Characterizing geodesic deviations in a Topological Star spacetime: massive, charged, spinning and stringy-like objects

Donato Bini; Giorgio Di Russo

REVIEW 2 major objections 6 minor 2 cited by

Characterizing geodesic deviations in a Topological Star spacetime: massive, charged, spinning and stringy-like objects

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Pith's one-line read In a Topological Star spacetime, every probe class — orbits, spinning bodies, charges, strings — deviates from Schwarzschild through one extra scale $r_b$, with the boundary skimmed tangentially instead of crossed.

desk verdict Solid geodesic-deviation core with a clean TS/Schwarzschild epicyclic discriminator; the spin section undermines itself by plotting s-hat=1 in a linear-in-spin MPD expansion. read the letter →

arxiv 2505.13020 v1 pith:GAWYKSJT submitted 2025-05-19 gr-qc

Donato Bini , Giorgio Di Russo This is my paper · ORCID

classification gr-qc PACS 04.20.-q04.70.-s11.25.-w

keywords TopologicalStargeodesicdeviationepicyclicfrequencyMathisson-Papapetrou-DixonequationsspinningparticleschargedstringyprobesSchwarzschildcomparison

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

The paper sets out to establish that the Topological Star (TS), a horizonless spacetime whose boundary is a cap at $r=r_b$, can be told apart from a Schwarzschild black hole by how test orbits deviate from geodesics. It derives explicit TS corrections to Schwarzschild motion for four families of probes: nearby geodesics, spinning bodies, magnetically charged particles, and electrically charged strings wound on the extra dimension. The pattern is that every deviation is controlled by the single dimensionless ratio $\alpha = r_b/r_s$, entering through the factor $\sqrt{f_b(r_0)} = \sqrt{1 - \alpha r_s/r_0}$. If the paper is right, the main observable signature is a small downward shift in the epicyclic frequency of quasi-circular orbits, of order $\frac12\,\alpha r_s/r_0$, and a qualitative difference at the boundary: black holes capture infalling probes, whereas the TS lets them skim the cap smoothly.

What carries the argument

The load-bearing object is the Topological Star metric $ds^2 = -f_s(r)\,dt^2 + dr^2/[f_s(r)f_b(r)] + r^2(d\theta^2+\sin^2\theta\,d\phi^2)+f_b(r)\,dy^2$, with $f_s(r)=1-r_s/r$ and $f_b(r)=1-r_b/r$, where $r_s<r_b$ so $r=r_b$ is a smooth cap replacing the horizon. This single metric supplies the whole comparative apparatus: every deviation formula in the paper reduces to the Schwarzschild expression with the factor $\sqrt{f_b(r_0)}$ (or powers of it) attached, so the parameter $\alpha=r_b/r_s$ organizes all four probe classes. The secondary machinery is the linear-in-spin Mathisson-Papapetrou-Dixon system with the covariant spin condition for spinning bodies, the Lorentz force for magnetic charges, and the worldsheet action with a coupling to the electric three-form potential for winding strings; its job is to turn each probe's internal structure into a calculable deformation of the reference geodesic.

What would settle it

An orbital-timing measurement around a compact object of known mass, at radius $r_0$, that matches the Schwarzschild epicyclic frequency to fractional precision better than $\tfrac12 r_s/r_0$ would exclude every stable TS configuration, since stability restricts $1<\alpha<2$.

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Extended reading notes

Core claim

The authors establish that in a Topological Star spacetime every modification of geodesic motion — for neutral neighbours, spinning bodies, magnetically charged particles, and electrically charged winding strings — is organized by the same extra scale $r_b$ encapsulated in $f_b(r)=1-r_b/r$. Concretely, the epicyclic frequency of circular equatorial geodesics becomes $\Omega_c = \Omega_{\rm Schw}\sqrt{f_b(r_0)}$, so for large radii $|\Delta\Omega_c|/\Omega_{\rm Schw}\simeq \frac12\,\alpha r_s/r_0$; the maximum relative enhancement of the radial deviation for unbound orbits is $2v^2\alpha\,\varepsilon$, where $\varepsilon = r_s/(2bv^2)$ is the post-Minkowskian parameter; the spinning-body system is sixth order with three regimes separated at $r^*=(5+\sqrt{2})r_s/4$, and its negative-spin solution reaches the cap $r=r_b$ tangentially; and the stringy circular-orbit Lyapunov exponent reduces to the Schwarzschild shadow value when $r_b=m=q=0$. The paper summarizes the results as: a black hole captures particles, while the TS only allows passages that smoothly skim the cap, with all TS deviations being small corrections to the Schwarzschild case.

