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The paper shows that in static spherically symmetric spacetimes, the constant-speed brachistochrone is the null geodesic of the optical metric, and solves it explicitly for the singular isothermal sphere, Schwarzschild exterior, and relativ

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2026-08-03 10:27 UTC pith:ISHHOTOD

load-bearing objection A clean, modest paper: the general result is Fermat's principle in disguise, but the analytic SIS solution and the Plummer application are genuinely new and worth publishing. the 1 major comments →

arxiv 2607.29265 v1 pith:ISHHOTOD submitted 2026-07-31 gr-qc

The brachistochrone problem for a constant velocity traveler in static and spherically symmetric spacetimes

Longfei Wang , Junji Jia This is my paper
classification gr-qc PACS 04.20.-q04.20.Cv
keywords brachistochrone problemstatic spherically symmetric spacetimeFermat's principleoptical metricsingular isothermal sphererelativistic Plummer profilenull geodesiccoordinate time minimization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper asks which spatial curve a traveler moving at fixed local speed should follow to minimize coordinate time in a static, spherically symmetric spacetime. It establishes that, if the traveler is free to steer without energy or acceleration limits, the brachistochrone is exactly the null geodesic of the optical (Fermat) metric, so every constant subluminal speed v shares the same fastest route, with duration scaled by 1/v. For the singular isothermal sphere interior the route is solved in closed form and always bends, with both turning radius and travel time growing with the equation-of-state index w. In a Schwarzschild exterior the route also always bends, and heavier central masses make it bend more and take longer. For the relativistic Plummer model, routes starting and ending within the core radius are straight lines through the center, while routes outside the core bend and bend more for more concentrated cores.

Core claim

The central claim is that minimizing coordinate time for a fixed-speed traveler in the SSS metric (1) collapses to one first-order equation, Eq. (13), dθ/dr = ±√[A(r)B(r)C(r0)/(C(r)(C(r)A(r0) − C(r0)A(r)))], and that the same equation holds for any constant subluminal speed v, with total time scaled by 1/v. The paper derives this from the Euler–Lagrange equations and identifies the minimizer with the null geodesic of the optical metric, a form of Fermat's principle. In the SIS interior the equation integrates in closed form to r(θ)=r0 cos^{−1/β}[β(θ−θ0)/α]; r0 is always the minimum radius and grows with w. Outside, in Schwarzschild spacetime, the same equation produces always-bent trajectori

What carries the argument

The load-bearing object is Eq. (13), the first-order brachistochrone equation dθ/dr = ±√[A(r)B(r)C(r0)/(C(r)(C(r)A(r0) − C(r0)A(r)))], a null geodesic equation of the optical/Fermat metric — the spatial metric obtained from ds²=0, in which curves that minimize coordinate time are geodesics. After a planar reduction (φ=0), this ODE encodes the whole variational problem and fixes the turning radius r0 at dr/dθ=0; its ∫ form (14) then gives the path for any SSS spacetime. For subluminal speeds the time integrand differs only by the constant factor 1/v, so the identical equation governs the route. This reduces the brachistochrone problem to quadrature: explicit integration gives Eq. (22) for the

Load-bearing premise

The load-bearing premise is that the traveler can steer along any smooth curve at a fixed local speed, without limits on how hard it accelerates or on its energy; if such limits are imposed, the fastest path will not generally be the null geodesic of the optical metric.

