REVIEW 5 minor 14 references
A Simple Pendulum in Schwarzschild Spacetime
T0 review · 0 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read Exact formula gives a pendulum's period near a black hole
desk verdict A clean, self-contained derivation of the Schwarzschild pendulum period for a fixed proper-length rope, with an honest analysis of its main idealization; worth publishing. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the static-slice spatial metric $h_{ij}$ with $h_{ij}dx^i dx^j=dr^2/N^2+r^2 d\theta^2$, together with its geodesics: by stress-energy conservation, a massless taut rope must lie on one, so the rope's shape is fixed by the geodesic equation. The load-bearing identity is the fixed-proper-length constraint, $\delta r(\theta)=N_2 r_s(N_1-N_2)\theta^2/4$, which converts the bob's two-dimensional motion into a one-dimensional harmonic oscillator. Everything else follows from expanding the point-particle Lagrangian in $\theta$ and reading off the frequency; the lapse $N(r)$ is the same function that appears in redshift and clock-rate formulas.
What would settle it
A numerical solution of an elastic rope with finite tension-wave speed, in the same static-slice geometry, should approach Eq. (42) only as the wave speed tends to $c$ and the rope mass tends to zero; if the limiting period still differs from Eq. (42), or if it matches the coordinate-length prediction $T_{MD,(2)}\propto N_2$ instead, the quasi-static geodesic assumption would be falsified.
Extended reading notes
Core claim
The paper's central claim is that the physical period of a fixed-proper-length pendulum in Schwarzschild spacetime, measured by a static observer at the bob at $r=r_2$ with the fulcrum at $r_1$, is $T_{(2)}=4\pi r_2^2\sqrt{N_2(N_1-N_2)}/(c r_s)$. After the transients decay, a taut, very light, inextensible rope of fixed proper length coincides with a geodesic of $h_{ij}$, the metric induced on the static hypersurface. The conserved quantity $r^2\,d\theta/ds=J$ combines with the arc-length normalization to produce a one-parameter family of geodesics; matching the fixed length $L_0$ and the endpoint angle $\vartheta$ gives the radial lift $\delta r=N_2 r_s(N_1-N_2)\vartheta^2/4$ to quadratic order. Substituting this holonomic constraint into the point-particle Lagrangian of the bob yields a harmonic-oscillator Lagrangian whose coordinate-time frequency corresponds to $T_t=4\pi r_2^2\sqrt{(N_1-N_2)/N_2}/(c r_s)$. Since a static observer at $r_2$ measures proper time $d\tau=N_2\,dt$, the local period is $T_{(2)}=N_2 T_t=4\pi r_2^2\sqrt{N_2(N_1-N_2)}/(c r_s)$.
Load-bearing premise
The result assumes the rope always settles instantly into a spatial geodesic of fixed proper length, which requires a tension wave to cross the rope many times during one swing; for slow tension waves or very long ropes the period formula would break down and an elastic-string model would be needed.
Editorial extensions
If this is right
- In the Newtonian limit, with $r_{1,2}\gg r_s$ and $L_0\ll r_2$, the period reduces to $T=2\pi\sqrt{L_0/g}$ with $g=GM/r_2^2$, exactly reproducing the classical pendulum result.
- Close to the horizon, the period scales as $T_{(2)}\propto\sqrt{N_2}$, and a local observer sees an effective pendulum length $\ell_{\mathrm{eff}}=2N_1 r_s$; combining that length with the local gravitational acceleration reproduces the limiting period.
- A static observer at the fulcrum measures a period longer by the ratio $N_1/N_2$, so the pendulum itself exhibits the same redshift factor that appears in clock-comparison experiments.
- The fixed-proper-length model and the coordinate-length model of earlier work agree only in the weak-field limit; near the horizon they predict different scalings, $T_{(2)}\propto\sqrt{N_2}$ versus $T_{MD,(2)}\propto N_2$, so they describe physically distinct pendula.
