Scattering angle in a Topological Star spacetime: a self-force approach

Donato Bini; Giorgio Di Russo; Massimo Bianchi

REVIEW 3 major objections 5 minor 3 cited by

Scattering angle in a Topological Star spacetime: a self-force approach

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A neutral scalar probe scattering around a Topological Star now has an analytic first-order self-force scattering angle, with the geodetic limit recovering Schwarzschild.

desk verdict A competent, honest computation of a new scattering-angle observable for Topological Stars, with one genuine caveat (the unfixed regulator sigma_phi) that the authors themselves flag. read the letter →

arxiv 2506.04876 v1 pith:XEDKCD7L submitted 2025-06-05 gr-qc

Massimo Bianchi , Donato Bini , Giorgio Di Russo This is my paper

classification gr-qc MSC 83C1083C2583C57 PACS 04.25.-g04.70.-s

keywords scatteringangleself-forceTopologicalStarscalarprobepost-MinkowskianexpansionPapapetroufieldsuper-energytensorsgeodesicmotion

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 claims that the scattering angle of a neutral, spinless, scalar-charged test particle in unbound equatorial motion around a Topological Star can be computed analytically, both at the geodesic level and to first order in the scalar self-force. The geodesic result is a power series in inverse angular momentum with coefficients that are polynomial in the energy and in the topological-star parameter $\alpha$, and its $\alpha \to 0$ limit reproduces the Schwarzschild scattering angle. The self-force contribution splits into a conservative and a dissipative part, given explicitly at the stated low post-Minkowskian and post-Newtonian orders; the conservative part contains a regularization cut-off $\sigma_\varphi$ that the paper leaves free. If the result is correct, it provides a concrete, gauge-invariant observable for distinguishing smooth horizonless compact objects from black holes in scattering experiments.

What carries the argument

The argument runs on two pieces. First, a large-angular-momentum expansion of the geodesic equations in the variable $w=1/\hat r$, with rescaled energy $\gamma$ and angular momentum $j$, turns the exact integral for $\chi$ into a series in $1/j$ whose coefficients are read off from the behavior of the effective potential near its largest root. Second, the self-force correction is computed by solving the first-order deviation of the orbit: the radial deviation $\delta r(T)$ is written as an integral over the force components $\hat F_t$, $\hat F_r$, $\hat F_\varphi$ weighted by geodesic kernels, and the scattering angle shifts by $\delta\chi = -2\hat L_-\int \delta r/r_g^3\,d\tau + \int \mathcal{L}(\tau)/r_g^2\,d\tau$, separating conservative (even in velocity) from dissipative (odd in velocity) pieces. The force components themselves are imported from earlier work on scalar perturbations and radiation losses in Topological Star spacetimes.

What would settle it

Recompute the conservative part of the self-force correction at the order where $\sigma_\varphi$ appears (the $\epsilon^2 v^4$ term) with a regularization scheme that introduces no such cut-off, or integrate the scalar self-force numerically for one fixed set of $\gamma$, $j$, and $\alpha$; if no choice of $\sigma_\varphi$ can bring the analytic and numerical results into agreement, the central formula in Eq. (5.50) is wrong.

Watch

Extended reading notes

Core claim

The central object is the scattering angle $\chi$ for a massive probe on an equatorial unbound orbit. The paper derives the geodetic expansion $\chi(\alpha,\gamma,j)+\pi/2 = \sum_{k\ge 1} \chi_k(\alpha,\gamma)/j^k$ with coefficients listed in Table I, exact in the topological-star parameter $\alpha$ and valid as a large-angular-momentum expansion. It then obtains the first-order self-force corrections $\delta\chi_{\rm cons}$ and $\delta\chi_{\rm diss}$ in Eq. (5.50), built from the conservative and dissipative components of the scalar self-force. A notable point is that setting $\alpha=0$ in the self-force result does not reproduce Schwarzschild, because the boundary conditions differ (regularity at the cap versus ingoing at the horizon); only the geodetic part has the Schwarzschild limit. The paper also identifies the background's electromagnetic source as a Papapetrou field and characterizes the Topological Star through sectional curvature invariants, geometric transport along circular orbits, and Bel super-energy tensors.

Load-bearing premise

The self-force components $\hat F_t,\hat F_r,\hat F_\varphi$ and the unbound-orbit expansions are taken from the authors' own earlier papers without being rederived here, and the conservative correction depends on an undetermined cut-off $\sigma_\varphi$; any error or ambiguity in that imported data would propagate directly into the central formula.