What would settle it

Numerically solve the same variational problem for a relativistic Plummer profile with both endpoints inside the core (e.g., r_i = r_f = 2M, b = 4M) using an optimizer that enforces a maximum proper acceleration. If a bent path under that acceleration cap reaches the endpoint in less coordinate time than the straight polar diameter, the claim that the straight line is the brachistochrone would fail in the constrained regime. Independent numerical integration of Eq. (13) for the SIS interior should reproduce Eq. (22); any mismatch would pinpoint an error in the closed form.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • For equal-radius endpoints in the SIS interior, the brachistochrone always dips to a turning radius r0 < ri, and both r0 and the total coordinate time increase as the equation-of-state index w rises.
  • In the Schwarzschild exterior, the fastest route always bends, and a larger central mass M yields a larger turning radius and a longer travel time.
  • In the relativistic Plummer model, fastest routes with both endpoints inside the core are straight lines through the center; with endpoints far outside the core they bend, and more concentrated cores bend them more.
  • Any constant sub-light speed v uses the exact same spatial route as the light-speed traveler; only the duration changes, by a factor 1/v.
  • The derived Eq. (13) agrees with the null geodesic equation of the optical metric, confirming the generalized Fermat principle for SSS spacetimes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Inference beyond the paper: because the optimal route does not depend on the traveler's speed, a route-planning system could compute the fastest spatial path once per spacetime and reuse it at any constant cruise speed; only the clock's rate changes.
  • Inference beyond the paper: the Plummer bend-to-straight transition suggests a general criterion for regular mass profiles — if the core scale exceeds the endpoint radii, the optical metric is effectively weak enough that the central straight diameter wins, which could be tested against other smooth density profiles.
  • Inference beyond the paper: with a proper-acceleration ceiling imposed, the optimal path should deviate from the optical-metric geodesic, so the size of the deviation is a direct measure of how much acceleration constraints matter for real spacecraft.
  • Inference beyond the paper: applying the same variational reduction to axisymmetric stationary spacetimes should yield non-planar fastest paths with azimuthal drift; the SSS solutions here provide a benchmark for numerical solvers before tackling the less symmetric case.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper studies the brachistochrone problem for a traveler moving at constant local speed in static, spherically symmetric spacetimes. It derives the Euler–Lagrange equations for an ultra-relativistic (null) traveler, reduces the motion to a plane, and obtains the first-order equation dθ/dr = sqrt(A B C0 / [C(C A0 - C0 A)]). This is applied to the singular isothermal sphere (SIS) interior, where an analytic solution r(θ)=r0 cos^{-1/β}[β(θ−θ0)/α] is found, to the exterior Schwarzschild vacuum, and to the relativistic Plummer profile. The paper then shows by Fermat's principle that for any constant subluminal speed v the minimizing spatial path is unchanged and the total time is scaled by 1/v.

Significance. The paper has three clear strengths: a clean variational derivation of the brachistochrone equations, an elegant analytic solution for the SIS interior, and a simple scaling argument connecting subluminal constant-velocity travelers to null geodesics of the optical metric. The explicit connection to Fermat's principle is a useful conceptual contribution, and the Plummer example demonstrates a qualitatively different regime in which the fastest path is straight through the center. The stated scope condition — no bound on proper acceleration and a fixed local speed — is explicit and is not an internal inconsistency. If the error in the Schwarzschild exterior section noted below is corrected, the paper will be a solid contribution to the variational treatment of travel-time problems in curved spacetime.

major comments (1)
  1. [Section III B, Eq. (28)] The Schwarzschild integral is incorrect. Substituting A=1−2M/r, B=1/A, C=r^2 into Eq. (13) gives dθ/dr = 1/[r sqrt(1−2M/r) sqrt( r^2(1−2M/r0)/(r0^2(1−2M/r)) − 1 )]. The factor sqrt(1−2M/r') belongs in the denominator, not the numerator. As written, Eq. (28) is (1−2M/r') times the correct integrand. Since Eqs. (29) and the numerical results in Fig. 2 are built on this integral, the Schwarzschild exterior results need to be recomputed. The qualitative trends may survive, but the quantitative claims for r0 and Δt in Section III B are not supported by the equations as stated.
minor comments (5)
  1. [Section III A, Eqs. (22) and (27)] The formulas contain 1/β, which is singular at w=1 (β=0), yet Fig. 1 includes w=1. The limit β→0 should be displayed or at least described so that the plotted w=1 curve is derived, not assumed.
  2. [Section III A, Eq. (24a)] The notation cos^{-1} is ambiguous. Use \arccos for the inverse cosine to distinguish it from a power of cos.
  3. [Section IV, Eq. (33)] The regime r0=0 is discussed as a straight-line solution, but the integrand in Eq. (33) is singular at r0=0. The limiting procedure should be stated.
  4. [Section V] The statement that the BT is 'the same' for subluminal velocity could be sharpened: the spatial projection is the same geodesic of the optical metric, while the parameterization and coordinate time differ. The text mostly says this, but an explicit sentence would prevent confusion.
  5. [References] Reference [14] lacks page numbers and appears incomplete as formatted.