Reading between the lines
- One consequence not drawn in the paper is that any constrained system whose shape follows a spatial geodesic should show the same lapse-combination structure, so Eq. (42) may be a special case of a general proper-length small-oscillation theorem in static spacetimes.
- A natural numerical check beyond the paper's scope would simulate a finite-tension-wave-speed rope with small mass: the period should approach Eq. (42) as the wave speed tends to $c$ and the mass goes to zero, while the coordinate-length model will not.
- The quasi-static consistency condition implies a practical bound for real pendula near a horizon: the fulcrum cannot be too distant ($r_1-r_s\ll 2\pi r_s\epsilon^{-1/4}$), which could guide whether tabletop black-hole analogues can realize the predicted period.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper determines the small-oscillation period of a pendulum in Schwarzschild spacetime under the following idealization: a taut, massless, inextensible rope of fixed proper length L0, with the fulcrum at radius r1 and the bob in equilibrium at radius r2, restricted to a meridional plane. Section 2.1 proves from stress-energy conservation that a massless tension-only rope coincides with a geodesic of the spatial metric of the static slice; the fixed-proper-length condition then implies that a small angular displacement ϑ is accompanied by a radial lift δr = N2 rs ϑ²/[4(N1−N2)] (Eqs. (28)–(31)). Substituting this holonomic constraint into the free-particle Lagrangian and expanding to quadratic order gives a harmonic oscillator with coordinate-time period Tt = 4π r2²/(c rs) √[(N1−N2)/N2] (Eq. (41)), and the central result is the period measured by a static observer at the bob's equilibrium position, T(2) = 4π r2²/(c rs) √[N2(N1−N2)] (Eq. (42)); a static observer at the fulcrum measures T(1) = (N1/N2) T(2). The paper recovers the Newtonian period 2π√(L0/g) in the weak-field short-rope limit (Eq. (53)), gives a near-horizon interpretation in terms of the local acceleration g ≃ c²/(2rsN2) and an effective length ℓeff = 2N1rs (Sec. 4.2), compares the fixed-proper-length model with the coordinate-length model of Ref. [1] and shows they predict different near-horizon scalings (T ∝ √N2 versus T ∝ N2, Eq.
Significance. If Eq. (42) holds, it is a notable result: a closed-form, parameter-free expression for the period of a textbook system in Schwarzschild spacetime, written in terms of the lapse function that governs clock rates and redshifts. The derivation is fully analytic, and I checked the algebra independently: the radial-lift relation (31), the reduced Lagrangian (40), and the resulting period (41)–(42) are internally consistent, as are the Newtonian limit (53) and the effective-length argument (54)–(63). Strengths worth naming: the result is falsifiable in a precise sense, because the fixed-proper-length and fixed-coordinate-length models predict different near-horizon scalings for the same experimental setup (Eq. (77)); the comparison with Ref. [1] is an independent check rather than an input; and the paper does not overclaim the quasi-static rope assumption, since Sec. 6 gives a quantitative necessary condition, shows the approximation improves as the bob approaches a fixed fulcrum, and identifies the near-horizon subcase where the signal time can reach a non-negligible fraction of the period (Eq. (91)).
minor comments (5)
- [§2.3, Eq. (31)] At Eq. (31), the static quadratic constraint is carried over to the dynamical problem with the phrase "for an instantaneous bob coordinate θ(t)"; because this step is precisely the quasi-static assumption whose validity is analyzed only in Section 6, a forward reference at this point would help the reader recognize that the central result is conditional on that assumption.
- [§4.2, Eqs. (57)–(58)] The ordering condition (58) deserves one interpretive sentence: for a fixed amplitude ϑ, the near-horizon limit r2→rs can be taken only while the inequality holds, which for N1 of order unity means roughly ε ≡ (r2−rs)/rs ≫ ϑ⁴; Eq. (54) is therefore an intermediate-asymptotic statement in which ϑ and ε are sent to zero in a correlated way rather than a limit at fixed ϑ. Without such a remark the two limiting statements in Sec. 4.2 can easily be misread as commuting.
