{"id":"55c2e022-88f3-4989-9e24-53e68a50fc09","arxiv_id":"2505.23874","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For the rainbow function g0=g1=(1-E/EP)^{-1}, the center-of-mass energy of a collision diverges as one particle's energy approaches the Planck energy, at any radius.","lead":"This paper studies particle collisions around black holes in gravity's rainbow, a modified gravity where the metric depends on the colliding particle's energy. It finds that for one well-known rainbow function, the center-of-mass energy diverges when a particle's energy nears the Planck scale, even at collisions far from the horizon.","discovery_kind":"incremental","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed outside-horizon divergence relies on computing a two-particle invariant with metrics evaluated at two different energies; in gravity's rainbow no single spacetime defines both four-velocities, so E_cm is not well-defined.","rationale":"I agree with the reader's weakest_assumption, specifically the second half: even if E in g(E/E_P) is identified with the conserved energy of each test particle, the two-particle center-of-mass energy requires one metric to define the inner product. Equations (8)-(9) mix g_ab(r,E_1) and g_ab(r,E_2), and the paper supplies no justification that this mixed expression is an invariant under the relevant diffeomorphisms or a limit of a well-defined collision process. This is not a question of being outside current consensus; it is an internal consistency gap in the derivation. The paper's own conclusion ('due to the inherent divergence of rainbow function') is a limitation statement that supports this reading. I therefore do not change the reader's REJECT verdict; the concrete test above would settle the issue if the authors want to rehabilitate the claim.","tokens_in":10374,"tokens_out":7569,"duration_ms":73995,"concrete_test":"Recompute E_cm^2 from Eq. (8) using a single common metric for both particles, e.g., g_ab(r,E_1) for both four-velocities, and evaluate the limit E_1→E_P at a fixed r_P>r_H. If the result does not diverge (or diverges only at r_H), the outside-horizon divergence in Eq. (13) depends on the unjustified mixed-energy contraction. As a supplement, check whether Eq. (13) can be derived from a single two-particle action; if no such derivation exists, the formula is a convention rather than a prediction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central result is the divergence of E_cm for collisions at r_P > r_H when g0=g1=(1-E/E_P)^{-1} and E_i→E_P (Eqs. (13), (23)). The load-bearing step is the treatment of the metric as particle-dependent in Eq. (3), g_ab(r,E_i), followed by the contraction in Eqs. (8)-(9) that evaluates one four-velocity in g_ab(r,E_1) and the other in g_ab(r,E_2) inside the single invariant E_cm^2=-(P1+P2)^2. In gravity's rainbow there is no unique spacetime manifold for two particles with different energies: each defines its own geometry, so the inner product P1·P2 is not defined unless a common 'frame' metric is chosen. The paper gives no such prescription and does not address the known multi-particle consistency problem of rainbow gravity. The claimed divergence is thus an artifact of the mixed-energy bookkeeping. The conclusion confirms this reading by attributing the effect to 'the inherent divergence of rainbow function,' i.e., to the chosen g0/g1 blowing up at E_P, not to a BSW-type critical angular momentum. Since the divergent limit sits at the boundary E/E_P=1 of the stated physical domain, the result is at best an unattainable singular limit of an ambiguous construction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10697,"tokens_out":3256,"duration_ms":34310,"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":[{"comment":"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.","section":"Sec. 3, Eqs. (8)-(9)"},{"comment":"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.","section":"Sec. 3, Eq. (14) and Sec. 4, Conclusion"},{"comment":"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.","section":"Sec. 3, Eqs. (5), (12)-(13)"},{"comment":"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.","section":"Sec. 4, Eqs. (22)-(23)"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Sec. 2, physical domain"},{"comment":"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.","section":"Sec. 3, Eq. (14)"},{"comment":"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.","section":"Acknowledgements"}],"recommendation":"reject","confidential_remarks":"The paper's core claim is vitiated by the ill-defined two-particle center-of-mass energy in gravity's rainbow and by the circular divergence of the chosen rainbow function. These are not presentation issues but fundamental flaws in the derivation. The paper does provide a systematic survey of rainbow functions, which might be useful as a reference if the conceptual problems are addressed, but the current manuscript does not meet the standards for publication in this journal. I would also note that the paper does not engage with the known multi-particle consistency problem in rainbow gravity, which is directly relevant to its central calculation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper claims to extend the BSW effect to gravity's rainbow and finds that with g0=g1=(1-E/EP)^{-1}, the center-of-mass energy diverges for collisions outside the horizon. That claim is not wrong algebraically, but it follows directly from the chosen rainbow function's pole at the Planck energy, not from a BSW-type mechanism. The stress-test is on target: the calculation inserts metrics evaluated at two different particle energies into a single invariant, and in rainbow gravity there is no single spacetime geometry that defines the inner product of two four-velocities. The paper gives no prescription to handle this, so the central result is not well-defined.