The Gravity Tunnel Superhighway

Alexander R. Klotz

REVIEW 3 major objections 5 minor 13 references

The Gravity Tunnel Superhighway

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper derives the exact radius of a constant-radius arc that minimizes travel time in a two-shaft gravity tunnel, under both constant-gravity and uniform-density models of Earth's interior.

desk verdict The final formulas are correct and the paper is a clean classroom problem; just fix the garbled derivation steps in Eqs. (5) and (14) before signing off. read the letter →

arxiv 2411.18758 v1 pith:RVCKUHKI submitted 2024-11-27 physics.pop-ph

Alexander R. Klotz This is my paper · ORCID

classification physics.pop-ph

keywords gravitytunnelshaft-and-arctraveltimeminimizationbrachistochroneconstant-gravitymodeluniform-densityphysicseducationcalculusofvariations

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 asks how fast a two-shaft-plus-arc gravity tunnel could be if the connecting segment is a circular arc at constant radius, and derives the exact radius that minimizes the end-to-end travel time. Under two idealized models of Earth's interior — constant gravitational acceleration and uniform density — the optimal radius and the minimum time come out in closed form: $r_g = 2R(\theta-2)/(\theta-4)$ and $T_g = \sqrt{(R/g)(8\theta - 2\theta^2)}$ for the constant-gravity model, and $r_\rho = R\sqrt{1-\theta/2}$ and $T_\rho = \sqrt{R/g}\left(\sqrt{2\theta-\theta^2}+2\arcsin\sqrt{\theta/2}\right)$ for the uniform-density model. These times are about 10% slower than the corresponding brachistochrone, and the same minimization can be averaged over all angular separations to give the best radius for a single 'superhighway' ring. The paper's motivation is pedagogical: the optimization uses only calculus, not variational methods, making it a gentler entry point to minimization ideas.

What carries the argument

The load-bearing object is the total travel time $T(r,\theta)=2T_f(r)+T_s(r,\theta)$, where the fall time $T_f$ is obtained from energy conservation in the shaft and the slide time is $r\theta/v(r)$ at the constant speed reached at the bottom. Minimizing this one-variable function with respect to $r$ — a single derivative, not a functional — is what produces the closed-form radii and times. For the uniform-density model the same structure is used with the simple-harmonic-motion solution of the shaft, and the slide speed comes from $v=\sqrt{gR(1-(r/R)^2)}$.

What would settle it

Simulate or construct a small-scale shaft-and-arc tunnel with a rounded entrance, and measure the time-minimizing radius; if it deviates from $2R(\theta-2)/(\theta-4)$ (constant gravity) or $R\sqrt{1-\theta/2}$ (uniform density) beyond the smoothing correction, the ideal-junction assumption is falsified for that geometry.

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Extended reading notes

Core claim

The central claim is that for a gravity tunnel made of two radial shafts joined by a constant-radius arc, the travel time as a function of connecting radius $r$ has a single physical minimum inside the planet, and that minimum can be written in elementary functions for the two standard interior models. For constant gravity the minimal radius is $r_g = 2R(\theta-2)/(\theta-4)$, which tends to zero as the surface separation approaches 2 radians and gives time $T_g=\sqrt{(R/g)(8\theta-2\theta^2)}$; for uniform density the minimal radius is $r_\rho = R\sqrt{1-\theta/2}$ with time $T_\rho = \sqrt{R/g}\left[\sqrt{2\theta-\theta^2}+2\arcsin\sqrt{\theta/2}\right]$. For a ring intended to serve all pairs of points, averaging the time over all angular separations yields the 'superhighway' radii $r_{av,g}=2R(4-\pi)/(8-\pi)\approx 0.35R$ and $r_{av,\rho}=R\sqrt{4-\pi}/2\approx 0.46R$. The author also compares the shaft-and-slide path with the brachistochrone and finds it about 10% slower, which he frames as a cheap price for a much simpler construction.

Load-bearing premise

The derivation assumes the traveler enters the arc at the full free-fall speed and continues at that constant speed with no loss at the smoothed junction, and that rotation and friction are absent.