Editorial extensions

If this is right

The geodetic series in Eq. (4.69) extends to arbitrary order in $1/j$ with the coefficients in Table I, giving a systematic post-Minkowskian prediction for Topological Star scattering that requires no further input beyond the energy and angular momentum of the probe.
The scalar self-force corrections, though derived at low post-Minkowskian and post-Newtonian orders, are explicit enough to serve as templates for distinguishing a Topological Star from a Schwarzschild black hole through scattering observables.
Because the $\alpha\to 0$ limit of the self-force sector does not equal Schwarzschild, any comparison between Topological Star and black-hole results must account for the different boundary conditions; the mismatch itself is a signature of the horizonless structure.
The super-energy density of a Topological Star factors as the Schwarzschild value times the amplifying factor $A(\alpha x)$, providing a compact diagnostic of the background energy content in the same calculation.

Reading between the lines

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

One could extend the same perturbative scheme to a charged probe, replacing the scalar field with the Papapetrou field; the structure of Eq. (5.50) suggests the dissipative sector would again be free of the cut-off $\sigma_\varphi$.
The appearance of $\sigma_\varphi$ at second order in the conservative angle is a concrete place where an independent post-Minkowskian or numerical self-force computation could fix the regularization scale; until then, comparisons involving the conservative $\epsilon^2$ term should treat that coefficient as scheme-dependent.
If Topological Stars are used as black-hole mimickers in gravitational-wave searches, the 'TS plus $\alpha\to0$ is not Schwarzschild' caveat implies that small-$\alpha$ expansions of self-force observables may be misleading, and the full parameter space should be probed directly.
A natural test of the imported force data is to compute the periastron advance for bound orbits in the same spacetime, which would provide a second gauge-invariant quantity that must be consistent with Eq. (5.50) in the unbound limit.

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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 paper is a straightforward, competent computation of the scattering angle for a scalar-charged probe around a Topological Star, self-force included. The genuinely new part is the geodetic large-j expansion in Section IV: explicit coefficients to order 1/j^6 in Table I, with the alpha->0 Schwarzschild limit as a check. That part is self-contained and looks right.

The self-force part builds on the authors' own earlier work for the force components and unbound-orbit data, so it is not independently derived here. That is a bit of a house-internal chain, but it is not fatal: the method is standard, and the paper is transparent about what comes from where.

The real soft spot is the one the authors themselves put in plain sight. The conservative self-force scattering angle, Eq. (5.50), contains the cut-off regulator sigma_phi at O(eps^2, v^4) and v^6. As written, the conservative piece is scheme-dependent at the advertised order; it becomes a prediction only after external matching or numerics fixes sigma_phi. The dissipative part is unaffected, and the authors explicitly say the regulator must be fixed by comparison with another approach. So this is an honest limitation, but it does undercut the phrasing 'complete analytic result' for the conservative sector. A referee should ask them to either fix sigma_phi or state the claim as conditional.

The characterization sections (Papapetrou field, super-energy, parallel transport) are auxiliary. Fine, but they do not carry the paper.

Overall: credit where it is due. The geodetic expansion is new and clean; the self-force extension is a plausible, well-documented application of prior results; the regulator issue is real and acknowledged. I would send it to a referee, because the observable is new, the method is standard, and the caveat is explicit. The referee should insist on a clearer statement about the status of the conservative result.

Referee Report

3 major / 5 minor

Summary. This paper studies unbound equatorial scattering of a massive scalar-charged test particle in the Topological Star spacetime, with the five-dimensional metric reduced to four-dimensional leaves. The geodesic scattering angle is derived in a large-angular-momentum expansion, Eq. (4.69), with coefficients listed in Table I, and the α→0 limit is reported to reproduce the known Schwarzschild geodetic result. The paper then includes first-order scalar self-force corrections, separating the result into conservative and dissipative parts in Eq. (5.50), using self-force components and unbound-orbit expansions imported from the authors' earlier works Refs. [40,42]. The spacetime is further characterized by a Papapetrou-field description of the electromagnetic source, sectional curvature invariants, super-energy tensors, and parallel and Fermi-Walker transport along circular orbits. The central claim is an explicit analytic PM/PN expression for the scattering angle at first self-force order.

