Zero Values of the TOV Equation

Jonas Pleyer

REVIEW 4 major objections 4 minor 5 references

Zero Values of the TOV Equation

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This thesis argues that the first zero of the TOV pressure is smaller than the Lane-Emden zero for identical polytropic parameters, and it backs the claim with numerical scans and initial proof steps.

desk verdict A sincere, readable master's thesis whose central numerical claim is invalid because the TOV integrations cross 2m/r=1, and whose existence proof omits p0; the useful bits (EoS, series, code) are standard or partial. read the letter →

arxiv 2411.15264 v1 pith:VHK3WSHP submitted 2024-11-22 gr-qc

Jonas Pleyer This is my paper · ORCID

classification gr-qc

keywords TOVequationLane-Emdenpolytropicofstatezerovaluesstellarstructurespecial-relativisticSchauderfixed-pointtheoremexactseriessolution

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

At the center of the thesis is a comparison between two stellar-structure equations: the relativistic TOV equation and its Newtonian limit, the Lane-Emden (LE) equation. Using a polytropic equation of state $\rho=A p^{1+1/n}$, the author integrates both equations from the same initial pressure $p_0$ and finds that the TOV pressure reaches zero at a smaller radius than the LE pressure for identical parameters. If true, this means general-relativistic corrections make polytropic stars more compact, so estimates of stellar radii from equations of state must shift when relativity is included. The thesis develops a special-relativistic equation of state for a non-interacting gas, offers first proof steps toward the TOV zero-value theorem, and contributes exact LE and TOV solutions that can be used as benchmarks.

What carries the argument

The arguments are carried by the TOV system $\partial_r m=4\pi\rho r^2$ and $\partial_r p=-(m\rho/r^2)(1+p/\rho)(4\pi r^3 p/m+1)(1-2m/r)^{-1}$ together with the polytropic EoS $\rho=A p^{1+1/n}$, and by its Newtonian reduction to the Lane-Emden equation $\xi^{-2}\partial_\xi(\xi^2\partial_\xi\theta)+\theta^n=0$ through the rescaling $\xi=r/\kappa$ with $4\pi\kappa^2=(n+1)K\rho_0^{1/n-1}$. The zero value is defined as the first radius at which the pressure vanishes, and the comparison uses the conversion back to the physical radius $r=\kappa\xi$, matching the two equations at the same central density and pressure. The fixed-point machinery is the Schauder theorem applied to the integral form of each equation; the LE zero-value theorem further uses the nonexistence of global solutions for $n\ge5$ taken from the cited literature.

What would settle it

Recompute the zero-value curves with an adaptive high-order integrator for fixed $A$ and $p_0$ over $n\in(0,5)$, converting TOV radii by $r=\kappa\xi$ and recording the first pressure node; the central inequality fails if any run yields $r_0^{\mathrm{TOV}}\ge r_0^{\mathrm{LE}}$ for the same parameters. A second check is to compare the new $n=2$ LE series against the numerically integrated LE zero radius near $\xi\approx3.918$.

Watch

Extended reading notes

Core claim

The central claim is that, for every polytropic equation of state $\rho = A p^{1+1/n}$ and identical parameters $A$ and $p_0$, the first zero of the TOV pressure occurs at a smaller radius than the first zero of the Lane-Emden solution. The manuscript establishes this as a numerical result across families of $(A,p_0,n)$ and leaves it as an open theorem to prove completely. It also proves the Lane-Emden finite-boundary statement for $0\le n<5$, derives a special-relativistic equation of state for a non-interacting gas, finds an exact LE series solution at $n=2$, and obtains an exact TOV solution in the $A\to 0$ limit.

Load-bearing premise

The load-bearing premise is that unique local TOV solutions exist from the initial data $m(0)=0$, $p(0)=p_0>0$ for every polytropic index used in the scans, since the numerical comparison and the zero-value hypothesis presuppose that existence; the Schauder fixed-point argument in Section 5.2 is the part of the manuscript intended to supply it.

