct)/dt$ is not an isometry of the original meVSL metric, the consistency claim reduces to a statement about the $\tilde c$-metric rather than the original $c(t)$-metric; checking whether a coordinate transformation connects the two would settle this.
  • The same Gauss-Codazzi projection route could be applied to the original metric with $c(t)$ left untouched, and the difference between the resulting equations and Eqs. (90) and (96) is a concrete test of the paper's central identification.
  • If the formulation holds, perturbation theory in the meVSL model can be built on this 3+1 background, giving a gauge-ready starting point for the forthcoming perturbation paper announced in the conclusion.
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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

3 major / 3 minor

Summary. The paper develops a 3+1 (ADM) decomposition of the Robertson–Walker metric within the minimally extended varying speed of light (meVSL) model. It claims that, because the speed of light varies with cosmic time, the RW line element should be written with a modified speed ~c = d(ct)/dt, yielding a time-dependent lapse function N = ~c/c = 1 + (b/4)Ht and a vanishing shift vector. The paper then derives the intrinsic and extrinsic curvatures, the Gauss–Codazzi relations, the Hamiltonian and momentum constraints, and the evolution equations, and asserts that these are identical to the Einstein equations of the meVSL model. The central consistency claim, stated in the abstract and repeated in Sections 4 and 5, rests on identifying the metric in Eq. (42) with the meVSL metric of Eq. (9).

Significance. If the central equivalence were valid, the paper would provide a useful interpretation of the meVSL model's varying speed of light as a lapse function and would offer a template for 3+1 calculations in VSL cosmologies. The treatment of embedded hypersurfaces, induced covariant derivatives, Eulerian observers, and the Gauss–Codazzi relations is systematic and could be of pedagogical value. However, the claimed consistency is not established: the derivation substitutes ~c for c without proof, and the resulting equations differ from the model's own Friedmann equations, including the absence of the cosmological-constant term. The paper contains no machine-checkable proofs or numerical code that could independently verify the central claim.

major comments (3)
  1. [Sec. 3.5, Eq. (42)] The replacement of the speed of light c by ~c = d(ct)/dt in the line element is not a coordinate transformation. With T = ct, the original meVSL metric Eq. (9) becomes ds^2 = -(c/~c)^2 dT^2 + a^2 dl^2, not -dT^2. Therefore Eq. (42), ds^2 = -~c^2 dt^2 + a^2 sigma_ij dx^i dx^j, is a different metric unless ~c = c, i.e., unless b = 0. All subsequent equations — Eqs. (59)–(61), (71), (90), and (95)–(96) — describe this different metric. The manuscript never proves or even states the equivalence between Eq. (9) and Eq. (42), so the assertion in Sec. 4.3.1 that Eq. (90) is 'of course, identical to the Einstein equations in the meVSL model' is unsupported.
  2. [Sec. 4.3.1, Eq. (90) and Sec. 4.3.3, Eq. (96)] Even if one accepted the ~c substitution, the derived equations do not match the quoted meVSL Friedmann equations. Eq. (90) reads dot-a^2/a^2 + k~c^2/a^2 = 8 pi G rho / 3, whereas the meVSL Friedmann equation, Eq. (14), contains -Lambda c^2/3 and a sum over components. Eq. (96), dot-dot-a/a = -4 pi G/3 (rho + P/~c^2) + H^2 d ln ~c/d ln a, differs from Eq. (16), which contains +Lambda c^2/3 and H^2 d ln c/d ln a. Thus the claimed identity with the model's Einstein equations fails independently of the ~c-versus-c issue.
  3. [Sec. 3.6, Eq. (56)] The lapse function N = 1 + (b/4)Ht is not an independent result of the 3+1 formalism; it is the definition N = ~c/c combined with the model's ansatz c = c0 a^{b/4} and Eq. (41). The apparent agreement between the 3+1 constraint/evolution equations and the meVSL Friedmann equations is therefore built into the input: the extra terms H^2 d ln c/d ln a and the modified coefficients in Eqs. (90) and (96) arise from the same scaling. A genuine consistency check would derive the 3+1 equations from the metric in Eq. (9) without presupposing this scaling.
minor comments (3)
  1. [Sec. 3.6, Eqs. (49)–(50)] The metric and inverse-metric components in Eqs. (49)–(50) are inconsistent: with g00 = -~c^2 in Eq. (47) and N = ~c/c, the inverse metric should have g^00 = -c^2/~c^2 rather than -1, and the spatial inverse should be a^{-2} sigma^{ij}, not a^2 sigma^{ij}.
  2. [Throughout] There are several typos: 'this session' in Sec. 2 should be 'this section'; the bibliography heading reads 'Refrences'; and after Eq. (96) 'dentical' should be 'identical'. Eq. (76) also contains an apparent typo '− 6 6' before the expression for the Ricci scalar.
  3. [Sec. 5] The statement that the lapse function variation 'can be interpreted as a change in the speed of light on the hypersurfaces' is a useful physical interpretation, but it should be explicitly tied to the fact that N = ~c/c holds only for the new metric (42), not for the original metric (9).

