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On the perturbed Friedmann equations in Newtonian Gauge

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Pith's one-line read Newtonian mechanics plus thermodynamics reproduces the perturbed Friedmann equations of general relativity at first order.

desk verdict A transparently heuristic Newtonian route to the first-order perturbed Friedmann equations; the final equations are right, but two key inputs are imported from relativity rather than derived. read the letter →

arxiv 2412.15139 v3 pith:DRJNMNVJ submitted 2024-12-19 gr-qc astro-ph.COmath-phmath.MP

Jaume de Haro , Emilio Elizalde , Supriya Pan This is my paper · ORCID
classification gr-qcastro-ph.COmath-phmath.MP PACS 04.20.-q04.20.Fy45.20.D47.10.ab98.80.Jk
keywords FriedmannequationsCosmologicalperturbationsNewtonianmechanicsgaugefirst-orderEulerequationthermodynamicsincosmologygeneralrelativity
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 tries to show that cosmology's two Friedmann equations, and their first-order perturbations in the Newtonian gauge, can be derived from Newtonian mechanics, the first law of thermodynamics, and Euler's equation, without starting from Einstein's field equations. The authors construct a perturbed scale factor $a_N = a(1-\Phi_N)$ and promote the classical Poisson equation to a wave equation for the Newtonian potential $\Phi_N$, with sound speed $1/\sqrt{3}$. At linear order the resulting equations coincide with the perturbed Einstein equations in the same gauge, including the conservation of stress-energy. If correct, this gives a pedagogical bridge: the relativistic perturbation equations become accessible through familiar Newtonian fluid reasoning.

What carries the argument

The load-bearing object is the perturbed scale factor $a_N = a(1-\Phi_N)$ and its square $a_N^2$, which plays the role of a dynamical scalar field. The derivation's decisive step is equation (17), where the classical Poisson equation is upgraded, in the spirit of early scalar theories of gravity, to the wave equation $\partial_t^2\Phi_N - \Delta_{\bar{x}}\Phi_N = -(4\pi G/3)\delta T$; the rescaled coordinates $\bar{x}=\sqrt{3}x$ make the sound speed $1/\sqrt{3}$, characteristic of a relativistic fluid. This wave equation, combined with a D'Alembertian form of the Friedmann equations, yields the geometric second Friedmann equation. The same construction also produces a four-Hubble vector $H_{g,N}=a_N^{-1}\mathrm{grad}_g(a_N)$, so the second Friedmann equation becomes $\mathrm{div}_g H_{g,N}=(4\pi G/3)T$, a divergence law structurally analogous to the Einstein equations.

What would settle it

Choose a fluid with equation of state parameter $w$ and pressure perturbation $\delta p = c_s^2\delta\rho$ with $c_s^2 \neq 1/3$, solve the linearized system from the paper's equations (51)-(52), and compare the resulting density contrast $\delta\rho/\rho_0$ and potential $\Phi_N$ with the standard linearized Einstein equations. Any disagreement at first order would falsify the central claim; reproducing the standard result only for $c_s^2=1/3$ would confirm that the wave-equation assumption is the load-bearing input.

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Extended reading notes

Core claim

The central claim is that the perturbed Friedmann equations in the Newtonian gauge have a geometric form whose first-order expansion reproduces general relativity. Writing the metric with the perturbed scale factor $a_N = a(1-\Phi_N)$, the second Friedmann equation becomes $$\frac{1}{$a_N^{2}$}\left(\$partial_n^{2}$ - \frac{1}{3}\mathrm{div}_\gamma\nabla - 2\bar{g}(H_N,H_N)\right)$a_N^{2}$ = \frac{8\pi G}{3}$a_N^{2}$ T,$$ where $n$ is the normal time-like vector of the conformal metric and $H_N$ is the total Hubble vector rate. This equation, together with the first Friedmann equation and the fluid equations obtained from the first law of thermodynamics and Euler's equation, is claimed to be fully equivalent to the first-order Einstein equations in the Newtonian gauge.