Significance. If the imported self-force data were independently validated, the result would be a useful analytic prediction for scattering off a horizonless compact object, complementing black-hole self-force results. The geodesic part is self-contained and checked against Schwarzschild, and the open discussion of the σφ ambiguity is a point in the paper's favor. However, the conservative sector of the central result currently contains an undetermined logarithmic scale, so the manuscript as it stands does not provide a complete parameter-free observable at the advertised order. The dissipative part and the geodesic expansion are nevertheless concrete, falsifiable outputs, and the paper contains several genuine geometric characterizations of the Topological Star spacetime.

major comments (3)

[§V.B, Eq. (5.50)] The conservative self-force correction contains the dimensionless scale σφ explicitly at O(ε², v⁴) and O(ε², v⁶), and the text states that a proper choice of σφ must be made only when another approach computes the same quantity. As written, Eq. (5.50) is a one-parameter family rather than a definite prediction, and the abstract's claim of computing the scattering angle including self-force effects is overstated. This is load-bearing because the conservative self-force correction is the main new observable; the paper should either fix σφ through a stated matching condition or explicitly qualify the result as a scheme-dependent expression awaiting external calibration.
[§V, Eq. (5.40) and Table III] The self-force components and unbound-orbit data are quoted from Refs. [40,42] without rederivation, and Eq. (5.50) is obtained by integrating those quantities. Any missing term or regularization convention in the imported data propagates directly into the central result. I recommend adding a compact derivation of at least the leading-order force components or a cross-check independent of the previous papers, such as a flux-conservation test for the dissipative sector or an explicit comparison with a known scalar self-force result in a limit where the boundary conditions coincide.
[§IV.D and Footnote 67] The only stated sanity check, the α→0 limit of the geodetic result, does not apply to the self-force sector, and Footnote 67 explicitly warns that “TS plus α→0 is not Schwarzschild” because of different boundary conditions. The paper should provide a separate validation of the self-force result, or clearly state that the self-force part is currently unvalidated, so that the reader does not infer a Schwarzschild check from the geodesic limit.

minor comments (5)

[Abstract and Introduction] The abstract says “scalar neutral probe” while the introduction speaks of a “neutral, massive, spinless test particle”; since the probe carries a scalar charge q in Eq. (5.39), the intended meaning is that it is electromagnetically neutral but scalar-charged. Please make this explicit.
[Sections II–III and IV.A–IV.C] The Papapetrou-field, super-energy, sectional-curvature, and geometric-transport material is not used in the scattering-angle calculation; consider either shortening it, moving it to an appendix, or stating explicitly how it feeds into the main observable.
[Notation throughout] The symbol γ is used both for the geodesic energy parameter and for the Lorentz factor in Section V; the notations γPN,n partially distinguish these uses, but a short notational remark would help the reader.
[Footnote 67] Footnote 67 qualifies the Schwarzschild limit and is important for interpreting Eq. (5.50); it should be promoted to the main text rather than left as a footnote.
[Equation (5.44)] The expression containing terms such as “− 1/180 f at0(45α −14)√(1+T²)” and related terms in δr(2,3) should be checked for typographical errors, and the typesetting of the f at1 notation should be made consistent with Table II.

Circularity Check

2 steps flagged · score 4.0 of 10

The geodetic scattering angle is derived in-paper and self-contained, but the self-force correction—a central advertised result—is imported wholesale from the authors' own Refs. [40] and [42]; the conservative SF part additionally depends on an unfixed regulator sigma_phi to be fixed by external matching.

self citation load bearing [Section V, Eq. (5.40) and Section V.B, Eq. (5.50)]
"by using the results of Ref. [40] the structure of the self force components is the following ... Using the results of Ref. [40] (see the associated supplemental material file), we have both a SF corrected conservative (v−even) and dissipative (v−odd) values of the scattering angle δχSF as summarized below"

The essential inputs to the self-force scattering angle—the components F_t, F_r, F_phi in Eq. (5.40) and the final conservative/dissipative values in Eq. (5.50)—are stated to come from Ref. [40], a prior paper by the same three authors. This paper does not rederive those components or provide an independent calculation of them. The final self-force part of the claimed scattering angle therefore inherits its values entirely from the authors' own earlier work, making the central result load-bearing on a self-citation. The α→0 Schwarzschild check is explicitly unavailable for the self-force sector, since footnote 67 says 'TS plus α→0 is not Schwarzschild.'
self citation load bearing [Section I (intro) and Section V, Eq. (5.41)/Table III]
"Then, in Section V, relying on our recent study for unbound orbits in the background of a Top Star [42], we will include scalar self-force effect that correct the geodetic result quite significantly."

