REVIEW 4 major objections 4 minor 2 references
Particle collisions around static spherically symmetric black hole and rotating black hole in gravity's rainbow
T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The paper claims that in gravity's rainbow, the Banados-Silk-West effect can produce infinite center-of-mass energy for collisions outside the event horizon when the rainbow function has a pole at the Planck energy.
desk verdict The claimed divergence is a pole inserted by hand, and the two-particle center-of-mass energy is not well-defined in rainbow gravity, so the BSW framing does not hold. 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 rainbow-function pair $g_0(E/E_P)$, $g_1(E/E_P)$, which rescale the time and spatial metric components according to the probing particle's energy. The load-bearing choice is $g_0=g_1=(1-E/E_P)^{-1}$, whose pole at $E=E_P$ enters the center-of-mass formulas (13) and (23) and produces the divergence. The calculation machinery is the geodesic Lagrangian in the energy-dependent metric, conserved energy and angular momentum, and the turning-point condition $R(r)=0$ that fixes the collision point and the required angular momentum.
What would settle it
Re-derive the two-particle center-of-mass invariant with separate metrics for the two particles, for example by evaluating $g_{ab}(E_1)$ and $g_{ab}(E_2)$ separately in the inner product rather than collapsing them to one metric; if $E_{\rm cm}$ remains finite as $E_1 \to E_P$, the divergence is an artifact of the single-metric identification. A numerical check at finite $E/E_P$ could also settle it.
Extended reading notes
Core claim
The central claim is that in gravity's rainbow, the Banados-Silk-West effect is not tied to the event horizon. With $g_0=g_1=(1-E/E_P)^{-1}$, the center-of-mass energy $E_{\rm cm}$ of two equal-mass particles colliding at a turning point outside the horizon diverges when the energy of either particle approaches the Planck energy $E_P$. The paper derives explicit formulas for $E_{\rm cm}$ using the radial turning-point condition $R(r)=0$, for a generic static spherically symmetric black hole and for the Kerr black hole, and shows the divergence appears as a factor $(1-E/E_P)^{-1}$ in the modified metric. Infinite energy is therefore a property of the rainbow function, not a near-horizon effect. It also shows that alternative rainbow functions from loop-quantum-gravity-motivated and gamma-ray-burst-motivated families yield finite $E_{\rm cm}$.
Load-bearing premise
The load-bearing premise is that the energy in the rainbow functions can be identified with each particle's conserved energy while the two-particle center-of-mass formula still uses one common metric; if that identification is wrong, the claimed divergence is not established.
Editorial extensions
If this is right
- For the rainbow function $g_0=g_1=(1-E/E_P)^{-1}$, particle collisions outside the event horizon can reach arbitrarily high center-of-mass energy when one particle's energy approaches the Planck scale.
- The divergence does not require the collision radius to approach the horizon, so in this model the Banados-Silk-West effect is decoupled from near-horizon geometry.
- For the two alternative rainbow-function families considered, center-of-mass energies stay finite, so the divergence is specific to the chosen rainbow functions.
- The required angular momentum for such collisions is fixed by the turning-point condition, giving explicit motion parameters that realize the divergent collisions.
Reading between the lines
- Editorial extension: if the energy entering the rainbow functions is instead the total system energy or the collision energy itself, the pole at the Planck scale would couple to the collision outcome, and the divergence claim may not survive.
- Editorial extension: a version of the calculation that keeps the two energy-dependent metrics distinct would be a direct consistency check of the claimed infinite energy.
- Editorial extension: the contrast between divergent and finite rainbow functions suggests a model-selection test, since observations of near-Planck-scale collision energies could favor or disfavor particular modified dispersion relations.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the Banados-Silk-West (BSW) effect in gravity's rainbow for static spherically symmetric and rotating black holes. The authors compute the center-of-mass energy of two colliding test particles whose geodesic equations are governed by energy-dependent rainbow-deformed metrics, using the specific rainbow functions g0=g1=(1-E/EP)^{-1}. They claim that this choice produces an infinite Ecm for collisions occurring outside the event horizon, in contrast to the standard BSW effect which requires near-horizon collisions. The paper also surveys two other families of rainbow functions, finding finite Ecm in those cases. The central result is presented in Eqs. (13) and (23), and the conclusion states explicitly that the divergence arises from the inherent divergence of the rainbow function.
