Bounds on the minimum orbital period in the background of 5-dimensional charged black holes

Yan Peng

REVIEW 2 major objections 4 minor 23 references

Bounds on the minimum orbital period in the background of 5-dimensional charged black holes

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For 5D charged black holes, the minimum orbital period is pinned between two mass-only bounds: 6√π√M ≤ Tmin ≤ 8√6π/3 √M.

desk verdict A correct but thin 5D extension; the main missing piece is an analytic monotonicity proof, which is easy to supply. read the letter →

arxiv 2506.21871 v1 pith:LAXE5M6Y submitted 2025-06-27 gr-qc hep-phmath-phmath.MP

Yan Peng This is my paper

classification gr-qchep-phmath-phmath.MP PACS 11.25.Tq04.70.Bw74.20.-z

keywords minimumorbitalperiod5-dimensionalblackholeschargedboundscircularnullgeodesicsextremalhigher-dimensionalgravity

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 establishes that the minimum orbital period of a test object circling a 5-dimensional charged black hole is bounded on both sides by expressions involving only the black hole's ADM mass: 6√π√M ≤ Tmin ≤ 8√6π/3 √M. The upper bound is reached when the black hole is neutral, and the lower bound when it is maximally charged. The work extends to five dimensions the 4-dimensional story in which Hod's bound and an upper bound constrain the fastest way to circle a black hole. If correct, it shows that in 5D the gravitational physics alone fixes the quickest orbit time at the two charge extremes, giving a clean target for higher-dimensional gravity models.

What carries the argument

The machinery is the circular null orbit period function T(r)=2πr/√(1 − 2M̃/r² + Q̃²/r⁴), obtained from ds²=0 on equatorial circular orbits. Minimizing this function over r>r_h yields the critical radius r_c = √(2M̃ + √(4M̃² − 3Q̃²)) and the exact minimum period expression (9). The bounds then follow from the monotonicity of Tmin with Q̃ and the horizon condition Q̃≤M̃.

What would settle it

Numerically search for periodic geodesics in the metric (1) that complete one revolution in coordinate time below 6√π√M for a given mass M; finding such an orbit would falsify the lower bound. Alternatively, evaluate the expression (9) and find a parameter region where Tmin increases with Q̃, contradicting the upper-bound assignment.

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Extended reading notes

Core claim

The central result is a pair of analytical bounds on the minimum orbital period Tmin of circular orbits around the 5D charged black hole described by the metric ds² = −(1 − 2M̃/r² + Q̃²/r⁴)dt² + (1 − 2M̃/r² + Q̃²/r⁴)⁻¹dr² + r²dΩ₃². The author derives the exact period formula T(r)=2πr/√(1 − 2M̃/r² + Q̃²/r⁴) for equatorial circular null orbits, minimizes it over r outside the horizon, and finds that Tmin decreases monotonically as the charge parameter Q̃ grows. Consequently the neutral case Q̃=0 gives the upper bound and the extremal case Q̃=M̃ gives the lower bound. Rewriting the mass parameter in terms of the ADM mass M yields the compact bounds 6√π√M ≤ Tmin ≤ 8√6π/3 √M, so at the two extremes the fastest orbit time is fixed by the mass alone.

Load-bearing premise

The derivation assumes that the globally shortest orbital period is attained by a light-speed circular orbit in the equatorial plane; if a non-circular, non-equatorial, or slower-than-light orbit could complete a revolution in less coordinate time, the bounds would not follow.

Editorial extensions

If this is right

At zero charge, the fastest orbit around a 5D Schwarzschild-type black hole takes exactly 8√6π/3 √M coordinate time.
At maximal charge, the fastest orbit takes exactly 6√π√M, and any charged black hole in between has a minimum period between these two values.
Because the bounds are fixed by the ADM mass alone, they provide a consistency condition for any proposed 5D gravity model that produces charged black holes.
The minimum period shrinks as charge increases, so charge acts to speed up the fastest possible orbit.
The result generalizes the 4D bounds 4πM ≤ Tmin ≤ 6√3πM to five dimensions, with a different mass scaling due to the extra spatial dimension.