Circularity Check

0 steps flagged

No significant circularity — the variational derivation is self-contained and the Fermat-principle extension is a consistency check, not an imported result.

full rationale

The paper's central derivation is self-contained. The brachistochrone equation Eq. (13) is obtained by applying the Euler–Lagrange equations to the effective action L = tdot defined in Eq. (4), with tdot fixed by the null condition ds^2 = 0 in Eq. (3). The first integrals, including Eqs. (9)–(12), are derived from the variational principle rather than assumed. The SIS analytic solution Eq. (22) follows by direct integration of Eq. (19) and the endpoint conditions Eqs. (23)–(24); no parameter is fitted to the target prediction. The subluminal extension in Section V factors a constant 1/v out of the time integral in Eq. (40), making the minimization equivalent to that of the null case; this is a straightforward mathematical consequence, not a circular redefinition. The citations to Fermat's principle and generalized Fermat principles (Refs. [24–26]) are used as a consistency check and for interpretation, not as load-bearing derivations, and the authors do not rely on self-citations. The explicit scope assumption that the traveler maintains constant local speed without an energy constraint is stated in Section I as part of the problem setup, so it is an honest limitation rather than a hidden circular step. No equation or fitted quantity is asserted to be equivalent to the result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper introduces no free fitted parameters: w, M, and b are model inputs from prior literature, and r0, θ0 are integration constants fixed by boundary conditions. No new particles, forces, or modified gravity are postulated; only existing spacetime models and a kinematic traveler assumption are used.

axioms (5)
  • ad hoc to paper The traveler can traverse any smooth curve at a fixed local speed with unlimited proper acceleration and no energy constraint.
    Sec. I explicitly chooses fixed local velocity over fixed energy; this is the defining premise of the model and excludes acceleration- or energy-bounded travelers.
  • standard math Euler–Lagrange equations are valid for the coordinate-time functional under the null constraint.
    Used in Sec. II to derive Eqs. (7)-(10).
  • domain assumption The SIS metric (16) from Remmen [22], with Schwarzschild matching and M = 2wR/(1+6w+w^2), correctly describes the spacetime.
    Sec. III relies on this metric for all SIS results; physical validity is inherited from the cited solution.
  • domain assumption The relativistic Plummer metric (31) from Tabatabaei et al. [23] is a valid, stable, BH-free spacetime for b > M.
    Sec. IV uses this metric; the b > M stability condition is taken from the cited paper.
  • domain assumption Fermat's principle in conformally stationary spacetimes, and its extension to massive signals, is applicable.
    Sec. V invokes Perlick [24,25] and Alsing [26] to identify the BT with the null geodesic of the optical metric and to extend to constant v < 1.

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Cite this review

Pith. "Pith review of The brachistochrone problem for a constant velocity traveler in static and spherically symmetric spacetimes." pith.science (2026). https://pith.science/paper/ISHHOTOD

@misc{pith2026260729265,
  author       = {Pith},
  title        = {Pith review of: The brachistochrone problem for a constant velocity traveler in static and spherically symmetric spacetimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ISHHOTOD}},
  note         = {Machine review of arXiv:2607.29265}
}
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read the original abstract

This work investigates the brachistochrone problem for a traveler with constant local velocity within static and spherically symmetric (SSS) spacetimes. The brachistochrone trajectory (BT) equations for ultra-relativistic travelers are derived for general SSS metrics, and the solution is formally obtained in an integral form. We then apply the result to two representative spacetimes corresponding to the singular isothermal sphere (SIS) with a finite boundary and the relativistic Plummer mass profile, respectively. For the SIS spacetime, the BT inside the boundary is solved analytically and found always to bend. As the equation of state index $w$ increases, the turning radius $r_0$ of the BT, and consequently the total time, also increase. For the BT outside the boundary, it is found that the heavier the central object, the larger the $r_0$ and the total travel time. For the relativistic Plummer model, the BT will be a straight line passing through the origin when the initial and final points' radii are comparable or smaller than the size of the core region of the mass distribution. When the end points lie well outside the core region, the BT bends, exhibiting a larger turning radius for a more concentrated core. We then extend the consideration to travelers with subluminal constant velocity $v$ and show through the generalized Fermat's principle that the BT will be the same geodesic in the optical metric as ultra-relativistic travelers, with the total travel time scaled by a factor of $1/v$.

Figures

Figures reproduced from arXiv: 2607.29265 by Junji Jia, Longfei Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. BTs (colored lines) within the SIS connecting: (left [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. BTs (colored lines) within the Schwarzschild re [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: as functions of the parameters b/M and ri,f /M. It is seen that for any constant b/M, as ri,f /M decreases, the minimal r0 decreases monotonically. Moreover, at a value of ri,f roughly the same order as b, the r0 becomes zero identically. This implies that when the initial and final points of the trajectory are within the core radius of the relativistic Plummer mass distribution, the straight line directly… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. BTs (colored lines) within the relativstic Plum [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

discussion (0)

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Reference graph

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