- [§5, Eq. (72)] In Eq. (72), the remainder notation O(α²) is potentially misleading because the omitted terms are not uniformly of order α² in this two-parameter expansion: they are α² times functions that are less singular in (r2−rs) but still nonvanishing. Please either give the subleading terms or use a two-parameter order symbol; the surrounding text states the point in words, but the equation itself can be misread.
- [§5, Eq. (78)] The criticism of Ref. [1]'s Appendix B would be substantially easier to verify if the corrected Christoffel symbol (78) were derived from the coordinate transformation (64) in a few displayed lines, or if the explicit metric components in the (r′,θ′) coordinates were listed; as written, the reader must reconstruct the reference's full calculation to confirm the allegedly omitted factor. Since this is the paper's only direct charge of an error in another work, completeness matters here.
- [Abstract and Sec. 1] The abstract's statement that "after transients decay, the displaced rope coincides with a geodesic of the induced spatial metric" reads as a dynamical fact; I suggest qualifying it with "under the quasi-static rope assumption," since Section 6 makes clear that a real rope with finite tension-wave speed need not satisfy this condition, and the bound derived there is only a necessary condition for the approximation.
Circularity Check
No circularity: Eq. (42) is derived self-consistently from the Schwarzschild metric and a fixed-proper-length constraint, with no fitted parameters and no load-bearing self-citation.
full rationale
The paper's central result, Eq. (42), is derived from first principles within the paper rather than being imported from its inputs. The derivation chain is self-contained: the massless-taut-rope stress-energy conservation argument in Sect. 2.1 proves that the rope lies on a spatial geodesic (Eqs. (14)-(19)); the fixed proper length condition, Eqs. (20) and (24), yields the radial lift δr in terms of the angular amplitude, Eq. (30) and then Eq. (31); substituting that holonomic constraint into the free-particle Lagrangian gives the harmonic-oscillator Lagrangian, Eq. (40), and hence the period, Eqs. (41)-(42). No parameter is fitted to the period itself, and the comparison with the coordinate-length model of Ref. [1] is an independent check, not an input. The Newtonian limit and near-horizon ∝√N2 scaling are derived consequences, not assumptions. The paper's own Section 6 explicitly identifies the quasi-static rope assumption as an idealization with a stated validity condition; this is an honest scope limitation, not a circular step. There is no self-citation chain on which the derivation depends. The result is therefore fully self-contained under the stated idealization.
Assumptions & free parameters
assumptions (5)
- domain assumption Schwarzschild metric is the fixed background; pendulum elements are test objects whose backreaction is neglected.
- domain assumption The rope is massless, taut, and inextensible, with zero energy density, so its spatial stress tensor is S^ij = -T t^i t^j and it follows a spatial geodesic.
- domain assumption After transients decay, the displaced rope coincides with a spatial geodesic joining fulcrum and bob; during motion it adjusts quasi-statically.
- standard math Small-angle expansion is valid; terms beyond quadratic order in the action and cubic in the constraint are neglected.
- standard math F(N) = N/(1-N^2) + arctanh N is strictly increasing on (0,1), so N1 = F^{-1}(F(N2) + L0/r_s) is unique.