Reading between the lines

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

A natural extension would be to check whether the monotonicity of Tmin with Q̃, which the paper verifies numerically, can be proven analytically from Eq. (9); if it fails for any parameter range, the bounds would need refinement.
The same minimization logic could be applied to D-dimensional charged black holes, potentially producing a family of mass-only bounds with dimension-dependent coefficients.
If a 5D black hole ever becomes observationally relevant, the lower bound could serve as a sharp test: any observed orbital period shorter than 6√π√M would rule out the 5D charged black hole metric.
The paper restricts to equatorial circular null orbits; whether non-equatorial or non-circular orbits can beat the claimed minimum is left open, and checking that would either confirm or tighten the bounds.

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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

Short paper, one idea: extend the minimum-orbital-period bounds from 4D to the 5D charged Schwarzschild metric. The method is the same as the author's previous 4D papers: set ds^2=0 for null equatorial circular orbits, compute T(r)=2πr/sqrt(1-2M~/r^2+Q~^2/r^4), minimize, then evaluate at the endpoints of the allowed charge range. The derivation of the critical radius rc is correct, and the endpoint bounds (10) and (11) are correct. The advertised bounds in terms of ADM mass, Eq. (12), also check out under the paper's convention for M~.

The soft spot is the load-bearing monotonicity assumption. After finding T_min, the author says 'we numerically find' that T_min decreases with Q~, and uses that to conclude the upper bound is at zero charge and the lower at maximal charge. That step is not proven anywhere in the text. It is easy to prove—the stress-test note gives a one-line rescaling showing T_min = 2π sqrt(3/2) sqrt(M~) (a+2)/sqrt(a+1) with a = sqrt(4-3q^2), which is strictly decreasing in q—but as written the paper rests on a numerical observation. A referee should ask for this proof. Also, Eq. (9) is misprinted: as printed it does not reduce to the correct value at Q~=0 (there's an extra factor in the denominator). The exact expression needs to be corrected.

The significance is modest. This is a direct extension of results already known in 4D, using the same technique; there is no new formalism or unexpected physics. But the calculation is clean and the bounds are explicit, and a 5D version is a legitimate addition to the literature. The paper is honest in scope and the citations are appropriate, mostly to the author's own prior work in the same program.

A serious referee can fix this in an afternoon. I'd send it to peer review with a request for an analytic proof of monotonicity and a corrected Eq. (9). With those changes it's publishable in a specialist journal.

Referee Report

2 major / 4 minor

Summary. The manuscript studies the minimum orbital period of test objects on circular lightlike trajectories in the 5-dimensional charged black hole metric (1). Starting from the null condition ds²=0 on the equatorial plane, the author derives the coordinate orbital period T(r)=2πr/sqrt(1−2M~/r²+Q~²/r⁴), finds the critical radius r_c=sqrt(2M~+sqrt(4M~²−3Q~²)) by differentiation, and writes the resulting minimum period T_min in Eq. (9). Numerical plots are used to argue that T_min decreases with the charge parameter Q~, which leads to endpoint bounds at Q~=0 and Q~=M~. These are then converted into the mass-only bounds 6√π√M ≤ T_min ≤ 8√(6π)/3 √M in Eq. (12). The paper claims these are precise analytical bounds for 5-dimensional charged black holes.

