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Distinguishing scale-dependent Planck stars from renormalization group improved Schwarzschild black holes by Gravitational waves

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Pith's one-line read The paper claims that gravitational waves from extreme-mass-ratio inspirals can observationally distinguish scale-dependent Planck stars from renormalization-group-improved Schwarzschild black holes, with the former falling below detector…

desk verdict Real waveform computation for two RG-improved metrics, but the headline detectability dichotomy rests on uncontrolled segment-length comparisons and no matched-filter SNR. read the letter →

arxiv 2506.17909 v1 pith:LEL33GGB submitted 2025-06-22 gr-qc

Li Huang This is my paper · ORCID
classification gr-qc
keywords extreme-mass-ratioinspiralsgravitationalwavesscale-dependentPlanckstarsrenormalizationgroupimprovedSchwarzschildblackholesanalytickludgewaveformscharacteristicstrainperiodicorbitsspace-basedwavedetectors
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

Using the gravitational waves emitted by a stellar-mass object spiraling into a supermassive black hole (an extreme-mass-ratio inspiral), this paper asks whether observations can tell apart two quantum-gravity-modified versions of the Schwarzschild spacetime: scale-dependent Planck stars and renormalization-group-improved Schwarzschild black holes. The paper computes orbital frequencies, energy and angular momentum fluxes, and analytic-kludge waveforms for timelike test particles on periodic orbits around each metric. It finds that the two spacetimes leave distinct imprints: waveforms from renormalization-group-improved Schwarzschild black holes have strain around $10^{-21}$ and spectral power concentrated between roughly 7 and 28 mHz, while Planck-star signals are an order of magnitude weaker and confined below about 2 mHz. Comparing the characteristic strain with detector sensitivity curves, the paper concludes that LISA, BBO, and DECIGO could detect the former but not the latter, making gravitational waves a direct observational test of which spacetime description is realized.

What carries the argument

The load-bearing object is the parametrized metric function $f(x)=1-\frac{2}{x}\left(1+s\Omega x^{-2}+\gamma s\Omega x^{-3}\right)^{-1}$, with $s=-1$ selecting scale-dependent Planck stars and $s=+1$ selecting renormalization-group-improved Schwarzschild black holes. Around this metric, the argument runs through the analytic-kludge waveform model: the quadrupole energy and angular momentum flux formulas $dE/dt$ and $d\ell/dt$ drive the adiabatic orbital evolution; the transverse traceless polarizations $h_+$ and $h_\times$ are built from the orbital phase; a discrete Fourier transform gives the frequency-domain spectra; and the characteristic strain $h_c(f)=2f\sqrt{|\tilde{h}_+(f)|^2+|\tilde{h}_\times(f)|^2}$ is compared against detector sensitivity curves. The large-eccentricity (EL) and small-eccentricity (ES) orbital-evolution approximations are auxiliary machinery used to cross-check geodesic orbits and to show that the two spacetimes deviate differently from the Schwarzschild limit.

What would settle it

Recompute the same energy and angular momentum fluxes with a second-order self-force or Teukolsky-equation code for the orbital parameters used in Figures 3 and 4. If the characteristic strain for scale-dependent Planck stars rises above the LISA, BBO, or DECIGO sensitivity curves, the paper's non-detection claim is contradicted; alternatively, a matched-filter search of mock LISA data with injected Planck-star waveforms would settle whether such signals are truly undetectable.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the sign parameter $s$ in the metric function $f(x)=1-\frac{2}{x}\left(1+s\Omega x^{-2}+\gamma s\Omega x^{-3}\right)^{-1}$ controls the observability of EMRI signals. For $s=-1$ (scale-dependent Planck stars), with $M=10^7M_\odot$, $\gamma=9/2$, $\lambda_-=-1$, and angular momentum $l=8.6$, the characteristic strain stays below the sensitivity curves of LISA, eLISA, TianQin, BBO, DECIGO, EPTA, IPTA, SKA, LIGO, aLIGO, and LIGO A+. For $s=+1$ (renormalization-group-improved Schwarzschild black holes), with $l=3.6$, the characteristic strain rises above the LISA, BBO, and DECIGO curves. Time-domain waveforms show zoom-whirl glitches whose number matches the periodic-orbit classification $(z,w,v)$, and frequency-domain spectra occupy different millihertz bands for the two spacetimes. The paper therefore claims that both time-domain and frequency-domain gravitational-wave observations can distinguish the two metrics, and that the renormalization-group-improved case is the one future space-based detectors are likely to see.