Load-bearing premise

The spinning-particle results stand on the assumption that spin is small enough to ignore spin-squared effects, which the displayed spins $s=\pm1$ (with $s/(m r_s)$ as the small parameter) do not establish.

Editorial extensions

If this is right

A quasi-circular orbit in a TS has epicyclic frequency $\Omega_c = \Omega_{\rm Schw}\sqrt{1-\alpha r_s/r_0}$, a relative shift of about $\frac12\alpha r_s/r_0$ at large radius that precision timing of orbital oscillations could in principle resolve.
Unbound equatorial encounters develop a maximum TS-induced radial deviation of relative size about $2v^2\alpha\,\varepsilon$, giving scattering-like trajectories a way to reveal the extra radius.
Spinning-body deviations exhibit three regimes depending on $r_0$ relative to $r^*=(5+\sqrt2)r_s/4$; in the TS case the inward-moving negative-spin particle approaches $r=r_b$ tangentially, whereas in Schwarzschild it falls through the horizon at an angle.
Magnetically charged probes feel the TS field only through $\hat q^2$ terms at leading order, while electrically charged strings feel it linearly in their winding charge $q$, and both displace the orbit in the same $\varepsilon$ expansion used for geodesics.
The Lyapunov exponent of unstable circular string orbits reduces to the Schwarzschild shadow value when $r_b=m=q=0$, so the TS correction to photon-sphere instability is a continuous function of $\alpha$.

Reading between the lines

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

The factorized form of the frequency shift suggests a mass-independent ratio test: timing two quasi-circular orbits at different radii could isolate $\alpha$ without knowing the central mass.
If the recently introduced rotating Topological Star inherits the cap geometry, these deviation computations should reorganize around the same factor $\sqrt{f_b}$, with frame-dragging mixing the radial and azimuthal channels.
The cap-skimming versus horizon-crossing contrast indicates that boundary-sensitive observations such as gravitational-wave echoes or tidal encounters may carry a sharper signature of horizonlessness than the small orbit shifts the paper emphasizes.

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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

Quick take on arXiv:2505.13020. The paper computes four types of geodesic deviation in topological-star spacetime and contrasts with Schwarzschild. The geodesic-deviation section is the real meat: the y-equation is trivial, theta decouples, and the radial equation gives Omega_c = Omega_Schw sqrt(fb(r0)), so |Delta Omega_c|/Omega_Schw ~ (1/2) alpha rs/r0 at large radius. That is a clean, falsifiable discriminator, and the rb=0 limit recovers Schwarzschild. The PM expansion for unbound orbits and the maximum radial-deviation enhancement ~2 v^2 alpha epsilon are also useful. Credit where due: the central derivation is concise and physically transparent, no constants are fitted, and the Schwarzschild baseline is legitimate.

Now the soft spots. The spin section is the weak point. The MPD equations are solved only to linear order in spin, which the paper states, and the expansion parameter is s-hat = s/(m rs). But Figs. 3 and 4 use s-hat = +/-1. At that value the neglected spin-squared/quadrupolar terms are the same order as the retained linear terms, so the plotted trajectories are not a controlled prediction. The advertised contrast—negative spin approaches the cap tangentially in TS versus crossing the horizon in Schwarzschild—is read off these large-spin runs. The geodesic section independently supports cap-skimming for unbound geodesics, so the overall picture may survive, but the spin-based version is not established as presented. Also, eq. (4.29) has a typo (a third derivative written as a second derivative), and the long PM/string tables are unverified; no symbolic file is shipped. These are limitations, not fatal flaws: the core geodesic-deviation result is unaffected.

