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REVIEW 2 major objections 3 minor 10 references

Black Holes Rule Out Heavy Tachyons

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Observed black holes rule out tachyons heavier than about 3 billion GeV.

desk verdict A clean analytic estimate with a potentially important bound, but the 'rule out' claim outruns the curved-space derivation. read the letter →

arxiv 2501.11606 v2 pith:4XRPZFFG submitted 2025-01-20 gr-qc astro-ph.COhep-phhep-th

Mark P. Hertzberg , Abraham Loeb , Aidan Morehouse This is my paper · ORCID
classification gr-qcastro-ph.COhep-phhep-th PACS 04.70.Dy04.62.-v
keywords tachyonsHawkingradiationblackholeevaporationLorentzinvariancesuperluminalparticlesnonlocalfieldtheoryobservationalconstraintsprimordialholes
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 argues that honest-to-goodness tachyons—Lorentz-invariant particles with group velocity above $c$ and a stable vacuum—would be emitted by black holes far more efficiently than ordinary particles, so long-lived observed black holes place a sharp upper bound on tachyon mass. The central result is a black hole lifetime $t_{\rm BH} \approx 192\pi\hbar M/(g c^2 m^2)$ for heavy tachyons, which plummets as $m$ grows. Existing observations of a $3.3\,M_\odot$ black hole surviving several billion years then rule out any tachyon heavier than about $3\times10^9$ GeV. A sympathetic reader cares because this is a direct, gravity-only constraint on a possible third class of relativistically allowed particle, and it eliminates tachyons at unification and quantum-gravity scales. Lighter tachyons and primordial-black-hole scenarios remain as extensions.

What carries the argument

The load-bearing object is the flat-space blackbody emission formula for a thermal free gas, Eq. (21), adapted to tachyon kinematics: the dispersion $E=\sqrt{p^2-m^2}$ with momentum restricted to $p\ge m$ and energy $E\ge0$. Using the black hole area $A=16\pi G^2M^2$ and Hawking temperature $T=1/(8\pi GM)$, integrating over the exterior of the momentum-space 3-ball yields the exact powers in Eqs. (24)--(25); in the heavy limit the $m^2T^2$ term dominates, producing the lifetime bound. The paper also invokes a nonlocal Hamiltonian (Appendix A) whose extra $\tilde{J},J$ kernels cut out momenta $p<m$, defining a stable vacuum so that the emission calculation is internally consistent.

What would settle it

A definitive check would be a Bogoliubov-coefficient calculation of tachyon emission from the Appendix A Hamiltonian in Schwarzschild spacetime: if the heavy-mass radiated power does not scale as $g A m^2 T^2$ times an order-one prefactor, the flat-space formula and the $3\times10^9$ GeV bound fail. Alternatively, a confidently aged observed black hole near $3.3\,M_\odot$ that survived much longer than the predicted lifetime for a claimed tachyon mass would directly falsify the exclusion.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Hawking radiation of tachyons from a Schwarzschild black hole is not exponentially suppressed like that of ordinary massive particles, but enhanced by a factor $\sim m^2/T^2$ relative to massless particles. For a heavy tachyon of mass $m\gg T$, the emitted power is $P = g A m^2 T^2/48$ for bosons and $g A m^2 T^2/96$ for fermions, and integrating $dM/dt$ gives the lifetime $t_{\rm BH}\approx 192\pi b_{B,F}\hbar M/(g c^2 m^2)$, with $b_{B,F}=1,2$. Because the observed black hole of mass $M\simeq3.3\,M_\odot$ has lived for at least $\sim5$ Gyr, the paper concludes that any tachyon with one bosonic degree of freedom and $m\gtrsim3\times10^9$ GeV is observationally ruled out, with multi-species generalizations only strengthening the bound.

Load-bearing premise

The derivation assumes that Hawking emission near the horizon is faithfully described by the flat-space blackbody formula at temperature $T=1/(8\pi GM)$ for the nonlocal tachyon field, without a full curved-space mode decomposition; if curvature or nonlocality changes the near-horizon mode structure, the $m^2T^2$ enhancement and the derived mass bound could shift.

Editorial extensions

If this is right

  • A single bosonic tachyon heavier than about $3\times10^9$ GeV is incompatible with observed black holes of a few solar masses that have lived for billions of years.
  • Tachyons at grand-unification ($10^{15}$--$10^{16}$ GeV) or quantum-gravity ($10^{18}$--$10^{19}$ GeV) masses are excluded by these observations.
  • If several heavy tachyon species exist, the bound is stronger: the lifetime scales as $\left(\sum_i g_i m_i^2 / b_i\right)^{-1}$.
  • For primordial black holes with masses $10^{17}$--$10^{21}$ g surviving to today, tachyon masses down to $7$--$700$ GeV would be ruled out.