Significance. If the claimed bounds are established, they constitute a mass-only universal constraint on the fastest orbital time in this higher-dimensional charged black hole background and extend a line of 4-dimensional results (Hod's bound and its variants) to five dimensions. The derivation up to the critical radius is elementary and self-contained, with no fitted parameters; the endpoints of the charge interval are evaluated correctly from the corrected minimum-period formula. The main advertised result, however, is not yet fully proven because the monotonicity in charge is only supported numerically, and the printed Eq. (9) contains algebraic errors. Both issues are localized and readily fixable by including a short analytic monotonicity proof and correcting the displayed formula.

major comments (2)

[§II, Eq. (9) (and Fig. 1)] Equation (9) is not the correct minimum-period expression as printed. Substitution of r_c²=2M~+Δ, Δ=sqrt(4M~²−3Q~²), into Eq. (6) gives T_min = 2π(2M~+Δ)^{3/2} / sqrt(4M~²−2Q~²+2M~Δ). The printed expression has only a single power (2M~+Δ) in the numerator, includes an extra factor sqrt(2M~+Δ) in the denominator, and the last square root contains Q~³ instead of Q~². With these errors, Eq. (9) does not reduce to the stated endpoint bounds (10) and (11); for example, at Q~=0 the printed form yields √2 π/√M~ rather than 4√2 π√M~. The authors should correct Eq. (9) and confirm that Fig. 1 and the endpoint calculations are based on the corrected formula.
[§II, after Eq. (9) and Fig. 1] The claim that T_min/√M~ decreases monotonically with Q~ is supported only by a numerical plot, but this monotonicity is load-bearing for the two-sided bounds (10)–(12). If an interior charge gave a larger or smaller T_min than the endpoints, the advertised bounds would fail. Since an exact expression is available, the authors should supply an analytic proof, for example by setting q=Q~/M~ and a=sqrt(4−3q²), writing T_min = 2π√M~ (2+a)^{3/2}/sqrt(4−2q²+2a), and showing that the derivative with respect to q is strictly negative on 0≤q≤1. Until that proof is included, the 'analytical' upper and lower bounds rest on a numerical observation rather than a derivation.

minor comments (4)

[§II, Eq. (1)] The event-horizon equation has two roots; the text should explicitly distinguish the outer horizon r_h=sqrt(M~+sqrt(M~²−Q~²)), which defines the exterior region used in the analysis.
[§II, Eq. (3) and abstract] The bound is derived for circular lightlike trajectories, but the paper speaks of 'test objects' and 'the minimum orbital period' without qualification. Please state explicitly that the result concerns lightlike circular orbits, or justify why no timelike orbit can have a shorter coordinate period.
[§II, Fig. 1] The figure panels should be labeled clearly with the values of M~ (1 and 3), and the axes should be labeled with the plotted quantity T_min/√M~ and Q~ rather than appearing as unlabeled ranges.
[§II, Eqs. (10)–(11)] The factors such as 4√2π and 3√3π should be written as 4√2 π and 3√3 π (or with explicit multiplication) to avoid ambiguity, especially since the mass-parameter relation M~=4/(3π)M is also easy to misread.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bounds are derived by direct minimization of the orbital period in a fixed external metric.

full rationale

The paper's derivation is self-contained once the five-dimensional charged black hole metric (1) from [22] is adopted. The orbital period T(r) in Eq. (6) follows algebraically from the null condition ds^2=0 and the period integral over one full revolution. Minimizing T(r) gives the critical radius rc in Eq. (8), and substitution yields the claimed minimum period expression. The endpoint bounds are obtained by setting Q~=0 and Q~=M~ respectively, which are the extremal charge values allowed by the horizon condition. The paper supports the charge-monotonicity step with numerical observation rather than an analytic proof, and Eq. (9) contains a typographical error in the printed denominator; these are correctness or rigor concerns, not circularity. No parameter is fitted to the output, no input quantity is defined in terms of the claimed bound, and no load-bearing result is imported solely from the author's prior work. The self-citations [19]-[21] are contextual comparisons and do not carry the derivation. Thus the central claim does not reduce by construction to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the cited 5D charged metric and on the null-orbit assumption. There are no fitted free parameters and no new physical entities introduced.

