REVIEW 2 major objections 4 minor 4 cited by
Size and Shape of Rotating Strings and the Correspondence to Black Holes
T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read For slowly spinning strings, the ratio of transverse to in-plane size falls as $1 - \gamma_\parallel J^2/S^2$, matching the black-hole ratio $1 - 4\pi^2 J^2/S^2$, so the paper takes this ratio as evidence for the rotating…
desk verdict A careful computation of rotating string sizes that finds the advertised J²/S² ratio match with black holes, but the correspondence step rests on an untested adiabatic-invariance assumption. 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 argument is carried by three linked tools. First, the operator method for rotating strings: a partition function $Z(x,\Omega)$ with an angular potential $\Omega$ is evaluated in the high-temperature limit, and the inverse Laplace/Fourier transforms from $(\beta,\Omega)$ to $(n,J)$ are done by saddle point, giving the density of states (2.43) and the sizes (2.54), (2.64). Second, the rhotation function $\rho(\beta,\Omega) = (1-\pi\Omega\cot(\pi\Omega))/\Omega^2$ controls the in-plane size; its Fourier transform yields the factors $\log(1+e^{-\beta|J|}) + \beta|J|/(1+e^{\beta|J|})$ that produce the parallel-size corrections. Third, a path-integral random walk model treats $J[X] = \frac{1}{2\pi\alpha'}\int (X^1\dot X^2 - X^2\dot X^1)\,ds$ as an enclosed-area constraint and reproduces the string sizes for all $J$. The ratio $\langle \bar r_\perp^2\rangle_n/\langle \bar r_\parallel^2\rangle_n$ is the central object because it is expected to cancel the leading effect of self-gravitation, making it the approximate adiabatic invariant that permits a comparison across the correspondence point.
What would settle it
Compute the sizes of a self-gravitating rotating string ball at the correspondence coupling (for example, a spinning string condensate with $g^2 S \sim 1$) and compare its $r_\perp/r_\parallel$ at fixed $J/S$ with the free-string value. If turning on self-interaction changes the ratio by an $O(1)$ factor, the adiabatic-invariance premise fails and the small-spin agreement between strings and black holes would not be a meaningful correspondence.
Extended reading notes
Core claim
The paper's central claim is that the shape of a highly excited rotating string, measured by the ratio $\langle \bar r_\perp^2\rangle_n/\langle \bar r_\parallel^2\rangle_n$, tracks the shape of a rotating black hole in the correspondence regime $|J|\lesssim S$. For strings with $J=O(1)$ the authors derive $\langle \bar r_\perp^2\rangle_n/\langle \bar r_\parallel^2\rangle_n \propto 1 - \gamma_\parallel J^2/S^2$ with $\gamma_\parallel = \sqrt{a}(2\log 2 - 1)/(8\log 2)>1$ (Eqs. (2.69)-(2.70)), and for Myers-Perry black holes the exact relation $r_\perp^2/r_\parallel^2 = S^2/(S^2+4\pi^2 J^2)$ gives $1 - 4\pi^2J^2/S^2$ at small spin (Eq. (3.13)). The functional form $J^2/S^2$ is the same on both sides; the numerical coefficients are $O(1)$ and the paper does not expect them to match exactly. The calculation also shows that no terms of order $J^2/\sqrt{n}$ appear in the string expansion, while terms of order $J^2/n$ do, which the authors interpret as matching the semiclassical black-hole structure in which $J^2/S^2$ is classical and $J^2/S^3$ is a quantum correction. They conclude that fundamental-string microstates carry spin in a way that, at very large mass, can be put in correspondence with semiclassical black holes.
Load-bearing premise
The comparison assumes that self-gravitation changes a string's perpendicular and parallel sizes by roughly the same factor, so that the ratio $r_\perp/r_\parallel$ is an approximate adiabatic invariant between the free-string and black-hole regimes; the paper states this as a reasonable expectation, not as a derived result.
Editorial extensions
If this is right
- For small angular momentum, rotation flattens a highly excited string: the ratio of transverse to in-plane average size decreases as $1 - \gamma_\parallel J^2/S^2$, with $\gamma_\parallel > 1$.
- The same $1 - O(1)J^2/S^2$ decrease follows from the exact black-hole relation $r_\perp^2/r_\parallel^2 = S^2/(S^2+4\pi^2J^2)$, supporting the correspondence between slowly rotating strings and black holes.
- At intermediate spins $J = O(\sqrt{n})$ the string ratio stays $<1$ and $O(1)$ through the slowly varying factor $C_\parallel > 1$, similar to black holes with $|J|\sim S$.
- At the largest spins $J=O(n)$ the string ratio behaves as $S/|J|$, whereas black holes give $S^2/J^2$; the paper attributes this difference to the absence of a single-string/stationary-black-hole correspondence in the ultraspinning regime.