Who is this for? People working on exotic compact objects and LISA-era discriminators. The epicyclic frequency result and the PM corrections are citable even if the spin section needs revision. My recommendation: send it to peer review. A serious referee should ask the authors to redo the spin plots at s-hat << 1 or explicitly discuss the uncontrolled regime, and to fix (4.29). The core is worth the referee time.

Referee Report

2 major / 6 minor

Summary. This paper studies deviations from geodesic motion in the five-dimensional Topological Star spacetime for four classes of probes: nearby geodesics, spinning test bodies, magnetically charged particles, and electrically charged stringy objects. The central derivations are an analytic expression for the modified epicyclic frequency of circular geodesic deviation, Omega_c = Omega_Schw sqrt(fb(r0)), a Post-Minkowskian expansion of unbound geodesic deviations, a linear-in-spin Mathisson-Papapetrou-Dixon treatment of circular orbits, a charge-expanded treatment of magnetic Lorentz-force motion, and a string-worldsheet analysis with a Lyapunov exponent for unstable circular orbits. The paper repeatedly contrasts these behaviors with Schwarzschild, and the Discussion claims that a black hole captures particles while a topological star only allows smooth tangential skimming of the cap at r=rb.

Significance. If the results hold, the paper provides a useful catalogue of analytic probe effects in a horizonless fuzzball-like geometry, with concrete observables such as the relative epicyclic shift |Delta Omega_c|/Omega_Schw ~ (1/2) alpha rs/r0 at large radius and the maximum radial-deviation enhancement 2 v^2 alpha epsilon for unbound orbits. The geodesic-deviation section is clean and recovers Schwarzschild in the limit rb=0; the PM tables and the charged/string extensions substantially enlarge the literature on topological-star dynamics. The main weakness is that the spin sector is presented in a regime where the linear-in-spin truncation is not controlled, which undermines the qualitative cap-skimming contrast that the Discussion highlights. The paper is forward-modeling throughout, with no constants fitted to data, and the Schwarzschild limit serves as a well-defined external baseline.

major comments (2)

[Section IV, Eqs. (4.6)-(4.7), (4.20)-(4.35); Figs. 3-4] Section IV solves the MPD equations to linear order in spin, and the paper itself states that 'we therefore limit our considerations below to the linear-in-spin case.' The dimensionless spin is s-hat = s/(m r_s), so the truncation requires s-hat << 1. However, Figs. 3 and 4 plot 'spins s = +/- 1 (in dimensionless units)' with r0 = 1.25, r_s = 0.8, r_b = 1, i.e. s-hat = +/- 1. At this value the omitted terms of order s-hat^2, coming from expanding the connection and the Riemann tensor off the reference circular geodesic and from the U^alpha U^beta term in Eq. (4.13), are of the same order as the retained linear terms. The qualitative conclusion drawn from these plots---that a negative-spin body approaches r = r_b tangentially in a TS while it crosses r = r_s in Schwarzschild---is therefore not a reliable prediction of the linearized MPD system. The authors should either restrict the spin-deviation plots and claims to s-hat << 1, or include the second-order spin corrections and demonstrate that the qualitative behavior persists.
[Section VII and Section III] The concluding claim that 'while a BH captures particles, the TS only allows for passages which (smoothly) skim the cap' is presented as a general result, but the only explicit illustrations in this paper are the large-spin MPD trajectories of Fig. 3(a). The geodesic-deviation analysis of Section III concerns relative deviations from a reference geodesic and does not by itself exhibit an unbound or radial geodesic that reaches r = r_b. Because cap-skimming should be a property of the spacetime at the geodesic level (e.g., a radial geodesic or an unbound orbit turning at r = r_b), the authors should demonstrate it with a direct geodesic integration, or qualify the Discussion to say that the claim is supported only by the spin-deviation examples.

minor comments (6)