Load-bearing premise

The derivation assumes that the perturbed Newtonian potential obeys the wave equation $\partial_t^2\Phi_N - \Delta_{\bar{x}}\Phi_N = -(4\pi G/3)\delta T$; if this guessed wave equation is not the right effective equation for the potential, the claimed equivalence with general relativity collapses.

Editorial extensions

If this is right

  • At first order, the perturbed Friedmann equations (51)-(52) reproduce the standard linearized Einstein equations in the Newtonian gauge, so cosmological perturbation theory can be formulated with Newtonian fluid variables.
  • The continuity and Euler equations derived here reduce at first order to $\nabla_\mu T^{\mu\nu}=0$, giving a thermodynamic route to stress-energy conservation.
  • Combining the Friedmann equations with conservation yields the constraint $\mathrm{div}_\gamma(\nabla\partial_t a_N)=4\pi G a_N^3(\rho+p)\mathrm{div}_\gamma v$, a first-order relation tying metric and velocity perturbations.
  • The second Friedmann equation can be written in the divergence form $\mathrm{div}_g H_{g,N}=(4\pi G/3)T$, making the parallel with the Einstein equations explicit.

Reading between the lines

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

  • The guessed wave equation (17) fixes the sound speed to $1/\sqrt{3}$; if the derivation were pushed to fluids with a different adiabatic sound speed, the Newtonian route would need an extra input or would break, so the claimed equivalence is tied to relativistic fluids.
  • Because the derivation works only at linear order and in a privileged foliation, it implicitly suggests that second-order perturbations, where vector and tensor modes mix, are where the Newtonian analogy would fail.
  • A testable extension: apply the same thermodynamic-plus-Newtonian derivation to scalar-tensor or modified-gravity cosmologies and compare the resulting perturbation equations with the linearized field equations of those theories.
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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 / 5 minor

Summary. The paper offers a pedagogical, heuristic derivation of the background Friedmann equations from Newtonian mechanics and thermodynamics, and then constructs a set of perturbed equations in the Newtonian gauge. The background derivation follows the standard McCrea–Milne route via the Lagrangian L_N = \dot a^2/2 + (4\pi G/3)a^2\rho_0. For perturbations, the authors introduce a perturbed scale factor a_N = a(1-\Phi_N), write a "reconciled" first Friedmann equation (14), postulate a scalar wave equation for \Phi_N (17) and a "simplest dynamical equation" (19), and claim that the resulting system (20) coincides with the linearized GR equations (21)-(22). They also derive the first-order conservation equation from the first law of thermodynamics and a modified Euler equation from the classical Euler equation with relativistic replacements. Section V rewrites the equations in geometric language, with Eq. (49) as the main result. The final equations are correct at first order, as verified by linearizing around FLRW, but the route to Eqs. (17) and (19) is not a derivation from Newtonian mechanics; it is a reconstruction guided by the known GR equations.

Significance. The paper's main positive contribution is a clear, self-contained demonstration that a certain set of Newtonian-gauge perturbation equations, written in terms of a_N, are equivalent at first order to the standard linearized Einstein equations. The geometric rewriting in Section V, especially Eq. (49), is elegant and may be useful for pedagogical purposes. The derived conservation and Euler equations, Eqs. (27) and (39), are correct and match GR. However, the central methodological claim that the perturbed Friedmann equations are "derived" from Newtonian mechanics plus thermodynamics is not supported. The wave equation (17) and the "simplest dynamical equation" (19) are imported assumptions whose coefficients encode the relativistic answer. Thus the paper's value is as a consistent heuristic reformulation, not as a derivation from Newtonian first principles. If the authors explicitly reframe the paper accordingly, it could be a worthwhile pedagogical contribution.