Editorial extensions

If this is right

For identical polytropic parameters, TOV pressure-zero radii are smaller than LE radii, so general-relativistic corrections shrink the predicted radius of a star with a given equation of state.
If Hypothesis 5.8 is right, each pair $(A,p_0)$ has a critical index $n_0$ beyond which TOV solutions never reach zero pressure, a relativistic analogue of the LE threshold at $n\ge5$.
The exact $n=2$ LE series gives a quantitative check for LE solvers, and the $A\to0$ TOV solution $p=p_0/(2\pi r p_0+1)$ gives a limit-case check for TOV solvers.
The special-relativistic EoS $\rho(p)$ is a bijection on $p>0$, so it can be inverted and tabulated uniquely for TOV integrations.
The Lane-Emden finite-boundary theorem for $0\le n<5$ provides a rigorous base for the analogous TOV hypothesis, while the cited global-nonexistence results mark the expected boundary at $n\ge5$.

Reading between the lines

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

A testable extension the paper leaves open is to scan zero values for the derived special-relativistic EoS $\rho(p)$ across $B$ and $p_0$; if the ordering persists, the small-radius effect is not an artifact of the polytropic form.
The unexplained bump in the $p_0=0.1$ TOV zero-value curve near high $n$ suggests a regime where relativistic pressure terms and polytropic structure compete; mapping that region could yield a sharper estimate of $n_0(A,p_0)$.
If a comparison proof is attempted, the $A\to0$ exact solution could serve as a lower-bound pressure profile; sandwiching the TOV pressure between that profile and the LE solution would give a direct argument for smaller zeros without global existence.
The mass bound $M<4R/9$ implies that zero values cannot grow without bound for fixed mass; combining it with the bump structure might predict where TOV solutions lose their zeros.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Read this one because the abstract promises a theorem about TOV zero radii. The honest summary: it is a master's thesis with one real (if modest) piece of work, one overstated numerical claim, and two technical defects that sink the central thesis in current form.

What is actually there: the special-relativistic EoS is derived cleanly from the partition function for a Jüttner gas; that derivation is textbook, but it is done carefully and the limiting checks are right. The n=2 Lane-Emden series solution is a legitimate recurrence and gives decent numerics, though calling it an 'exact solution' overstates it: it is a convergent power series, not a closed form, and the author admits the series struggles near the zero. The code is on GitHub, which is good practice. The author is also honest in Section 5 that the zero-value theorem is only a hypothesis and explains why the LE proofs don't transfer.

The soft spots are not minor. The numerical zero-value comparison, which is the paper's central claim, is invalid as presented. In the Section 4.1 example the text reports a TOV pressure zero at r ≈ 6.80, but Figure 3's bottom-right panel has m/r ≈ 0.75, i.e. 2m/r > 1, while Section 3.1 correctly states e^{-λ} = 1−2m/r > 0. The integration is being run past the Schwarzschild limit. With the same parameters, 2m/r reaches 1 near r ≈ 0.32, so the reported zero at 6.80 is not a physical stellar radius. If Figure 6 contains such points, the claimed comparison is not between TOV stellar solutions and LE solutions.

The existence proof in Section 5.2 has a separate concrete error: U2 omits the initial pressure p0, so the fixed-point equation U(m,p)=(m,p) would force p0=0. That is repairable, but it means the paper currently has no theorem backing the numerics. The abstract's 'results show' overstates what is an unproven hypothesis.

Who gets value: a patient reader interested in series methods for Lane-Emden or in how not to design TOV numerical scans. The thesis is readable and the author is engaged, but the central claim does not hold as written.

Recommendation: desk reject as a journal submission. If revised, the numerical scans need to stop at the surface before 2m/r=1, or use a parameter range where physical solutions exist, and the fixed-point operator needs the p0 term; after that the zero-value comparison might become a modest but sound numerical study.

Referee Report

4 major / 4 minor

Summary. The manuscript derives a special-relativistic equation of state for a non-interacting gas from the partition function, applies it together with polytropic equations of state to the Tolman-Oppenheimer-Volkoff equations, and compares the first zero of the pressure with the Lane-Emden solution. The central claim is that TOV zero values are smaller than LE zero values for identical parameters. The paper also presents an attempted local existence proof for the TOV system, a conjecture about a critical polytropic index, and a new series solution of the Lane-Emden equation at n=2.