The unbound hyperbolic-orbit parametrization used in the self-force calculation—rgeo(T), φ(T), ut(T) in Eq. (5.41) and the v-expanded coefficients in Table III—is imported from Ref. [42], another paper by the same authors, rather than derived in the present work. These imported geodesic data feed into δr(T), Eqs. (5.43)-(5.44), and hence into the self-force correction to the scattering angle. Together with Ref. [40], the self-force sector of the paper is a chain of self-cited prior results with no independent verification supplied here.

full rationale

The geodetic part of the paper is genuinely self-contained: starting from the TS metric (4.1), the in-paper derivation leads to the scattering-angle expansion (4.69) with coefficients in Table I, and the α→0 limit reproduces the known Schwarzschild result. The generic self-force framework in Eqs. (5.1)-(5.36) is also derived in the paper. However, the central self-force result is not: the force components in Eq. (5.40) and the final values in Eq. (5.50) are explicitly said to use Ref. [40], and the unbound-orbit expansion (5.41)/Table III rests on Ref. [42], both by the same authors. The paper itself flags after Eq. (5.50) that the conservative part contains the explicit cut-off scale σφ at O(ε²,v⁴) and O(ε²,v⁶), and that 'if/when another approach will compute the same quantity, a proper choice of σφ can be made to produce agreement'; I treat this as an acknowledged limitation and incompleteness rather than as a definitional equality, but it reinforces that the conservative self-force prediction is not yet a standalone number until σφ is fixed externally. Weighing all of this, the load-bearing self-citation affects the central SF claim, while the geodetic derivation and the in-paper integration framework provide independent content, so the circularity score is 4 rather than higher.

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

The central computation imports the Topological Star geometry from prior literature and the self-force results from the authors' own Refs [40] and [42]. The only genuinely undetermined constant in the final observable is the regulator sigma_phi. No new particles, fields, or forces are introduced.

free parameters (1)

sigma_phi = not determined
Cut-off regulator in the phi-component of the self-force, appearing in the conservative scattering angle Eq. (5.50) at O(epsilon^2, v^4) through ln(2 v sigma_phi). It is not fixed by the paper and requires an external matching or numerical input.

assumptions (5)

domain assumption The Topological Star metric Eq. (4.1) is an exact, stable solution of D=5 Einstein-Maxwell theory in the parameter range used.
The background is taken as given from Refs [34-36], including stability conditions rs < rb < 2rs, without re-derivation in this paper.
ad hoc to paper The scalar self-force components in Eq. (5.40) from Ref [40] are correct and complete at the required PN and PM orders.
The paper states 'by using the results of Ref. [40]' and does not reproduce the derivation; the final formula Eq. (5.50) depends entirely on those components.
domain assumption Boundary conditions for the self-force are regularity at the topological star cap, not horizon ingoing conditions.
The authors emphasize in footnote [67] that 'TS plus alpha to 0 is not Schwarzschild' because of different boundary conditions, which is central to why the regulator sigma_phi is not fixed by the Schwarzschild limit.
domain assumption The probe is a neutral spinless scalar charge with small mass, and its backreaction is treated only at first self-force order O(q^2).
The equations in Section V, especially Eqs. (5.1) and (5.39), assume this perturbative treatment and leave gravitational and electromagnetic self-force to future work.
standard math The large-angular-momentum expansion and the commuted PM and PN expansions are valid in the regime considered.
The whole calculation uses epsilon = rs/(2 b v^2) and expands in 1/j; the paper notes potential subtleties in exchanging PM and PN expansions in footnote [68].

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Pith. "Pith review of Scattering angle in a Topological Star spacetime: a self-force approach." pith.science (2026). https://pith.science/paper/XEDKCD7L

@misc{pith2026250604876,
  author       = {Pith},
  title        = {Pith review of: Scattering angle in a Topological Star spacetime: a self-force approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XEDKCD7L}},
  note         = {Machine review of arXiv:2506.04876}
}

read the original abstract

We compute the scattering angle for a scalar neutral probe undergoing unbound motion around a Topological Star, including self-force effects. Moreover we identify the `electro-magnetic' source of the background as Papapetrou Field compatible with the isometries and characterize Topological Stars by studying their sectional curvature, geometric transport along special curves and the gravitational energy content in terms of the super-energy tensors.

Figures

Figures reproduced from arXiv: 2506.04876 by the authors.