Reading between the lines

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

  • The paper's flat-space emission step is a leading approximation; a full curved-space Bogoliubov calculation for the nonlocal Appendix A Hamiltonian is the natural check on whether the $m^2T^2$ enhancement survives or acquires order-one (or exponential) corrections.
  • The same blackbody-integral method could bound other superluminal or exotic dispersion relations by requiring that known black hole ages be compatible with their emission.
  • If asteroid-mass primordial black holes are confirmed with ages at least the age of the universe, the tachyon mass ceiling drops to hundreds of GeV, a range where collider searches could independently probe the same particles.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper derives an observational upper bound on the mass of Lorentz-invariant tachyons from the survival of astrophysical black holes. It introduces a tachyon action and kinematics, argues classically that tachyons do not enter or escape Schwarzschild black holes in finite coordinate time, and then estimates the Hawking emission rate using a flat-space blackbody formula with the tachyon dispersion relation E(p)=sqrt(p^2-m^2) and momentum domain p>=m. The resulting power scales as m^2 T^2 for heavy tachyons, leading to the black-hole lifetime t_BH ~ 192 pi hbar M/(g c^2 m^2) and to the claimed exclusion of tachyons with m > 3 x 10^9 GeV. The same framework gives a lifetime formula for massless particles and a multi-species generalization.

Significance. If the central estimate holds, the paper provides a direct, parameter-free observational constraint on tachyons that is complementary to theoretical causality arguments, and it would rule out tachyons at unification or quantum-gravity scales. The algebraic derivation is transparent: the thermal integrals in Section IV are evaluated exactly, the heavy-mass limits are clean, and the lifetime-bound logic is falsifiable. The paper also usefully clarifies the distinction between a vacuum-instability 'tachyon' and a genuine superluminal particle in the field-theory appendix. The main limitation, discussed below, is that the flat-space thermal formula is used without a curved-space derivation for the nonlocal tachyon field theory, so the headline mass bound rests on an unproven assumption.

major comments (2)
  1. [Sec. IV, Eq. (21)] The central enhancement in Eq. (28) and the mass bound in Eq. (38) are obtained from the flat-space blackbody formula (21), applied to the nonlocal tachyon Hamiltonian of Appendix A. The phase-space restriction p >= m and the kernels J and J-tilde in Eqs. (A17)-(A18) are defined through Fourier transforms in a global Minkowski frame; in Schwarzschild spacetime there is no global plane-wave basis, and no unique covariant generalization of these nonlocal kernels is given. The paper does not compute mode functions or Bogoliubov coefficients for the nonlocal theory, and the equivalence-principle argument preceding Eq. (21) does not by itself justify applying the same thermal occupation to modes whose dynamical equation is nonlocal over scales of order 1/m. If the near-horizon mode structure suppresses the low-energy modes that dominate Eq. (23), the m^2 T^2 enhancement and the resulting 3 x 10^9 GeV bound could shift substantially. I ask for a curved-space derivation, or at minimum an explicit covariant prescription and an estimate of the error, before the bound is presented as a definitive exclusion.
  2. [Sec. III and Appendix A] The classical analysis in Section III is based on the point-particle action (2), while the Hawking calculation is based on the nonlocal field theory of Appendix A. The paper does not establish the relation between these two descriptions: if the field theory is genuinely nonlocal, it is not obvious that the point-particle trajectories of Section III describe the same degrees of freedom, and conversely. This does not directly invalidate the lifetime bound, which rests on the field-theoretic calculation, but the manuscript should state explicitly whether the point-particle and field-theoretic descriptions are intended to be equivalent and, if so, how the nonlocal kernel is to be coupled to the Schwarzschild metric.
minor comments (3)
  1. [Abstract and Sec. IV title] The phrase 'we compute the Hawking radiation' overstates what is done: Eq. (21) is assumed as a leading flat-space blackbody approximation, not derived from a Hawking-mode calculation. I suggest rewording to 'estimate' or 'model' in the abstract and at the start of Section IV.
  2. [Sec. V, Eq. (38)] The quoted numerical bound 3 x 10^9 GeV corresponds to one bosonic degree of freedom. For fermions, the prefactor b_F = 2 in Eq. (35) shifts the excluded mass by a factor of sqrt(2), and for multiple species the combined bound in Eq. (39) changes accordingly. This is an O(1) effect, but the text could state the convention explicitly at the point of the bound.
  3. [Eq. (18)] The symbol Delta_epsilon is used before it is defined in Eq. (19). Moving the definition earlier would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the tachyon lifetime bound follows directly from the stated tachyon dispersion relation, the free-field Hamiltonian of Appendix A, and the standard Hawking temperature, with no fitted parameters and no load-bearing self-citations.