Significance. If the central claim were correct, it would identify a qualitatively new mechanism for unbounded center-of-mass energies outside black-hole horizons, with potential implications for Planck-scale physics probes and for the phenomenology of gravity's rainbow. The paper usefully catalogs several rainbow functions and shows which yield finite or divergent Ecm, which could serve as a reference for future work. However, the claimed divergence is not a new physical effect: it is inserted by hand through a rainbow function that diverges at the Planck scale, and the two-particle center-of-mass energy used to exhibit it is not well-defined in the framework of gravity's rainbow. These load-bearing issues mean the paper's main result is unsupported.
major comments (4)
- [Sec. 3, Eqs. (8)-(9)] The center-of-mass energy is defined through the invariant E_cm^2 = -(P_1+P_2)^2, but the metric in gravity's rainbow depends on the probing particle's energy, g_ab(r,E_i). In Eq. (8), the contraction of the two four-velocities is performed using g_ab(r,E_1) for one velocity and g_ab(r,E_2) for the other, inside a single spacetime invariant. This assumes a common spacetime geometry for two particles with different energies, which gravity's rainbow does not provide. No prescription for a common 'frame' metric is given, so E_cm is not well-defined for E_1 ≠ E_2. This issue is load-bearing because the divergence claimed in Eqs. (13) and (23) relies on E_1 and E_2 approaching E_P independently.
- [Sec. 3, Eq. (14) and Sec. 4, Conclusion] The claimed divergence is circular: the chosen rainbow functions g0=g1=(1-E/EP)^{-1} diverge as E→EP, and this divergence is inserted directly into the Ecm formula. Indeed, the conclusion states that Ecm diverges 'due to the inherent divergence of rainbow function.' This is not a BSW-type effect driven by critical angular momentum, but simply the divergence of an input function. Consequently, the statement that infinite Ecm can be achieved outside the horizon is a restatement of the chosen rainbow function's singular behavior, not a derived property of the collision kinematics.
- [Sec. 3, Eqs. (5), (12)-(13)] The energy E appearing in the rainbow functions is identified, without discussion, with the conserved energy E_i of each particle in the geodesic equations (Eq. (5)). This identification is nontrivial: in gravity's rainbow, E is the energy of the probing particle, but the conserved energy in the geodesic equation is derived from a Lagrangian that already contains the deformed metric. Moreover, the turning-point condition R(r_p)=0 imposes a relationship among E_i, the angular momentum, and r_p; the paper then takes E_i→EP independently, without verifying that this limit is compatible with the turning-point condition. This unexamined compatibility is essential for the claimed outside-horizon divergence.
- [Sec. 4, Eqs. (22)-(23)] The rotating black hole section uses a Kerr metric in Eq. (22) that is incompletely specified: g_tt is written as -(1-2M/r), but g_tφ and g_φφ are not fully given, and the expression is garbled. The horizon radius and the condition r_p ≥ r_+ are used without demonstrating that the energy-dependent deformed metric possesses the same horizon structure. As in the static case, the divergence in Eq. (23) follows directly from inserting the divergent rainbow function, independent of the Kerr metric details, so the rotating case adds no independent support for the central claim.
minor comments (4)
- [Throughout] The manuscript contains many typographical and typesetting errors, including duplicated equation numbers (Eq. (12) appears twice) and a missing Eq. (24) despite Eq. (25) being cited. The formulas in Eqs. (2), (20), and (22) are particularly hard to read due to garbled symbols and missing parentheses.
- [Sec. 2, physical domain] The stated physical domain 0≤E/E_P≤1 for relativistic particles is not reconciled with the limits E_i→EP used later; the divergent limit sits exactly at the boundary of this domain, which suggests the result is a singular boundary limit rather than a physically accessible regime.
- [Sec. 3, Eq. (14)] The phrase 'the theory with constant velocity of light and also solves the horizon problem' is not explained; the connection between the chosen rainbow functions and these properties is asserted without derivation or citation to a specific mechanism.
- [Acknowledgements] The acknowledgements contain a personal note about the birth of the authors' daughter; while this is a human sentiment, it is outside the usual scope of an academic paper and should be removed or moved to a personal dedication.