Editorial extensions

If this is right

The optimal shaft-and-slide tunnel is a viable pedagogical proxy for the brachistochrone: it teaches minimization with only calculus, and its optimal depth and time are closed-form in both standard Earth models.
The shaft-and-slide path is at most about 10% slower than the brachistochrone between the same surface points, so a constant-radius arc is a cheap approximation to the true fastest path.
For antipodal travel ($\theta=2$), the optimal arc radius collapses to zero, recovering the fall-through-the-center time, which is 38 minutes under constant gravity and 42 minutes under uniform density.
If a single ring is dug to connect many cities, the best depth under the averaged-time criterion is around $0.35R$ (constant-gravity model) or $0.46R$ (uniform-density model), both lying within the outer core.
The same down-over-up minimization appears in a flat vertical-field analogue whose optimal drop height is $D/4$, giving a factor of $4/\pi \approx 1.27$ depth and $\sqrt{4/\pi}\approx 1.12$ time penalty compared to the cycloid brachistochrone.

Reading between the lines

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

In real-Earth terms the constant-gravity radius $0.35R$ is the more physically relevant target, since the author's earlier work showed constant $g$ matches seismic-density fall times better than uniform density; this paper does not itself draw that engineering conclusion.
A finite smoothing radius at the shaft-to-arc junction, or any dissipative loss, would break the constant-speed assumption; a natural extension is to recompute the optimum under a constraint on centripetal acceleration or curvature.
The 10% gap between shaft-and-slide and brachistochrone suggests a design heuristic: when excavation costs rise steeply with depth, the shallow constant-radius tunnel may become the true cost optimum even though it is slower than the mathematical minimum-time curve.
The same averaging trick over $\theta$ could be applied to a finite set of cities rather than a uniform great-circle distribution, which would shift the optimal depth toward the most-traveled angular separations.

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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

The punchline: the final formulas are right, but the derivation as printed has two spots that are not. The stress-test note holds up. Differentiating Eq. (4) and Eq. (13) does give the claimed optima in Eqs. (6) and (15), but the displayed Eq. (5) is internally inconsistent—sqrt(g(r-R)) is imaginary for r

What's actually new: within the cited gravity-tunnel literature, nobody has optimized this particular piecewise path—two vertical shafts joined by a constant-radius arc. The paper gives clean closed forms for the optimal radius and minimal time under both a constant-g and a uniform-density Earth model, plus the "superhighway" radius that minimizes average travel time over a great circle under the stated weighting. The derivations are elementary one-variable minimization, exactly right for an undergraduate problem set. The energy-conservation step that sets the arc speed is explicit, and the no-friction, no-rotation, smoothed-transition idealization is standard for this genre and not load-bearing. The paper is honest about the auxiliary polynomial fit for the constant-g brachistochrone: it is just for plotting, fails at small angles, and the author says so.

Soft spots aside from Eqs. (5) and (14): Eq. (14)'s typesetting is genuinely hard to parse, and Section V (role in physics education) runs a little long. Neither changes the core contribution. The self-citation to the author's 2015 paper is appropriate background, not padding. The dream origin and Thanksgiving acknowledgment are charming, not flaws.

Verdict: this is a solid pedagogical contribution. It does not change physics, engineering, or measurement; its value is in the classroom and in recreational gravity-tunnel lore. The math is simple enough that an undergraduate can reproduce it, which is the point. The central argument holds up. A serious referee should engage with it, especially to clean up the printed derivation. I'd send this to review, not desk reject.

Referee Report

3 major / 5 minor

Summary. The paper studies a gravity tunnel consisting of two vertical radial shafts connected by a circular arc at constant radius. It derives the arc radius that minimizes the total fall-slide-rise time under two models of Earth's interior: constant gravitational acceleration and uniform density. Closed-form expressions are obtained for the optimal radius and the minimum travel time in each model (Eqs. (6)–(7) and (15)–(16)), and these times are compared with the corresponding brachistochrone. The paper also considers a circular "superhighway" and derives the radius that minimizes the average travel time over all angular separations (Eqs. (19)–(20)). Finally, it discusses the value of this problem in undergraduate physics education.