Circularity Check

1 steps flagged · score 6.0 of 10

The time-dependent lapse N = 1 + (b/4)Ht is definitionally built from the meVSL c(t) input via N ≡ \tilde c/c; the asserted consistency with the model's Einstein equations is supported only by self-citation while the compared equations differ (c vs \tilde c and missing Λ).

  1. self definitional [Sec. 3.5–3.6, Eqs. (41), (42), (48), (56)]
    "dx0 ≡ dT = d(c[t]t) = (1 + d ln c[t]/d ln a Ht)c[t]dt ≡ ˜c[t]dt ... ds2 = −˜c2dt2 + γijdxidxj ... where we define the lapse function N ≡ ˜c/c as x0(t+dt)−x0(t) = ... ≡ ˜c[t]dt ≡ N[t]c[t]dt ... In the meVSL model, c = c0ab/4 and thus N[t] = 1 + b/4 Ht."

    The advertised result N(t) is not an independent output of the 3+1 formalism. It is introduced by definition: N ≡ \tilde c/c, with \tilde c itself constructed from the model's c(t) in Eq. (41). Substituting the meVSL input c = c0 a^{b/4} then makes N = 1 + (b/4)Ht an identity. The central display result is therefore a restatement of the model's own speed-of-light scaling, and the later curvature, constraint, and evolution equations inherit this definition.

full rationale

The genuinely circular step is the derivation of the lapse function: Eq. (48) defines N ≡ \tilde c/c, Eq. (41) defines \tilde c from c(t), and Eq. (56) evaluates the definition with the model input c = c0 a^{b/4}. Thus the 'prediction' N = 1 + (b/4)Ht reduces by construction to the meVSL c-scaling. The 3+1 projection identities themselves are standard and not circular. The paper's further claim that the resulting equations are 'identical to the Einstein equations in the meVSL model' is not a circular reduction but a support/correctness problem: Eq. (90) contains \tilde c and no Λ, while Eq. (14) contains c and Λ, and Eq. (96) similarly differs from Eq. (16). Because those mismatches mean the asserted identity is not an algebraic consequence of the definitions, I do not count that assertion as a second circular step, but it reinforces that the self-cited 'consistency' check is not independent. Overall, one central result reduces to its input by definition, so partial circularity is present.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central result rests on one free parameter b and two ad hoc assumptions about the time coordinate and the substitution of tilde c for c. No new particles or forces are introduced.