**Figure 2.** FIG. 2: The LHS of Eq. (4.41) is plotted as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3: Scattering orbit in a TS spacetime for the following [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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Forward citations

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Reference graph

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In principle one can include a component Ay(r, θ), but we will refrain from doing so
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The relation (5.46) should not be confused with the or- bital average of Fα , given by ⟨Fα ⟩ = ∫ ∞ −∞ Fα (τ )dτ = b v ∫ ∞ −∞ Fα (T ) dT ut(T )
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TS plus α → 0 is not Schwarzschild

As already said, switching oﬀ the TS parameter α is not suﬃcient to reproduce the Schwarzschild situation [63, 64], because of the diﬀerent boundary conditions imposed in the Schwarzschild case (purely ingoing conditions at the horizon) and in the TS case (regularity at the ca...
[79]

Damour for raising this issue

We thank T. Damour for raising this issue

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

It is symmetric and traceless in all pair of indices

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[2] [2]

It is divergence-free ∇αT (g)αβγδ = 0. Indeed, the analogous quantity (Bel-Robinson) deﬁned in terms of the Weyl tensor Cαβγδ and its dual carries a similar information, but without any reference to the spacetime sources. We will not insist on studying explic- itly the diﬀerences between these two tensors (already discussed in the literature [55, 58]), an...

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[3] [3]

(4.10) The associated Kretschmann invariant is (2)Kt−r = Rαβγδ Rαβγδ = r2 s (4r − 5rb)2 4r8 , (4.11) showing a geometrical role for the radius r = 5 4 rb

t − r section, with (Lorentzian) 2-metric (2)ds2 = −fs(r)dt2 + dr2 fs(r)fb(r) . (4.10) The associated Kretschmann invariant is (2)Kt−r = Rαβγδ Rαβγδ = r2 s (4r − 5rb)2 4r8 , (4.11) showing a geometrical role for the radius r = 5 4 rb

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[4] [4]

(4.12) The associated Kretschmann invariant is (2)Kr−φ = [(rs + rb)r − 2rsrb]2 r8 , (4.13) showing a geometrical role for the radius r = 2rsrb (rs+rb) = 2 1 rs + 1 rb

r − φ section, with (Euclidean) 2-metric (2)ds2 = dr2 fs(r)fb(r) + r2dφ2 . (4.12) The associated Kretschmann invariant is (2)Kr−φ = [(rs + rb)r − 2rsrb]2 r8 , (4.13) showing a geometrical role for the radius r = 2rsrb (rs+rb) = 2 1 rs + 1 rb

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[5] [5]

r − y section, with (Euclidean) 2-metric (2)ds2 = dr2 fs(r)fb(r) + fb(r)dy2 . (4.14) The associated Kretschmann invariant is (2)Kr−y = r2 b (4r − 5rs)2 4r8 , (4.15) showing a geometrical role for the radius r = 5 4 rs, namely rs ↔ rb exchange their roles when passing from (2)Kt−r to (2)Kr−y, as expected. B. Static observers, split of the Riemann and super...

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[6] [6]

Let us introduce an orthonormal frame adapted to the static observers (4.16) eˆt = u = 1√−gtt ∂t , e ˆr = 1√grr ∂r , e ˆθ = 1√gθθ ∂θ , e ˆφ = 1√gφφ ∂φ , e ˆy = 1√gyy ∂y

the orbits are geodesics. Let us introduce an orthonormal frame adapted to the static observers (4.16) eˆt = u = 1√−gtt ∂t , e ˆr = 1√grr ∂r , e ˆθ = 1√gθθ ∂θ , e ˆφ = 1√gφφ ∂φ , e ˆy = 1√gyy ∂y . (4.25) Instead of the Ω parametrization it can be convenient to have a velocity parametrization writing Ucirc = γ(eˆt + ve ˆφ) , (4.26) such that Γ = γ √−gtt , ...

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[7] [7]

(4.29) the transport is trivial for the θ and φ components of X and reduces to X θ(τ ) = X θ(0) and X y(τ ) = X y(0)

Parallel transport Consider a generic vector X = X α∂α and study the equation for parallel transport of X along Ucirc, dX α dτ + Γ αµνX µU ν circ = 0 . (4.29) the transport is trivial for the θ and φ components of X and reduces to X θ(τ ) = X θ(0) and X y(τ ) = X y(0). We can then assume X θ(0) = X y(0) = 0. The other components are such that dX ˆt dτ = −...

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[8] [8]

A spatial rotation from the initial data for all Ω such that ζ < 0

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