Significance. If the formulas are correct, the paper provides exact closed-form optima for a simple, physically motivated path family, derivable with elementary calculus and energy conservation. The comparison with brachistochrones is instructive, and the superhighway extension is a motivating capstone. The derivation is self-contained and does not rely on fitted parameters except for the auxiliary polynomial in Eq. (8), which is explicitly used only for plotting. The central arithmetic checks out, and the main weaknesses are the garbled derivative displays and a few clarity issues that are local and fixable.

major comments (3)

[Section II, Eq. (5)] The displayed derivative is internally inconsistent: for r<R the term sqrt(g(r-R)) is imaginary, and the signs of the terms do not follow from differentiating Eq. (4). The correct derivative is dT/dr = [(θ-2) + θ r/(2(R-r))] / sqrt(2g(R-r)), whose numerator vanishes at r = 2R(θ-2)/(θ-4), which is Eq. (6). As printed, Eq. (5) does not support the claimed result and must be repaired.
[Section III, Eq. (14)] The displayed equation cannot vanish for 0<r<R because the leading term is positive and all other terms are also positive (for the physical parameter range). The correct derivative of Eq. (13) yields a single physical root r = R sqrt(1 - θ/2), i.e., Eq. (15). The author should replace Eq. (14) with the correct expression and, if useful, show the intermediate algebra.
[Section IV, Eq. (18)] The claim of an "optimal depth" for the superhighway depends on the chosen definition of the average over angular separation. The uniform weighting over θ in [0, π] is a modeling assumption, not a consequence of the physics. The abstract and Section IV state "the optimal depth" without qualification. Please explicitly state that Eqs. (19) and (20) are optimal only under this uniform-angular-weighting assumption, and note that a different weighting (e.g., by population or city pairs) would give a different optimum.

minor comments (5)

[After Eq. (14)] The statement that "Equations such as (5) and (14) typically have two solutions" is not accurate for the corrected equations: the constant-g derivative has a single physical root, and the uniform-density case has a positive and a negative root, with the negative one unphysical. Please revise the sentence.
[Eq. (17)] The notation Tg,ρ for the brachistochrone time is confusing because g and ρ are used elsewhere to label the two interior models. A subscript such as "br" would be clearer.
[Figure 2] The left panel shows the optimal radius for both models, but the caption does not identify which curve corresponds to which model. Please add a legend or label the curves explicitly.
[Eq. (8)] The fourth-order polynomial appears to be a fit with empirically determined coefficients. Please state explicitly that the coefficients are fitted, that the approximation is used only for plotting, and specify the range of validity (the text says "for θ between 0 and 2" but does not state whether θ is in radians and whether the formula returns minutes).
[Section IV, Eq. (18)] The sentence "the constant term picks up a factor of π and the linear term a factor of π^2/2" is terse; consider showing the integration explicitly to help readers reproduce the average.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the optimal radii and minimum times follow by direct differentiation of the paper's own kinematically derived travel-time expressions.

full rationale

The paper's central results are self-contained minimizations. For constant gravity, Eq. (4) is assembled from free-fall kinematics and constant-speed arc motion, and differentiating it yields the optimal radius in Eq. (6); for uniform density, Eq. (13) is built from the simple-harmonic solution and differentiating it yields Eq. (15). No parameter is fitted to the target optimum, and no cited result is used to force the stationarity condition. The author's self-citation [1] only motivates why the constant-gravity assumption is reasonable; it is not load-bearing for the derivation of the optimum under that assumption. The polynomial fit in Eq. (8) is explicitly marked as a plotting aid and does not feed back into the optimization. The brachistochrone comparisons are external benchmarks rather than inputs. The printed derivative displays in Eqs. (5) and (14) contain sign/notation errors, but that is an internal correctness issue, not circularity: the quoted final radii check out against correct differentiation. The superhighway average in Section IV uses the already-derived travel times and minimizes the average directly. Overall, the derivation chain is independent of its conclusions.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central derivations use no fitted constants beyond Earth's standard radius and surface gravity. One auxiliary polynomial fit appears in Eq. (8) and is not load-bearing. The main assumptions are the two Earth-interior models and idealizations of a lossless, frictionless, non-rotating tunnel. No new physical entities are introduced.