Load-bearing premise

The detectability verdict rests on treating orbits very close to the black hole with weak-field, slow-motion quadrupole and analytic-kludge formulas, and the paper does not quantify how much error that approximation introduces.

Editorial extensions

If this is right

  • A null detection of the predicted Planck-star band by LISA, BBO, or DECIGO would be consistent with the paper's claim, while a detection above the noise curve in the 7-28 mHz band would support renormalization-group-improved Schwarzschild black holes over scale-dependent Planck stars.
  • The frequency-domain separation (0-2 mHz versus 7-28 mHz) gives EMRI searches a fast spectral pre-filter before matched filtering.
  • Because renormalization-group-improved Schwarzschild waveforms closely track Schwarzschild waveforms while Planck-star waveforms do not, future detectors that see ordinary Schwarzschild-like EMRI signals can already constrain the scale-dependent Planck-star branch.
  • The zoom-whirl glitch structure tied to the periodic-orbit labels $(z,w,v)$ provides a morphology-based discriminator that is independent of amplitude calibration.
  • The EL/ES comparison indicates that the two approximation schemes become unreliable for Planck stars at high eccentricity, so any future detection claim in that branch would need geodesic-level waveform modeling.

Reading between the lines

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

  • Inference: because the detectability comparison uses $l=8.6$ for Planck stars and $l=3.6$ for renormalization-group-improved Schwarzschild black holes to match the same radial frequency, the conclusion may depend on that parameter choice; a matched-parameter scan across semilatus rectum and eccentricity would test whether the strain gap persists.
  • Inference: if the weak-field quadrupole and analytic-kludge approximations lose accuracy near the innermost stable circular orbit, the predicted non-detection for Planck stars could be an artifact; checking the fluxes against a full Teukolsky or self-force calculation would settle that.
  • Inference: the same waveform pipeline could be applied to other modified Schwarzschild metrics, turning the frequency-band location (0-2 mHz versus 7-28 mHz) into a generic discriminator between horizonless compact objects and black holes.
  • Inference: abundance forecasts for EMRI sources could convert the detectable-versus-undetectable split into an observational prior: if LISA sees many millihertz EMRIs resembling renormalization-group-improved Schwarzschild black holes and none resembling scale-dependent Planck stars, that is evidence about which quantum-gravity branch is realized.
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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

5 major / 4 minor

Summary. The paper studies extreme-mass-ratio inspirals in two static, spherically symmetric metrics: scale-dependent Planck stars (s = -1) and renormalization-group-improved Schwarzschild black holes (s = +1), both written in Schwarzschild gauge with metric function parameterized by gamma and Omega. It derives post-Newtonian-style expansions for test-particle energy, angular momentum, orbital frequencies, and precession rate; introduces the EL and ES schemes for orbital evolution; computes quadrupole energy and angular momentum fluxes; generates analytic-kludge time-domain waveforms and their discrete Fourier transforms; and compares the resulting characteristic strains with sensitivity curves of ground-based and space-based detectors. The central claim is that LISA, BBO, and DECIGO could detect signals from the RG-improved Schwarzschild case while Planck-star signals remain undetected, and that time-domain and frequency-domain waveforms can distinguish the two spacetimes.

Significance. The question is well motivated: EMRIs are among the most promising probes of strong-field gravity, and identifying observational signatures that distinguish quantum-gravity-inspired metrics would be a valuable result. The manuscript contains useful checks: the Schwarzschild limit of the flux formulas reproduces the Peters formulas, and the orbital and waveform calculations are self-contained once the external metric is adopted. However, the quantitative detectability conclusion is built on weak-field flux formulas and on characteristic-strain comparisons whose normalization is unspecified, so the headline observational claim is not yet established.