Cite this review
Pith. "Pith review of A Simple Pendulum in Schwarzschild Spacetime." pith.science (2026). https://pith.science/paper/PM5RZSPQ
@misc{pith2026260800866,
author = {Pith},
title = {Pith review of: A Simple Pendulum in Schwarzschild Spacetime},
year = {2026},
howpublished = {\url{https://pith.science/paper/PM5RZSPQ}},
note = {Machine review of arXiv:2608.00866}
}
abstract
We determine the small-oscillation period of a simple pendulum in Schwarzschild spacetime. After transients decay, the displaced rope coincides with a geodesic of the induced spatial metric. This determines the radial lift of the bob to quadratic order in the angular amplitude and leads to the period measured by an observer at the bob's equilibrium position, $r=r_2$, with the fulcrum at $r_1$, \[ T_{(2)}=4\pi\frac{r_2^2}{c r_s}\sqrt{N_2(N_1-N_2)}, \] where $N_i=N(r_i)$ and $N(r)=\sqrt{1-r_s/r}$ is the Schwarzschild lapse. The result is therefore expressed in terms of the same function that determines clock rates and gravitational redshifts. In the Newtonian limit, we reproduce the classical result $T=2\pi\sqrt{L_0/g}$. We interpret the period close to the horizon in terms of an effective pendulum length. We compare our treatment with the coordinate-length constraint adopted in an earlier study and argue that the two problems are inequivalent away from the classical, weak-field limit.
Reference graph
Works this paper leans on
-
[1]
M. A. Martin-Delgado,A General Relativistic Pendulum: Isochronous vs. Geodesic Motion (2022),arXiv:2205.02509
work page Pith review arXiv 2022
-
[2]
Gourgoulhon,3+1 formalism and bases of numerical relativity(2007),gr-qc/0703035
E. Gourgoulhon,3+1 formalism and bases of numerical relativity(2007),gr-qc/0703035
arXiv 2007
-
[3]
A. R. Brown,Tensile Strength and the Mining of Black Holes, Phys. Rev. Lett.111, 211301 (2013),arXiv:1207.3342
arXiv 2013
-
[4]
Poisson and C
E. Poisson and C. M. Will,Gravity: Newtonian, Post-Newtonian, Relativistic, Cambridge University Press, Cambridge (2014)
2014
-
[5]
V. P. Frolov and A. Zelnikov,Introduction to Black Hole Physics, Oxford University Press, Oxford (2011)
work page 2011
-
[6]
P. Fromholz, E. Poisson, and C. M. Will,The Schwarzschild metric: It’s the coordinates, stupid!, Am. J. Phys.82, 295–300 (2014)
work page 2014
-
[7]
Particle hanging on a string near a Schwarzschild black hole
M. LaHaye and E. Poisson,Particle hanging on a string near a Schwarzschild black hole, Phys. Rev. D104, 044016 (2021),arXiv:2106.06968
work page Pith review arXiv 2021
-
[8]
A. P. Lightman, W. H. Press, R. H. Price, and S. A. Teukolsky,Problem book in relativity and gravitation(1979)
work page 1979
Show all 14 references
-
[9]
Lee,Introduction to Riemannian Manifolds, Springer, Cham, Switzerland, 2nd edition (2019)
J. Lee,Introduction to Riemannian Manifolds, Springer, Cham, Switzerland, 2nd edition (2019)
2019
-
[10]
F. D. Mazzitelli,The infalling photon, the infalling particle, and the observer at rest near the horizon of a black hole, Eur. J. Phys.41, 065601 (2020)
2020
-
[11]
L. D. Landau and E. M. Lifschitz,The Classical Theory of Fields, Butterworth-Heinemann, Amsterdam, 4th edition (1980)
1980
-
[12]
R. K. S. Hankin,Light Inextensible Strings (Thread) Under Tension in the Schwarzschild Geometry, The Physics Educator03, 2150005 (2021)
2021
-
[13]
Gourgoulhon,Relativité générale(2014),http://luth.obspm.fr/~luthier/gourgoulhon/ fr/master/relat.html
E. Gourgoulhon,Relativité générale(2014),http://luth.obspm.fr/~luthier/gourgoulhon/ fr/master/relat.html
2014
-
[14]
Hammerlindl, J
A. Hammerlindl, J. Bowman, and T. Prince,Asymptote: the Vector Graphics Language(2025), https://asymptote.ualberta.ca/. 15
2025
Reviewed August 15, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.