\n\nWhat the paper does well: it lays out the geodesic equations for a single particle in rainbow-deformed Schwarzschild and Kerr geometries, and it systematically checks three rainbow function families, showing the other two give finite E_cm. That comparison could be useful to someone cataloging rainbow function choices, and the single-particle kinematics appear straightforward.\n\nThe soft spots are serious. The divergence at E_i->EP is inserted by hand through the rainbow function, so it is not a new physical prediction; the conclusion even admits this by attributing the divergence to \"the inherent divergence of rainbow function.\" The BSW effect is about critical angular momentum near an extremal horizon; here the divergence has nothing to do with angular momentum and occurs at any radius because the chosen function blows up. The framing is misleading. Also, identifying the energy in the rainbow functions with the conserved energy of each geodesic is asserted without justification. The multi-particle geometry issue is not a technicality; it undermines the definition of E_cm itself.\n\nThere are also presentation problems: equation numbering is off (two Eq. (12)s), and the text is rough. Those are minor compared to the conceptual gap.\n\nBottom line: this is a straightforward exercise that does not deliver a well-defined result. It might be of marginal use as a catalog of rainbow-function choices, but as a claim about BSW or infinite energies, it is not convincing. I would not send it to peer review.","headline":"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.","tokens_in":11158,"tokens_out":2318,"would_cite":false,"duration_ms":23357,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C10","83D05"],"pacs":["04.70.-s","04.70.Bw","04.50.Kd","04.60.-m"],"model":"deepseek-v4-flash","headline":"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.","keywords":["Banados-Silk-West effect","gravity's rainbow","rainbow functions","center-of-mass energy","black hole collisions","Kerr black hole","modified dispersion relation","Planck energy"],"falsifier":"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.","tokens_in":10181,"feed_emoji":"🕳️","tokens_out":11452,"duration_ms":108629,"temperature":0.7,"pith_summary":"The paper asks whether the Banados-Silk-West effect — the production of arbitrarily energetic particle collisions near black hole horizons — survives in gravity's rainbow, a modified gravity whose metric depends on the probing particle's energy. It claims that for the rainbow functions $g_0=g_1=(1-E/E_P)^{-1}$, the center-of-mass energy of two colliding particles diverges as one particle's energy approaches the Planck scale, and that this divergence can occur for collisions outside the event horizon. The divergence is traced to the pole in the rainbow function rather than to the horizon geometry. The result is shown for both static spherically symmetric and rotating (Kerr) black holes, while two other common rainbow-function families give finite energies.","feed_headline":"Infinite collision energy moves outside the black hole horizon","feed_subtitle":"The Planck-energy pole of gravity's rainbow can make black-hole collision energy diverge outside the horizon.","key_machinery":"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.","core_discovery":"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 $E_{\\rm cm}$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the Banados-Silk-West near-horizon collision effect that this paper generalizes.","marker":"[1]"},{"why":"Supplies the energy-dependent metric construction of gravity's rainbow used to write the modified spacetimes.","marker":"[38]"},{"why":"Frames gravity's rainbow as doubly special relativity extended to curved spacetime, motivating the energy-dependent geometry.","marker":"[39]"},{"why":"Source of the rainbow function $g_0=g_1=(1-E/E_P)^{-1}$ whose Planck-energy pole drives the claimed divergence.","marker":"[50-53]"},{"why":"Source of the alternative rainbow-function family that the paper shows gives finite center-of-mass energy.","marker":"[54,55]"},{"why":"Source of the gamma-ray-burst-motivated rainbow function that the paper shows gives finite center-of-mass energy.","marker":"[57]"}],"fun_headline_variants":["Rainbow gravity moves infinite collision energy past horizon","Infinite collision energy escapes black hole in rainbow gravity","Planck-energy pole yields divergence outside the event horizon","BSW effect extends beyond horizon with gravity's rainbow","Gravity's rainbow: collisions diverge outside the black hole"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Rainbow gravity moves infinite collision energy past horizon","Infinite collision energy escapes black hole in rainbow gravity","Planck-energy pole yields divergence outside the event horizon","BSW effect extends beyond horizon with gravity's rainbow","Gravity's rainbow: collisions diverge outside the black hole"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000157,"raw_usage":{"total_tokens":1159,"prompt_tokens":823,"completion_tokens":336,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":439,"completion_tokens_details":{"reasoning_tokens":259}},"tokens_in":439,"tokens_out":336,"duration_ms":4059,"temperature":1.0,"reasoning_tokens":259,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:45:57.100274+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Bañados, J","cited_arxiv_id":null,"evidence_quote":"Defines the Banados-Silk-West near-horizon collision effect that this paper generalizes."}],"review_version":1}