free parameters (1)

fourth-order polynomial coefficients for constant-g brachistochrone time (Eq. 8) = -4.4, 22.7, -45.9, 51.4, 5.4
Fitted to the exact brachistochrone time for the constant-g model so it can be plotted in Figure 2. The paper admits it fails for small angles and says it does not convey insight. It is not used to derive the central optimum, but it supports the statement that constant-g tunnel times are about 10% slower.

assumptions (5)

domain assumption Gravitational field inside Earth can be approximated as constant (constant-g model).
Used throughout Section II; justified by Earth's density profile and cited to the author's 2015 paper; a modeling assumption, not derived here.
domain assumption Earth has uniform density, giving a linear restoring force via the shell theorem.
Used in Section III, Eq. (9); a standard textbook model, but an approximation to the real Earth.
domain assumption The transition between shaft and arc is smoothed, lossless, and takes negligible time; motion is frictionless and Earth is non-rotating.
Invoked in the Introduction ('after a smoothed transition', 'Ignoring rotation, friction') and in Eq. (3) energy conservation; determines arc speed and total time.
ad hoc to paper The superhighway average weights every angular separation theta uniformly over [0, pi].
Definition in Eq. (18); reasonable for random point pairs on a great circle, but not derived from city or population distributions; affects the superhighway radius result.
standard math Calculus minimization (set derivative to zero, select the physical root) and the shell theorem are valid.
Standard mathematical tools used in Eqs. (5)-(7) and (14)-(16).

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Cite this review

Pith. "Pith review of The Gravity Tunnel Superhighway." pith.science (2026). https://pith.science/paper/RVCKUHKI

@misc{pith2026241118758,
  author       = {Pith},
  title        = {Pith review of: The Gravity Tunnel Superhighway},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RVCKUHKI}},
  note         = {Machine review of arXiv:2411.18758}
}

read the original abstract

This manuscript discusses gravity tunnels formed by connecting two vertical shafts by a constant-radius tunnel within the Earth, which featured in a dream I had in September 2024. The total travel time through such a tunnel can be minimized with respect to the radius at which the shafts are connected. I derive this minimal radius and minimum time given two assumptions for Earth's interior, that of constant gravitational acceleration and that of uniform density. Both models have solutions in terms of basic functions, and are typically 10% slower than the brachistochrone curve between the same points. I also find the optimal depth of a "superhighway," which minimizes the average time to fall between any two points on a great circle. Finally, I discuss the role of problems like these in physics education.

Figures

Figures reproduced from arXiv: 2411.18758 by the authors.

**Figure 2.** FIG. 2. Left. Optimal radius of the angular portion of the tunnel to minimize travel time, as a function of the angle traversed. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Pith tools

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

[1] [1]

The gravity tunnel in a non-uniform earth

Alexander R Klotz. The gravity tunnel in a non-uniform earth. American Journal of Physics, 83(3):231–237, 2015

work page 2015

[2] [2]

A. Redier. De paris a rio-janeiro en 42m11s. La Nature, 520, 1883

work page

[3] [3]

Through the earth in forty minutes

Paul W Cooper. Through the earth in forty minutes. American Journal of Physics, 34(1):68–70, 1966

work page 1966

[4] [4]

Terrestrial brachistochrone

Giulio Venezian. Terrestrial brachistochrone. Am. J. Phys, 34(8):701, 1966

work page 1966

[5] [5]

From conservation of en- ergy to the principle of least action: A story line

Jozef Hanc and Edwin F Taylor. From conservation of en- ergy to the principle of least action: A story line. Amer- ican Journal of Physics, 72(4):514–521, 2004

work page 2004

[6] [6]

Classical mechanics

John Robert Taylor. Classical mechanics. 2005

work page 2005

[7] [7]

Gravity Tunnel Drag

Thomas Concannon and Gerardo Giordano. Gravity tun- nel drag. arXiv preprint arXiv:1606.01852, 2016

work page Pith review arXiv 2016

[8] [8]

Free fall through the rotating and inhomogeneous earth

Stefan Isermann. Free fall through the rotating and inhomogeneous earth. American Journal of Physics , 87(8):646–653, 2019

work page 2019