- The random walk model reproduces the sizes for all $J$ and gives the geometric picture of the rotation-plane area $|J|$ competing with the random-walk length $n-|J|$ available for transverse spread.
Reading between the lines
- As an editorial extension, if the size ratio is truly adiabatic, then higher multipole moments of the string's spatial distribution should likewise match black-hole moments at small $J/S$; the random-walk distribution provides a concrete way to compute them.
- The area-constrained random walk suggests a testable analogue in statistical mechanics: closed walks of fixed length square and fixed enclosed area in two dimensions should show the same flattening with $J^2/n$, which could be checked numerically.
- The subleading terms identified in the paper ($J^2/n^{3/2}$ on the string side) are predicted to correspond to one-loop quantum corrections on the black-hole side; computing those corrections would make the dictionary quantitative.
- The paper's focus on the ratio, rather than absolute sizes, implies that absolute string sizes at the correspondence point remain fixed by self-gravitational collapse, so future computations of absolute sizes must include self-interaction while the ratio remains the matching observable that is insensitive to the interaction details.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper analyzes the average spatial size and shape of highly excited fundamental strings with fixed angular momentum J, for excitation level n >> 1, in three regimes: J = O(1), J = O(sqrt(n)), and J = O(n). Using an operator method and saddle-point/large-n expansions, the authors obtain the average sizes transverse to and within the rotation plane, together with their ratio. For small J they find <rbar^2_perp>/<rbar^2_parallel> proportional to 1 - gamma_parallel J^2/S^2, and they compare this with the exact Myers-Perry black-hole result r_perp^2/r_parallel^2 = S^2/(S^2 + 4 pi^2 J^2). On the strength of the shared J^2/S^2 small-spin dependence, they argue that string microstates carry spin in a way that is in correspondence with semiclassical black holes. A path-integral random-walk model is also constructed that reproduces the string-size integrals.
Significance. The paper is careful and technically detailed: the saddle-point computations track subleading orders, the Fourier transforms between angular velocity and angular momentum are performed by residues, and the black-hole ratio in Eq. (3.13) is exact and dimension-independent. The random-walk model gives a geometric and potentially reusable representation of highly excited string states, and the authors are candid about the external normalization needed in Eq. (4.17). If the adiabatic-invariance assumption is accepted, the small-spin comparison is a nontrivial and interesting quantitative check of the black-hole/string correspondence; the paper also clearly delineates the large-J regime where no correspondence is expected.
major comments (2)
- [Section 2.2, before Eq. (2.70)] The claim that r_perp/r_parallel is an approximate adiabatic invariant under self-gravitation is asserted rather than derived. The comparison in Eq. (2.70) is made at zero string coupling, while Eq. (3.13) describes a self-gravitating black hole. The only bridge between the two regimes is the expectation that self-gravitation affects the two directions roughly equally. Since a J-dependent differential shrinkage of the two directions would change exactly the J^2/S^2 term being compared, this step is load-bearing for the central claim. Please provide a quantitative estimate or bound from a self-gravitating string-ball computation (for example, along the lines of Ref. [9]), or alternatively state explicitly that the matching concerns only the free-string and black-hole scalings and is not a prediction at the correspondence point.
- [Appendix C.3, Eq. (C.18)] The J -> 0 limits of the two string sizes are 2 log 2 and pi^2/6 for the normalized definitions, so their ratio is not 1 at J = 0, whereas the black-hole ratio in Eq. (3.13) is exactly 1 at J = 0. The paper explains that the two calculations impose different spin constraints, but this means that the object compared in Eq. (2.70) is not literally the same normalized ratio as the black-hole ratio. The comparison therefore requires the unknown O(1) constant to be absorbed. Please spell out why the J^2/S^2 functional dependence is robust under this renormalization, and how the adiabatic-invariance argument applies to the normalized ratio rather than to the separately computed sizes.
minor comments (4)
- [Section 2.2, Eq. (2.69)] The bullet after Eq. (2.70) states that gamma_parallel > 1, which contradicts the definition in Eq. (2.69): for all c >= 2, sqrt(a) (2 log 2 - 1)/(8 log 2) < 1, and for c = 2 it is approximately 0.13. Since only positivity is needed for the ratio to decrease, please replace the inequality by gamma_parallel > 0 or correct the formula.
- [Section 2.2, Eq. (2.70)] The entropy S in Eq. (2.70) is used without an explicit definition in the string section; the identification S^2 proportional to n and the string units should be stated before Eq. (2.70), since the black-hole comparison in Eq. (3.13) uses the actual entropy.
- [Section 4.1, Eq. (4.17)] The random-walk normalization e^{a/beta} is fixed by matching the string partition function in Eq. (4.17). The main text should state more prominently that the sizes reproduced in Section 4.2 are therefore a calibrated consistency check rather than an independent derivation, even though the authors do acknowledge this in the discussion below Eq. (4.17).