major comments (3)
  1. [IV, Eqs. (17) and (19)] The central route to the perturbed Friedmann equations is not a derivation. Eq. (17) is introduced "in the spirit of the first early scalar theories of gravity" as a wave equation for \Phi_N with sound speed 1/\sqrt{3} and source \delta T; nothing in Newtonian mechanics, the first law, or Euler's equation produces this equation. Eq. (19) is then declared "the simplest dynamical equation containing both equations," but its first-order content is precisely a combination of the linearized GR equations (21)-(22) that the authors want to recover. These two equations are load-bearing postulates, not consequences of the Newtonian framework, and the paper should label them as such. The Conclusion's statement that the authors have "heuristically derived the relativistic Friedmann equations at first-order perturbations, starting from the principles of Newtonian mechanics and the first law of thermodynamics" overstates the logical status of Eqs. (17) and (19); it should be revised to distinguish the genuinely derived conservation and Euler equations from the postulated wave equation.
  2. [IV.A, Eqs. (36)-(39)] The derivation of the Euler equation is a consistency argument, not a derivation from Newtonian mechanics alone. The transition from Eq. (30) to Eq. (36) uses the replacements \rho \to \rho+p, v \to au, and \nabla\Phi_N \to -(1/a_N)\nabla a_N, all of which are motivated by the relativistic conservation law \nabla_\mu T^{\mu\nu}=0 (Eqs. (33)-(35)), not by Newtonian physics. The authors should state explicitly that these replacements are an input from GR, so that the final first-order equation (39) is verified to match GR rather than derived from classical Euler. This is important because the abstract claims to "derive" the set of equations using the first law and Euler's equation.
  3. [IV, Eqs. (20)-(22)] The claim that Eqs. (20) coincide with the first-order GR equations (21)-(22) is stated without showing the relevant calculation. For a pedagogical paper, the explicit linearization of a_N^2 = a^2(1-2\Phi_N) should be included, demonstrating how the first equation in (20) reduces to the GR constraint (21) and how the second reduces to a combination of (21) and (22) (or to (22) plus the background equations). Without this calculation, the central equivalence claim rests on a citation to [17] and an assertion. I verified the claim and it is correct, so this is a completeness issue rather than an error, but it is central enough to warrant a revision.
minor comments (5)
  1. [IV, after Eq. (14)] The coordinates \bar{q} are never defined; the rescaling \bar{x} = \sqrt{3}x implies \bar{q} = \sqrt{3}q, but this should be stated explicitly to avoid confusion between \Delta_q and \Delta_{\bar{q}}.
  2. [IV, paragraph before Eq. (10)] The sentence "the metric (91)" should refer to Eq. (9) or (10), not to the metric in Appendix A.
  3. [V, Eq. (48)] The notation "div_{\bar{g}}\nabla" is missing its argument; it should read "div_{\bar{g}}(\nabla a_N^2)" or an equivalent explicit expression.
  4. [IV, before Eq. (19)] The phrase "containing both equations" is ambiguous; specify that it combines the background second Friedmann equation (15) with the wave equation (17), or state the intended combination explicitly.
  5. [Appendix A] There is a typo: "week limit" should be "weak limit."

Circularity Check

2 steps flagged · score 4.0 of 10

The perturbation 'derivation' imports the target as the wave ansatz (17)-(19); the conservation/Euler part is genuinely derived and the GR check is external, so circularity is partial.

  1. self definitional [Section IV, between Eqs. (16) and (17), and Remark IV.1]
    "In the spirit of the first early scalar theories of gravity [13–16], the natural generalization of this equation is ∂ 2 t2 Φ N − ∆ ¯xΦ N = − 4πG 3 δT. (17) ... where the sound speed is c s = 1√ 3 - characteristic of a relativistic fluid. Consequently, in this framework, the Newtonian potential Φ N is a sound wave traveling with velocity 1/ √ 3, and driven by the source perturbation δT."

    Eq. (17) is not derived from Poisson's law, Newtonian mechanics, or the first law of thermodynamics. It already contains the relativistic input c_s = 1/√3 and the GR source δT = δρ − 3δp. The final perturbed second Friedmann equation, Eq. (20)/(49), has the same D'Alembertian structure with coefficient 1/3 and source T = ρ − 3p. Thus the target first-order GR combination is loaded into the 'natural generalization,' and the subsequent derivation of the perturbed Friedmann equations recovers what was already assumed.

  2. fitted input called prediction [Section IV, Eq. (19) and the following paragraph]
    "Then, the simplest dynamical equation containing both equations is: □ ¯qa 2 N − 2H 2a 2 N = 8πGa 4 3 T, (19) ... It is important to realize that these equations coincide with the first order equations in GR."