assumptions (4)

domain assumption The 5D charged black hole metric (1) with \tilde M = 4/(3\pi)M and \tilde Q = 2\sqrt{3\pi}Q is a valid description of the spacetime.
Taken from ref. [22] without derivation; all subsequent results depend on this metric.
domain assumption The fastest orbital period occurs in the light-speed limit, so the period is computed from null circular orbits (ds^2=0).
Stated in Section II before Eq. (3); no proof is given that massive-particle orbits cannot have shorter coordinate periods.
domain assumption Equatorial circular orbits with \theta_1=\theta_2=\pi/2 and dr=0 are sufficient to find the global minimum.
The paper reduces to this case without proving that other trajectories yield smaller periods.
standard math T(r) tends to infinity at the horizon and at infinity, so the single critical point r_c is the global minimum.
Calculus argument in Section II; the limits follow from f(r_h)=0 and T~2\pi r at large r.

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Cite this review

Pith. "Pith review of Bounds on the minimum orbital period in the background of 5-dimensional charged black holes." pith.science (2026). https://pith.science/paper/LAXE5M6Y

@misc{pith2026250621871,
  author       = {Pith},
  title        = {Pith review of: Bounds on the minimum orbital period in the background of 5-dimensional charged black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LAXE5M6Y}},
  note         = {Machine review of arXiv:2506.21871}
}

read the original abstract

In this paper, we study the upper and lower bounds on the minimum orbital period of 5-dimensional charged black holes. Our results indicate that the upper bound of the minimum orbital period corresponds to non-charged black holes, while the lower bound is achieved in the case of maximally charged black holes. We further establish precise analytical expressions for the upper and lower bounds of the minimum orbital period. Our findings provide valuable insights into 5-dimensional charged black holes and help constrain theoretical gravity models.

Figures

Figures reproduced from arXiv: 2506.21871 by the authors.

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Works this paper leans on

23 extracted references · 21 canonical work pages

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Emparan and H

R. Emparan and H. S. Reall, Black holes in higher dimensio ns, Living Reviews in Relativity 11 (2008) 1

work page 2008

[2] [2]

G. T. Horowitz, Black holes in higher dimensions. Cambri dge University Press, 2012

work page 2012

[3] [3]

The event horizon is located at rh = √ ¯M ± √ ˜M 2 − ˜Q2, which is the real root of 1 − 2 ˜M r2 + ˜Q2 r4 = 0

(1) Here ˜M is the mass parameter expressed as ˜M = 4 3π M with M as the black hole ADM mass and ˜Q is the charge parameter expressed as ˜Q = 2√ 3π Q with Q as the black hole electric charge. The event horizon is located at rh = √ ¯M ± √ ˜M 2 − ˜Q2, which is the real root of 1 − 2 ˜M r2 + ˜Q2 r4 = 0. The existence of horizons requires ˜Q ⩽ ˜M . And the te...

work page

[4] [4]

Kanti, Black holes in theories with large extra dimens ions: A review, International journal of modern physics A 19 (2004) 4899

P. Kanti, Black holes in theories with large extra dimens ions: A review, International journal of modern physics A 19 (2004) 4899

work page 2004

[5] [5]

Aharony, S

O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri and Y. Oz , Large n ﬁeld theories, string theory and gravity, Physics Reports 323 (2000) 183

work page 2000

[6] [6]

M.Bardeen,W

J. M.Bardeen,W. H.Press,andS. A.Teukolsky, Astrophys . J. 178, 347 (1972)

work page 1972

[7] [7]

Chandrasekhar, The Mathematical Theory of Black Hole s (Oxford University Press, New York, 1983)

S. Chandrasekhar, The Mathematical Theory of Black Hole s (Oxford University Press, New York, 1983)

1983

[8] [8]

S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwar fs, and Neutron Stars: The Physics of Compact Objects (Wiley, New York, 1983)

work page 1983