- [Figure 3] The left and right panels of Figure 3 would benefit from a common horizontal axis or explicit annotation of the correspondence regime |J| <~ S, so that the visual resemblance can be quantified.
Circularity Check
The central string/black-hole ratio comparison is independently derived; the auxiliary random-walk 'reproduction' is a calibrated consistency check by construction.
-
self definitional
[Section 4.1, Eq. (4.17); Section 4.2, Eqs. (4.21)-(4.23) and text following them]
"the most important point at which one relates the random walk with the fundamental string is through the normalisation factor e^{a/β}. This factor is determined by imposing that the integral of d_{n,J}(x_i) over the entire space is equal to the number of strings at level n and angular momentum J in the thermodynamic limit... This expression for d_{n,J} is an external input that needs to be provided for the random walk to be consistent with the string calculation. ..."
Eq. (4.17) fixes the random-walk normalization e^{a/β} by equating the integrated walk count to the operator-method string density d_{n,J}. The size moments in Section 4.2, e.g. (4.21) and (4.23), are then the same Laplace/Fourier transforms of the same integrand (with only a power of β shifted) as the operator-method sizes (2.52) and (2.56), normalized by the same d_{n,J}. Hence the 'reproduction' of string sizes is an identity by construction: the random walk was calibrated to the string partition function, and the size integrals coincide term by term. The paper candidly labels d_{n,J} an 'external input', so this is a self-consistency check rather than an independent geometric prediction.
full rationale
The central correspondence claim—that (r⊥/r∥) for free strings behaves as 1 − γ∥ J²/S² with γ∥ > 0, matching the black-hole form S²/(S²+4π²J²)—is derived from the operator-method saddle-point calculation of Section 2 and the Myers-Perry metric in Section 3, with no fitted parameter connecting the two. The black-hole ratio (3.13) is an exact geometric identity; the string ratio (2.70) follows from the high-temperature partition function and size integrals. No input from the target black-hole result is used in the string computation. The bridge between the free-string computation and the self-gravitating black hole is the asserted approximate adiabatic invariance of r⊥/r∥ (Section 2.2); that is a heuristic expectation, not a derived result, and is a correctness risk rather than a circularity. The one genuinely by-construction element is the random-walk model of Section 4: its normalization is fixed by the string density of states (4.17) and its size moments are the same integrals as the operator method (4.21)-(4.23), so its successful 'reproduction' is a calibrated consistency check. Citations of the authors' earlier work [4] are used only to delimit the regime of correspondence, not as inputs to the small-J match. This affects only the auxiliary random-walk claim; the main string/black-hole comparison remains self-contained. Score 4 reflects partial circularity in that secondary model with independent central content.
Assumptions & free parameters
free parameters (2)
- Random-walk normalization N(β,k) = e^{a/β} =
e^{a/β}, with a = π² c/6 and c = D-2
- Random-walk parameter length ℓ =
π
assumptions (6)
- standard math Modular-transformed high-temperature form of the open string partition function: Z(β≈0, Ω) = const β^{c/2} e^{cπ²/6β} Ω/sin(πΩ)
- standard math Saddle-point (steepest descent) approximation is valid for large n including subleading corrections
- domain assumption Definition of string size via the mean squared spread of the operator X^i at τ=0 (Eqs. 2.3 and 2.30)
- domain assumption The ratio r⊥/r∥ is an approximate adiabatic invariant under self-gravitation
- ad hoc to paper Random-walk density normalized to the string partition function (Eq 4.17)
- domain assumption Correspondence principle: string states with g²S of order one become black holes (background from Susskind, Horowitz-Polchinski)
Cite this review
Pith. "Pith review of Size and Shape of Rotating Strings and the Correspondence to Black Holes." pith.science (2026). https://pith.science/paper/GXUGQYVD
@misc{pith2026241118690,
author = {Pith},
title = {Pith review of: Size and Shape of Rotating Strings and the Correspondence to Black Holes},
year = {2026},
howpublished = {\url{https://pith.science/paper/GXUGQYVD}},
note = {Machine review of arXiv:2411.18690}
}
read the original abstract
In light of the correspondence between black holes and fundamental strings with non-zero spin, we compute the sizes of rotating strings for small, moderate, and large values of the angular momentum and compare them to the sizes of rotating black holes. We argue that the ratio of the size perpendicular to the rotation plane to the size along the rotation plane is an approximate adiabatic invariant and can therefore be meaningfully compared for objects on different sides of the correspondence point. We show that the spin-dependence of this ratio for small angular momenta agrees for black holes and strings. When the spin is large, the ratios for these objects exhibit different behavior, but this is expected since for large angular momenta there is no direct correspondence between black holes and single-string states. We also develop a random-walk model that describes highly excited strings and accurately reproduces the sizes of rotating strings for all values of angular momentum.
Figures
Figures from the paper (3 more)
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Reviewed August 12, 2026 · model on record in the stance chip above.
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