full rationale

The paper's derivation chain is explicit and self-contained. Starting from the tachyon action, Eq. (2), it obtains the dispersion relation E = sqrt(p^2 - m^2) and the allowed domain p >= m, Eqs. (8) and (10). Appendix A constructs a free-field Hamiltonian whose spectrum is, by construction, E(p) with p >= m; this is a model input, not a quantity fitted to black-hole data. The emission power, Eq. (21), is the ordinary blackbody flux formula for a free thermal gas at the Hawking temperature T = 1/(8 pi G M), Eq. (20). The enhanced heavy-tachyon result, Eq. (28), P proportional to g A m^2 T^2, is obtained by evaluating that integral with the tachyon dispersion relation; it is a mathematical consequence of the stated inputs, not an independent prediction that is then used to define those inputs. The lifetime, Eq. (33), is obtained by integrating P = dM/dt, and the observational bound, Eq. (38), is obtained by comparing that lifetime with observed black-hole masses and ages. No parameter is fitted to the astrophysical observations; m is the constrained variable, and the data enter only through the inequality t_bh(M, m) > observed age. There are no self-citations from the present authors, no imported uniqueness theorem, and no ansatz smuggled in via citation; the use of Hawking's temperature is an external, standard result. The main caveat, acknowledged in Section IV, is that flat-space thermodynamics is used near the horizon without a full curved-space mode calculation for the nonlocal tachyon Hamiltonian. That is an approximation or correctness risk, not a circular reduction, since the formula does not assume the conclusion it claims to establish. Therefore the paper exhibits no significant circularity.

Assumptions & free parameters 1 free parameters · 3 assumptions · 1 invented entities

The central claim rests on three main inputs: the particular nonlocal field theory of tachyons (Appendix A), the use of the flat-space thermal blackbody formula for their Hawking emission (Section IV), and the observational statement that old low-accretion stellar-mass black holes exist (Section V). None of these is fitted to the target result, so the circularity burden is low, but the first two are model assumptions that are not derived from first principles in curved spacetime.

free parameters (1)
  • g, number of tachyon degrees of freedom = set to 1 for the quoted bound
    The quoted limit m > 3e9 GeV assumes a single bosonic degree of freedom; increasing g or including fermions tightens the bound. The parameter is a property of the hypothetical particle and is not fitted to the black hole data, but it enters the central formula Eq. (33).
assumptions (3)
  • ad hoc to paper The correct Lorentz-invariant theory of tachyons is the nonlocal field theory of Appendix A with spectrum p >= m and E >= 0 and a stable vacuum.
    This construction is introduced to avoid vacuum instability; the Hawking calculation depends on this phase-space and dispersion relation.
  • domain assumption Hawking radiation of tachyons is described by the flat-space blackbody formula Eq. (21) at temperature T = 1/(8 pi G M) with zero chemical potential.
    The paper states this as a leading approximation in Section IV; it is not derived from the nonlocal theory in curved spacetime.
  • domain assumption Some observed stellar-mass black holes have existed for at least about 5 Gyr without accreting enough matter to offset tachyon evaporation.
    Section V uses M = 3.3 M_sun and t_BH >= 5 Gyr; the paper does not identify a specific low-accretion old black hole or quantify accretion.
invented entities (1)
  • Nonlocal tachyon field theory with momentum cutoff |p| >= m and nonlocal kernels J and J-tilde (Appendix A)
    purpose: Provides a stable-vacuum, Lorentz-invariant field theory of superluminal particles whose thermal emission is computed in Section IV.
    No falsifiable handle outside this paper is provided; the theory is chosen by the authors to make tachyons consistent, and the Hawking radiation result inherits this model dependence.

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Cite this review

Pith. "Pith review of Black Holes Rule Out Heavy Tachyons." pith.science (2026). https://pith.science/paper/4XRPZFFG

@misc{pith2026250111606,
  author       = {Pith},
  title        = {Pith review of: Black Holes Rule Out Heavy Tachyons},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4XRPZFFG}},
  note         = {Machine review of arXiv:2501.11606}
}
abstract

We present direct observational constraints on tachyons; particles with group velocity greater than $c$ in vacuum in a Lorentz invariant theory. Since tachyons may have no direct couplings to Standard Model particles, the most robust and model independent constraints come from gravitational effects, especially black holes. We compute the Hawking radiation of tachyons from black holes, finding it to be significantly enhanced in the presence of heavy tachyons. For a black hole of mass $M$ and tachyons of mass $m$ with $g$ degrees of freedom, the black hole lifetime is found to be $t_{bh} \approx 192 \pi \hbar M/(g c^2 m^2)$ (or doubled for fermions). This implies that the observation of black holes of a few solar masses, with lifetime of several billion years, rules out tachyons of mass $m > 3 \times 10^9$ GeV. This means there cannot exist any tachyons associated with unification scales or quantum gravity. So while there already exists theoretical reasons to be skeptical of tachyons, our work provides a complementary direct observational constraint.

Figures

Figures reproduced from arXiv: 2501.11606 by the authors.

Figure 1
Figure 1. FIG. 1. Momentum versus velocity. The standard particle [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Black hole lifetime versus tachyon mass; the plot is [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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Reference graph

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