free parameters (1)
  • b (or beta = -b/4) = constrained by SNeIa, GRB, and QSO time dilation observations in prior papers; Table 1 lists 1+beta values around 1
    The meVSL model assumes c = c0 a^(b/4), and the paper's lapse formula N = 1 + (b/4)Ht depends on b. The value of b is not derived here; it is taken as a model parameter from the author's earlier fits to cosmological time dilation data.
assumptions (4)
  • domain assumption The RW metric with a time-dependent speed of light c(t) is a valid spacetime metric, so that ds^2 = -c(t)^2 dt^2 + a(t)^2 dL^2 and Einstein equations with a varying c apply at cosmological scales.
    Adopted from the author's prior meVSL papers and used throughout Section 2 and Section 3.5.
  • ad hoc to paper The foliation time coordinate may be chosen as T = c(t)t, with the spatial hypersurfaces defined by constant T.
    Introduced in Section 3.3 and Eq (41). This choice is what produces the lapse N = tilde c / c; it is not forced by the RW geometry.
  • ad hoc to paper Substituting tilde c = d(ct)/dt for c in the metric leaves the physics of the meVSL model unchanged.
    Eq (42) replaces the coefficient c^2 with tilde c^2 without a coordinate transformation or a justification that the models are equivalent. This is the load-bearing assumption of the 3+1 derivation.
  • standard math Standard decomposition formulas, Gauss-Codazzi relations, and the form of the perfect-fluid stress-energy tensor are valid.
    Used in Sections 3.7 to 4.3 without proof; these are textbook results in the 3+1 formalism.

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Pith. "Pith review of 3+1 formalism of the minimally extended varying speed of light model." pith.science (2026). https://pith.science/paper/3LJEJVYT

@misc{pith2026241219049,
  author       = {Pith},
  title        = {Pith review of: 3+1 formalism of the minimally extended varying speed of light model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3LJEJVYT}},
  note         = {Machine review of arXiv:2412.19049}
}
abstract

The $3+1$ formalism provides a structured approach to analyzing spacetime by separating it into spatial and temporal components. When applied to the Robertson-Walker metric, it simplifies the analysis of cosmological evolution by dividing the Einstein field equations into constraint and evolution equations. It introduces the lapse function $N$ and the shift vector $N^i$, which control how time and spatial coordinates evolve between hypersurfaces. In standard model cosmology, $N = 1$ and $N^i = 0$ for the Robertson-Walker metric. However, the $N$ becomes a function of time when we apply the metric to the minimally extended varying speed of light model. This approach allows for a more direct examination of the evolution of spatial geometry and offers flexibility in handling scenarios where the lapse function and shift vector vary. In this manuscript, we derive the model's $N$ and $N^i$, along with the constraint and evolution equations, and demonstrate their consistency with the existing Einstein equations. We have shown in a previous paper that the possibility of changes in the speed of light in the Robertson-Walker metric is due to cosmological time dilation. Through the $3+1$ formalism, we can make the physical significance more explicit and demonstrate that it can be interpreted as the lapse function. From this, we show that the minimally extended varying speed of light model is consistent.

Figures

Figures reproduced from arXiv: 2412.19049 by the authors.

Figure 1
Figure 1. At t = tk, physical quantities and constants, such as ak, ρk, Pk, Tk, ck, kk, and ℏk, are fixed and uniform across the spatial hypersurface defined by t = tk. As the universe evolves and expands, these quantities change to al , ρl , Pl , Tl , cl , kl , and ℏl at time t = tl . Importantly, the CP and Weyl’s postulate do not require the speed of light ck at time tk to be equal to cl at time tl ; instead, its value is … view at source ↗
Figure 2
Figure 2. Consequently, the metric simplifies to ds = ˜cdt = ˜cdτ , directly correlating the proper time with the coordinate time. This setup underscores the orthogonality of the galaxy worldlines to the hypersurfaces of constant time. Specif￾ically, a vector Aµ = (˜c dt, 0, 0, 0) along the worldline is orthogonal to a vector Bµ = (0, dx1 , dx2 , dx3 ) within the hypersurface at constant T (i.e., gµνAµBν = 0). Such a model pr… view at source ↗
Figure 2
Figure 2. A foliation of the spacetime of the RW metric. The hypersurfaces [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Alleviating the Hubble Tension via Cosmological Time Dilation in the meVSL Model

    physics.gen-ph 2025-09 reject novelty 3.0 of 10

    The meVSL model's parameter b reduces the baryon drag sound horizon, raising inferred H0, and changes the cosmological time-dilation exponent to n=1-b/4; the paper forecasts SN sample sizes to detect this.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.