[Section IV, after Eq. (4.29)] The displayed derivative 'd^3 eta_r / dtau^2' should read d^3 eta_r / dtau^3.
[Sections IV-VI and Fig. 4 caption] There are several typos: 'cons ideartions' should be 'considerations', 'strenght' should be 'strength', 'dimesionless' should be 'dimensionless', and 'Schwarzshild' in the Fig. 4 caption should be 'Schwarzschild'.
[Sections V-VI] The symbol hat-q is introduced as the charge-to-mass ratio in Section V, while Section VI reuses q for the winding charge n R_y and introduces the confusing phrase 'q = hat-q/m'. Distinct notation would avoid ambiguity.
[Section III, Eq. (3.24)] The quantity Delta Omega_c / Omega_c^Schw = sqrt(f_b(r0)) - 1 is negative for r0 > r_b; the text quotes the absolute value. Please state this sign explicitly.
[Tables I, III, IV] The Post-Minkowskian coefficients are presented without derivation or numerical cross-check. A supplementary file or a brief description of the recursive solution method would substantially improve verifiability of these long expressions.
[References] References [41] and [43] are the same paper and should be consolidated or cross-referenced once.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's quantitative results are derived from the TS metric and standard equations of motion, with no fitted parameters and no constructional equivalence.

full rationale

The derivation chain is self-contained. The epicyclic-frequency shift in Eqs. (3.22)-(3.25) is obtained by inserting the TS metric of Eq. (2.1) into the standard geodesic-deviation equation (3.1) and solving for the eigenfrequency; the Schwarzschild baseline is recovered at rb=0, and the large-radius limit |Delta Omega_c|/Omega_Schw ~ (1/2) alpha rs/r0 is an algebraic consequence, not a fitted value. The unbound-orbit PM expansion in Eqs. (3.36)-(3.46) is likewise a perturbative solution of the TS geodesic equations for prescribed boundary data, so the alpha-dependent corrections are genuine derived quantities. The spinning-probe section solves the Mathisson-Papapetrou-Dixon equations to linear order in spin with the covariant spin condition, the charged-probe section uses the Lorentz force with the background Maxwell field, and the stringy-probe section uses the standard world-sheet sigma-model action; none of these fits a parameter to data and then renames it a prediction. Self-citations [8,10-12] are context for self-force and charge-stability results but are not load-bearing for the present derivations, and no uniqueness theorem is imported from the authors' prior work. One limitation is explicitly stated in Section IV: 'We therefore limit our considerations below to the linear-in-spin case,' and the plots in Figs. 3-4 use dimensionless spin s-hat = +/-1, where the omitted spin-squared quadrupolar terms are not controlled; this is a validity concern for the large-spin illustration, not circularity, because the equations solved do not presuppose the cap-skimming conclusion.

Assumptions & free parameters 11 free parameters · 6 assumptions · 0 invented entities

The central formulas rest on a handful of background assumptions inherited from prior literature (TS metric, separability, geodesic results) and on two explicit modeling choices (linear-in-spin MPD, rigid string embedding). No new entity is postulated, and no parameter is fitted to data; all plotted parameters are inputs chosen for illustration. The spin magnitude s=+/-1 is the only place where a chosen value may violate the domain of the approximation.

free parameters (11)

alpha = rb/rs = 1.25 in plots (rb=1, rs=0.8); 5/4 in Sections V-VI
Dimensionless parameter characterizing the TS; chosen by hand, not fitted. Central deviation formulas depend on it.
rs (Schwarzschild radius) = 0.8 in plots
Common length scale of the spacetime; input metric parameter set to 0.8 for numerical examples.
rb (cap radius) = 1.0 in plots
TS cap radius; input metric parameter; alpha = rb/rs = 1.25.
M = rs/2 = 0.4 in Schwarzschild limit
Common mass scale used in the post-Minkowskian parameter epsilon = rs/(2 b v^2).
probe spin s (or s-hat = s/(m rs)) = s=+/-1 in Fig. 3
Spin magnitude of the extended body; MPD analysis is linear in spin, so s-hat should be small, but plotted values are not tied to a small s-hat.
background magnetic flux P = P=0.5 in Section V figures
Magnetic 2-form flux parameter of the TS background; chosen by hand.
probe magnetic charge q (or q-hat = q/m) = q=+/-0.3 in Fig. 5
Charge-to-mass ratio of the probe in the small-charge expansion; chosen by hand.
string winding charge q = n Ry = q=+/-0.3 in Figs. 6-7
Winding charge of the stringy probe around the compact y-direction; chosen by hand.
background electric flux Q = Q=0.5 in Figs. 6-7
Electric 3-form charge density parameter; restored from the Q=0 default in Section VI.
velocity v, impact parameter b = v=0.5, b=5 in unbound orbit plots
Orbital parameters for hyperbolic-like geodesics; define the post-Minkowskian parameter epsilon = rs/(2 b v^2).
integration constants C1, C2 and A_{i,j} = C1=C2=0 in circular case; boundary conditions set many A_{i,j}
Chosen to satisfy orthogonality and initial conditions; 9 independent constants remain in the general PM solution.