    Eq. (19) is chosen as the 'simplest dynamical equation containing both equations,' without deriving its coefficients (the 1/3 in front of the Laplacian, the −2H^2 term, and the source T) from Newtonian mechanics or thermodynamics. The coefficients are fixed so that linearization reproduces the known GR equations (21)-(22), and the statement that (20) 'coincide with the first order equations in GR' is a consistency check of a reverse-engineered ansatz rather than an independent first-principles derivation. The advertised derivation of the perturbed Friedmann equations therefore reduces by construction to the target equations.

full rationale

The background Friedmann derivation from the Lagrangian (1) is a standard heuristic McCrea-Milne style construction; although the Lagrangian is cited to the author's prior work [9], that self-citation is not load-bearing because the equations are explicitly checked against known GR results. The conservation and Euler equations of Section IV.A, Eqs. (27) and (39), are genuinely derived from the first law and Euler's equation and independently match GR, so the paper has substantive non-circular content. However, the central perturbation equation, Eq. (19), and its geometric form (49), are not consequences of Newtonian mechanics: Eq. (17) is a postulated wave equation whose sound speed and source are relativistic inputs, and Eq. (19) is the 'simplest' operator chosen to reproduce the known linearized GR equations. The subsequent verification against Eqs. (21)-(22) is external and correct, but the construction is guided by the target result rather than derived from the stated first principles. This is a partial circularity: the first-order Friedmann equations are effectively reverse-engineered, while the conservation/Euler part remains an independent derivation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the Newtonian gauge metric ansatz, the Poisson equation, the first law of thermodynamics, and two ad hoc postulates (eqs. 17 and 19) that determine the form of the perturbed equations. No free parameters are fitted to data, and no new entities are introduced.

assumptions (5)
  • domain assumption Newtonian gauge metric: ds^2 = (1+2Φ_N)dt^2 - a^2(1-2Φ_N)dq^2
    The metric ansatz is taken from GR (Landau-Lifshitz and Einstein's book) and justified in the appendix via the equivalence principle, but it is an external input, not derived within the paper's Newtonian framework.
  • domain assumption Poisson equation: Δ_x Φ_N = 4πG δρ
    Standard Newtonian gravity relating the potential to density perturbations; used as a bridge to the first Friedmann equation.
  • domain assumption First law of thermodynamics in integral form: d/dη ∫ ρ dV = -p d/dη ∫ dV
    Used to derive the continuity equation (26) for the energy density.
  • ad hoc to paper Scalar wave equation for Φ_N: ∂_t^2 Φ_N - Δ_x Φ_N = -(4πG/3)δT
    Introduced in Section IV as 'the natural generalization' of the Poisson equation, inspired by early scalar gravity theories; not derived from Newtonian mechanics and not directly implied by GR.
  • ad hoc to paper Postulated 'simplest dynamical equation' containing the first Friedmann and Poisson equations: □ a_N^2 - 2H^2 a_N^2 = (8πG/3)a^4 T
    Chosen so that it reduces to the desired limits; this is the key construction step, not a derivation.

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Cite this review

Pith. "Pith review of On the perturbed Friedmann equations in Newtonian Gauge." pith.science (2026). https://pith.science/paper/DRJNMNVJ

@misc{pith2026241215139,
  author       = {Pith},
  title        = {Pith review of: On the perturbed Friedmann equations in Newtonian Gauge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DRJNMNVJ}},
  note         = {Machine review of arXiv:2412.15139}
}
read the original abstract

Based on the Newtonian mechanics, in this article, we present a heuristic derivation of the Friedmann equations, providing an intuitive foundation for these fundamental relations in cosmology. Additionally, using the first law of thermodynamics and Euler's equation, we derive a set of equations that, at linear order, coincide with those obtained from the conservation of the stress-energy tensor in General Relativity. This approach not only highlights the consistency between Newtonian and relativistic frameworks in certain limits but also serves as a pedagogical bridge, offering insights into the physical principles underlying the dynamics of the universe.

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