major comments (5)
  1. [Sec. 4.3, Eq. (4.24), Figs. 3-4] The characteristic strains in Figs. 7-8 are computed from DFTs of waveforms with different durations: 20,000 s for the Planck star (Fig. 3) and 1,200 s for the RG-improved Schwarzschild black hole (Fig. 4). For a quasi-periodic signal, |tilde h(f)| scales with the integration time, so h_c(f) = 2 f sqrt(|tilde h_+|^2 + |tilde h_x|^2) is not a source-intrinsic quantity unless a common observation time T_obs is fixed and the DFT normalization is specified; neither is given. Rescaling both waveforms to a one-year observation multiplies the plotted Planck-star and RG-BH strains by very different factors (about 1.6e3 and 2.6e4, respectively), so the apparent order-of-magnitude gap can be dominated by the choice of segment length. The detectability claim in Sec. 5 should be based on a matched-filter SNR computed with a common, astrophysically motivated observation time and the detector noise PSD.
  2. [Sec. 4.1, Eqs. (4.14)-(4.15), Sec. 4.2, Eqs. (4.22)-(4.23)] The quadrupole flux formulas and the analytic-kludge waveform formulas assume v/c << 1 and Newtonian orbital dynamics, yet they are applied to eccentric zoom-whirl orbits in the strong-field regime. For the RG-improved Schwarzschild case the initial angular momentum is l = 3.6, close to the Schwarzschild ISCO value l ~ 3.46 at x = 6, and the Planck-star orbits also probe the strong-field region. No error estimate or comparison with black-hole perturbation theory is provided, and these fluxes drive the waveforms from which the detectability and distinguishability conclusions are read off. The authors should quantify the systematic error of Eqs. (4.14)-(4.15) and (4.22)-(4.23) in this regime before drawing quantitative conclusions.
  3. [Eqs. (4.18) and (4.21)] The decomposition dE/dt = dE/dt|_{Schw} + dE/dt|_{sOmega} is not what is claimed: the sOmega part in Eq. (4.18) contains two terms at orders p^{-6} and p^{-7} that have no sOmega factor, and Eq. (4.21) similarly contains two Omega-independent terms at p^{-9/2} and p^{-11/2}. These are pure Schwarzschild higher-order terms, not corrections from the modified metric, so the actual Omega-dependent correction is not cleanly identified. The fluxes are also stated without derivation. Since these fluxes govern the inspiral evolution and the resulting waveforms, this needs to be corrected or justified.
  4. [Eq. (2.22), Eqs. (2.25)-(2.33), Eqs. (3.1)-(3.8)] Eq. (2.22) gives l_Schw as a sum containing terms scaling as p^{1/2}, p^{1/2}, p^{3/2}, p^{5/2}, p^{7/2}, p^{7/2} followed by O(p^{-9/2}); this is internally inconsistent for a large-p expansion, and the powers are presumably missing minus signs. Similar exponent and notation issues appear in the frequency expansions, e.g., Eq. (3.8) contains a term proportional to (2 pi nu_x)^3 inside an expansion in nu_phi. These expressions are inputs to the EL/ES orbital evolution and to the frequency relations, so the authors should verify and correct them.
  5. [Sec. 3.3, Figs. 1-4] The two spacetimes are compared at the same radial frequency, Upsilon_x/(2 pi) = 0.2 mHz, but with different angular momenta (l = 8.6 for Planck stars versus l = 3.6 for RG-improved Schwarzschild black holes) and hence different orbital radii. Since the waveform amplitude depends on the orbital radius, this comparison conflates the effect of the metric with a difference in the chosen orbital configuration. The claim that waveforms distinguish the two spacetimes should be tested over a common parameter space (for example, fixed p and e, or fixed l) to verify that the apparent distinguishability is not an artifact of the chosen comparison.
minor comments (4)
  1. [Eq. (4.13)] Eq. (4.13) writes the quadrupole tensor as the sum of its components, Q_ij = Q_11 + Q_22 + Q_12 + Q_21 + Q_33; this is not a tensor equation and should be replaced by a componentwise definition of the tensor.
  2. [Sec. 3.3, Fig. 2 caption] The paragraph introducing Fig. 2 states that the RG-improved Schwarzschild case uses 'gamma = 9/2 and lambda_- = -1.0', while the caption correctly uses lambda_+ = 1.0; the text should be harmonized.
  3. [Sec. 5] The statement that gravitational waves have been 'observationally verified' to distinguish the two spacetimes is too strong: no real gravitational-wave observation is analyzed in this paper. The conclusion should be phrased as a prediction or a detectability forecast.
  4. [Abstract] The sentence 'These gravitational wave sensitivity curves can be experimentally tested for both spacetimes considered' is unclear and should be rephrased to describe the comparison between predicted characteristic strains and detector sensitivity curves.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained given the externally sourced metrics; the remaining concerns are approximation and comparison fairness, not reduction of outputs to inputs.