assumptions (6)

domain assumption The TS metric (2.1) is an exact solution of D=5 Einstein-Maxwell equations and is stable in the parameter range rs < rb < 2rs.
Taken from Refs. [1-3]; the paper's calculations assume the background is physical.
domain assumption Hamilton-Jacobi separability of TS geodesics with a Carter-like constant K, Eq. (2.18), holds.
Taken from prior geodesic analysis [1,3,4,9]; used to set K=L for equatorial orbits.
domain assumption Mathisson-Papapetrou-Dixon equations with covariant spin condition (4.3) are truncated to linear order in spin.
Stated in Section IV; quadrupolar spin-squared terms are omitted. Figures using s=+/-1 may violate the small-spin requirement.
domain assumption The string probe is rigid: embedding X^M(tau,sigma) with y + n sigma Ry and no worldsheet oscillations (Eq. 6.1).
Assumed in Section VI; no justification is given for neglecting oscillator modes.
domain assumption Post-Minkowskian expansion in epsilon = rs/(2 b v^2) and small-charge expansions converge for the orbit parameters studied.
Analytic results are truncated at O(epsilon^3) and O(q^3); the plots use v=0.5, b=5, so epsilon is moderately small in the chosen units.
standard math Standard results of geodesic deviation and Riemann curvature are used without proof.
Eq. (3.1) is the standard Jacobi equation; assumed as background.

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Pith. "Pith review of Characterizing geodesic deviations in a Topological Star spacetime: massive, charged, spinning and stringy-like objects." pith.science (2026). https://pith.science/paper/GAWYKSJT

@misc{pith2026250513020,
  author       = {Pith},
  title        = {Pith review of: Characterizing geodesic deviations in a Topological Star spacetime: massive, charged, spinning and stringy-like objects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GAWYKSJT}},
  note         = {Machine review of arXiv:2505.13020}
}

abstract

We study deviations from geodesic motions in a Topological Star spacetime for either massive, charged and spinning particles, elucidating different behaviours with the Schwarzschild spacetime. We also consider the deviations for the motion of electrically charged stringy probes in $D=5$, framing all cases within a unified picture.

Figures

Figures reproduced from arXiv: 2505.13020 by the authors.

**Figure 2.** FIG. 2: Spin Deviation from the circular geodesic chosen pa [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3: Comparison of the geodesic deviations of three parti [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4: Evolution in time of three particles with different sp [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: FIG. 5: Evolution in time of three particles with different ma [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6: Trajectories followed by the center of mass of the [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: FIG. 7: Evolution in time of the centers of mass of three strin [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

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Forward citations

Cited by 2 Pith papers

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

deviations

Denoting the simple root in (6.19) as r3 = 4AB − 9C − A3 A2 − 3B , (6.21) the polynomial P3(r) becomes P3(r) = ( r − rc)2(r − r3) , (6.22) where r3 = −2rc and rc ≈ 3rs 2 − q 3 √ 3Qrs 2J + q2 rs(32Q2 + 3rs(5rb + 9rs)) 8J 2 + m2 27r3 2 8J 2 , Ec ≈ 2J 3 √ 3rs + q 2Q 3rs + m2 √ 3rs 4J + + q2 4Q2 − 6rbrs + 9r2 s 12 √ 3J rs , (6.23) up to order O(m2, q2) includ...

work page

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Symmetric curl

deviations due to the TS structure correspond to small additional corrections to the Schwarzschild case, mostly interesting from a theoretical point of view. Finally, an appendix summarizes useful geometrical information for the ( t, r, y) part of the TS metric. Acknowledgments We thank A. Geralico for useful comments. D. B. acknowledges sponsorship of th...

work page

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arXiv 2023