Significance. If the quantitative comparison and the existence results were correct, the paper would be a useful reference on how general-relativistic corrections affect polytropic stellar radii. The manuscript has genuine strengths: the numerical code is referenced and appears intended to be reproducible, the Lane-Emden solver is validated against known exact solutions in Section 4.2, and the partition-function derivation in Chapter 2 is explicit. However, the load-bearing numerical and analytical claims fail for the reasons detailed below, so the stated results are currently unsupported.

major comments (4)

[§4.1, Figure 3] The reported TOV solution is continued beyond the domain in which the TOV equations are the stellar-structure equations. Equation (3.1.13) and the text below it require e^{-λ}=1−2m/r>0, i.e. m(r)<r/2. The bottom-right panel of Figure 3 shows m/r≈0.75 for the TOV curve by r≈2.5, so 2m/r>1 well before the reported zero at r≈6.80. The factor (1−2m/r)^{-1} in Eq. (4.1.2) has therefore changed sign in this regime, and the computed zero is not a physical TOV radius. The zero values plotted in Figure 6 inherit this problem, so the central comparison between TOV and LE is not a comparison of physical solutions.
[§4.1 vs. Abstract] The paper's own example contradicts the abstract. With the Table 2 parameters (A=2, p0=0.5, n=3), Section 4.1 reports the LE zero at r≈2.31 and the TOV zero at r≈6.80, so the TOV zero is larger, not smaller, as the abstract claims. Unless Figure 6 uses a different normalization or a different definition of 'identical parameters,' the abstract's central statement is false for this example.
[§5.2, Eq. (5.2.18)] The fixed-point operator U2 is defined without the initial pressure p0. For a genuine solution, p(r)=p0−∫(p+ρ)/r'²(4πρr'³+m)(1−2m/r')^{-1}dr', so U2((m,p))(r)=p(r)−p0, not p(r). The fixed-point equation U(m,p)=(m,p) then forces p0=0, contradicting the assumed p0>0. Consequently Lemma 5.9 does not prove local existence for the TOV system with p0>0, and the claimed theoretical foundation for Hypothesis 5.8 is absent.
[§4.4, Figure 6] The numerical zero-value scan has no error estimates, no convergence study for the TOV solver, and no check that the stopping point is reached before 2m/r=1. Section 4.2 validates only the Lane-Emden integrator against exact solutions. Since the central claim is quantitative, these omissions are substantial, and they are directly related to the unphysical continuation noted in the first major comment.

minor comments (4)

[Appendix B] The n=2 Lane-Emden solution is an infinite series whose convergence is proven only for ξ≤1 (Theorem B.2), while Figure 7 shows deviation from the numerical solution already before the zero near ξ≈3.9. Calling this an 'exact solution' in the abstract and conclusion is an overstatement.
[§2.2, Eqs. (2.2.11)–(2.2.12)] The non-relativistic limit of the derived internal energy does not reproduce Nmc²+(3/2)NkBT, and the factor N is dropped between equations (2.2.11) and (2.2.12). This appears to be an algebraic error in the derivative of log K2 and undermines Section 4.3's relativistic EoS, although it does not affect the polytropic TOV analysis.
[§3.3, Lemma 3.1] The statement assumes ρ is monotonically decreasing, but the proof says 'since ρ is monotonously increasing.' In the context of the proof, decreasing is the correct hypothesis; the text should be corrected.
[Abstract] 'Tollmann' should be 'Tolman,' and the phrase 'an additional exact solution and index n=2' is grammatically garbled.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: zero-value comparison is independent numerical integration; only self-reference is a code repository pointer.

full rationale

The main claim about TOV versus LE zero values is obtained by directly integrating the two ODE systems for fixed polytropic parameters (A, p0, n), with the LE solver benchmarked against the known exact solutions for n=0, 1, and 5 in Section 4.2. The equation of state in Section 2 is derived from the partition function without using any zero-value information, so the zero radii are outputs rather than inputs of the calculation. The only self-citation, [Ple21], points to the author's GitHub code and carries no theorem or fitted value; it is not load-bearing. The Section 5.2 existence proof does contain a genuine defect: U2 in equation (5.2.18) omits the initial pressure p0, so for a true solution U2 equals p-p0 and the fixed-point equation U(m,p)=(m,p) would force p0=0, contradicting p0>0; this invalidates the proof, but it is a mathematical-error issue rather than a circular reduction, and the numerical integrations do not depend on that proof. Similarly, the claim in Section 4.1 that the TOV pressure zero is at r≈6.80 while the LE zero is at r≈2.31 contradicts the abstract's ordering statement, but an inconsistent empirical claim is not an input-equivalent derivation. No fitted parameter is renamed as a prediction, and no result is imported from prior work by the same authors.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central numerical claim depends on the polytropic EoS parameters (A, p0, n) as inputs, on the standard TOV/LE equations, and on the unproven existence/uniqueness of solutions. No new entities are introduced.