Circularity Check
The claimed outside-horizon infinite E_cm is the chosen rainbow function's pole restated in E_cm variables, not a derived result.
-
self definitional
[Abstract; Section 3, Eq. (14) and Eq. (13)]
"By employing rainbow functions g0=g1=(1-E/EP)^-1, infinite Ecm can be achieved through collisions occurring outside the event horizon, rather than being confined solely to collisions near the event horizon."
The chosen rainbow function has a pole at E=EP by definition. Inserting g0=g1=(1-E/EP)^-1 into the static Ecm formula (Eq. 13) makes Ecm diverge exactly when Ei approaches EP, because 1-Ei/EP appears in the denominator. The conclusion states this directly: 'the Ecm diverges to infinity due to the inherent divergence of rainbow function.' No geodesic or BSW-type mechanism is derived; the divergence is the input function's singularity expressed in Ecm variables. The relevant limit is the boundary E/EP=1 of the stated domain 0<=E/EP<=1.
-
self definitional
[Section 4, Eq. (23) and following paragraph]
"As Eq. (23) show, if the energy of any one of these particles Ei -> EP, a divergent center-of-mass energy can be obtained."
This is the rotating-black-hole analogue of the same reduction: Eq. (23) is obtained by substituting the same singular rainbow function g0=g1=(1-E/EP)^-1 into the Ecm expression, and the divergence is again triggered solely by the pole at Ei=EP. The claimed outside-horizon result therefore does not come from the Kerr geodesic structure or from the BSW critical angular momentum; it is the chosen rainbow function's divergence carried through the algebra.
full rationale
The central derivation reduces to its input by construction. The paper chooses g0=g1=(1-E/EP)^-1 (Eq. 14), inserts it into the Ecm formulas (Eqs. 13 and 23), and takes Ei->EP; since the rainbow function itself diverges at EP, Ecm diverges. The authors' own conclusion confirms this reading: the divergence is 'due to the inherent divergence of rainbow function.' Thus the finding of infinite Ecm outside the horizon is not an independent physical prediction but a restatement of the chosen function's pole. A separate correctness issue, not needed for the circularity verdict, is that gravity's rainbow assigns different metrics to particles of different energies, so a two-particle invariant contracted in one common metric requires an additional prescription that the paper does not justify. There is no load-bearing self-citation chain here; the circularity is definitional. Score 9 rather than 10 only because the paper does perform nontrivial algebra in converting metric components into Ecm, but the singular conclusion is fixed at the input stage.
Assumptions & free parameters
assumptions (4)
- domain assumption The spacetime metric in gravity's rainbow is energy-dependent, with the replacement g_tt -> g_tt/g0(E/EP)^2 and spatial components scaled by 1/g1(E/EP)^2 (Eq. (1), (12)).
- ad hoc to paper The energy E appearing in the rainbow functions is identified with the conserved energy E_i of each particle in the geodesic equations (Eq. (5)).
- ad hoc to paper The two-particle center-of-mass energy can be computed by combining metric components evaluated at each particle's own energy (Eq. (9)-(11)).
- domain assumption The rainbow function g0=g1=(1-E/EP)^{-1} is adopted as a valid choice (Eq. (14)).
Cite this review
Pith. "Pith review of Particle collisions around static spherically symmetric black hole and rotating black hole in gravity's rainbow." pith.science (2026). https://pith.science/paper/YNCDMMDJ
@misc{pith2026250523874,
author = {Pith},
title = {Pith review of: Particle collisions around static spherically symmetric black hole and rotating black hole in gravity's rainbow},
year = {2026},
howpublished = {\url{https://pith.science/paper/YNCDMMDJ}},
note = {Machine review of arXiv:2505.23874}
}
read the original abstract
We extend the Banados-Silk-West effect to the static spherically symmetric black hole and rotating black hole in gravity's rainbow. Through systematic investigation that the effects of different rainbow functions on the center-of-mass energy of two test particles colliding outside the event horizon, we discussed the possibility of infinite center-of-mass energy Ecm and the corresponding motion parameters. By employing rainbow functions g0=g1=(1-E/EP)-1, infinite Ecm can be achieved through collisions occurring outside the event horizon, rather than being confined solely to collisions near the event horizon.
Reference graph
Works this paper leans on
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Reviewed August 7, 2026 · model on record in the stance chip above.
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