free parameters (3)

polytropic constant A = varied (0.1, 1, 8)
Polytropic constant in ρ=A p^{1/γ}; varied by hand in Section 4.4.
initial pressure p0 = varied (0.1, 1, 8)
Central pressure boundary condition; chosen by hand.
polytropic index n = varied over 0 to 5
Polytropic index scanned to generate Figure 6.

assumptions (6)

domain assumption The special-relativistic Hamiltonian H = mc^2 sqrt(1+p^2/m^2c^2) describes a non-interacting gas
Used in Section 2.2 to compute the partition function; the author states no source was found.
domain assumption Adiabatic condition δQ=0 holds during stellar evolution
Invoked in Section 2.3 to derive the relation between volume and temperature.
domain assumption Spherical symmetry and ideal-fluid stress-energy tensor
Basis of the TOV derivation in Section 3.1.
standard math Schauder fixed-point theorem
Used in Lemmas 5.6 and 5.12 to prove local existence.
standard math Quittner-Souplet theorem on non-existence of global solutions for semilinear elliptic equations
Used to establish the LE finite-boundary theorem in Section 5.1 and Appendix E.
domain assumption Polytropic EoS ρ=A p^{1/γ}
Used for all numerical TOV and LE comparisons.

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Cite this review

Pith. "Pith review of Zero Values of the TOV Equation." pith.science (2026). https://pith.science/paper/VHK3WSHP

@misc{pith2026241115264,
  author       = {Pith},
  title        = {Pith review of: Zero Values of the TOV Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VHK3WSHP}},
  note         = {Machine review of arXiv:2411.15264}
}

abstract

This thesis calculates the special-relativistic internal energy of a non-interacting gas. We derive an equation of state (EoS) which we apply to the Tollmann-Oppenheimer Volkoff (TOV) equation. Furthermore we present numerical results of the TOV equation for various configurations of a polytropic EoS. These results show that the zero values of the TOV equation compared to its non-relativistic counterpart, the Lane-Emden (LE) equation, are smaller for identical parameters. We present initial developments to a proof of this theorem. Furthermore, an additional exact solution and index $n=2$ of the LE equation was discovered.

Figures

Figures reproduced from arXiv: 2411.15264 by the authors.

**Figure 1.** Relativistic Equation of State The relativistic EoS ρ(p) is normalised such that values can be compared with a polytropic EoS. Graphs for the relativistic version are independent of the exponent n which is a degree of freedom intrinsic to the polytropic EoS. By normalisation, the graphs of the polytropic EoS are independent of the factor A. 11 [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗

**Figure 2.** Graph of exact LE solutions [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** Comparison of the TOV and LE equation. The images show the plots for the parameters of [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Validation of numerical LE results The first two plots show absolute and relative difference of known exact and numerically calculated results while the third plot shows the behaviour of the maximum value of ∆ as the stepsize decreases. 4.3 Relativistic EoS In the prev…

**Figure 5.** Figure 5: Comparison of TOV polytropic and relativistic EoS [PITH_FULL_IMAGE:figures/full_fig_p028_5.png]

**Figure 6.** Figure 6: Zero Values of TOV and LE equation In the first plot results for p0 = 8 have been omitted for better visual clarity. These behave similar to the p0 = 1 results. 25 [PITH_FULL_IMAGE:figures/full_fig_p030_6.png]

**Figure 7.** Figure 7: LE Solution for n = 2 We see the power the series calculated with the coefficients explained above. The far right tick is the last calculated value for Rm where plotting is stopped) and the sequence Rm = (|am|) −1/m which approaches the radius of convergence. These res…

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5 extracted references · 4 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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