full rationale

The paper's derivation chain is: adopt externally sourced metrics (Eqs. 2.1-2.6, from refs. [80-83]); solve geodesic equations and compute orbital frequencies; use EL/ES methods; compute quadrupole energy and angular momentum fluxes (Eqs. 4.14-4.21) with standard formulas; generate AK waveforms (Eqs. 4.22-4.23); and compare characteristic strains (Eq. 4.24) against detector sensitivity curves. Each step is a consequence of the stated metric and standard gravitational-wave formulas, with no parameter fitted to the predicted quantity. The author's prior work [83] is cited alongside independent references [80-82] for the metric and horizon properties, so the self-citation is not load-bearing. The different choices of orbital angular momentum (l=8.6 for Planck stars, l=3.6 for RG-improved Schwarzschild) are presented as a comparison at the same radial frequency, not as fitted outputs. The caveats noted by the reader and in Sec. 5 -- weak-field AK/quadrupole approximations and unequal DFT observation durations in Figs. 7-8 -- are accuracy and fairness issues for the detectability claim, but they do not amount to defining the conclusion in terms of its inputs. No circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a chosen metric family, the geodesic approximation, the quadrupole flux formulas, and the analytic-kludge waveform model. No new particles or forces are introduced. The free parameters are all hand-picked input values, not fitted to observational data.

free parameters (8)
  • gamma (dimensionless cutoff parameter) = 9/2
    Chosen by hand; the paper does not justify this value physically.
  • lambda- for scale-dependent Planck stars = -1.0
    Chosen to fix the sign and magnitude of Omega through eq. (2.5).
  • lambda+ for RG-improved Schwarzschild black holes = 1.0
    Chosen to fix Omega through eq. (2.6); the paper notes lambda+=1 gives one horizon.
  • Omega for Planck stars = 176.392
    Computed from eq. (2.5) with gamma=9/2 and lambda-=-1, not fitted to data, but the parameter combination is a modeling choice.
  • Omega for RG-improved Schwarzschild black holes = 0.204
    Computed from eq. (2.6) with gamma=9/2 and lambda+=1.
  • Initial l and E for Planck-star orbit = l=8.6, E=0.99446821
    Chosen to give Upsilon_x/(2*pi)=0.2 mHz; directly controls the waveform and strain amplitude.
  • Initial l and E for RG-improved Schwarzschild orbit = l=3.6, E=0.95891696
    Chosen to give the same Upsilon_x/(2*pi)=0.2 mHz; the difference in l drives the detectability result.
  • Inclination angle iota and longitude of pericenter zeta = pi/4, pi/4
    Chosen for the waveform computation; do not affect the qualitative distinction.
assumptions (5)
  • domain assumption The spacetimes are described by metric (2.1)-(2.4) from refs [80-83] with s=-1 for Planck stars and s=+1 for RG-improved Schwarzschild black holes.
    The paper adopts this metric family without independent derivation; its physics is taken from the cited literature.
  • standard math The test particle follows a geodesic of that metric (eqs. 2.7-2.10).
    Standard in the EMRI literature; the paper uses the Lagrangian and conserved E and l.
  • domain assumption The gravitational-wave energy and angular momentum fluxes are given by the quadrupole formulas (4.14)-(4.15).
    These are weak-field linearized formulas applied to strong-field near-ISCO orbits; the paper does not justify their accuracy here.
  • domain assumption The waveform polarizations are given by the analytic-kludge formulas (4.22)-(4.23).
    Cited to ref. [96]; a known approximation whose accuracy for these orbits is not quantified.
  • domain assumption Detector sensitivity curves from refs [31,32,33,35,36,74-79] are valid representations of the instruments.
    Taken from literature; the paper does not derive them.

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Pith. "Pith review of Distinguishing scale-dependent Planck stars from renormalization group improved Schwarzschild black holes by Gravitational waves." pith.science (2026). https://pith.science/paper/LEL33GGB

@misc{pith2026250617909,
  author       = {Pith},
  title        = {Pith review of: Distinguishing scale-dependent Planck stars from renormalization group improved Schwarzschild black holes by Gravitational waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LEL33GGB}},
  note         = {Machine review of arXiv:2506.17909}
}
read the original abstract

Extreme mass-ratio inspirals (EMRIs), consisting of a stellar-mass black hole orbiting a supermassive black hole, are among the primary targets for future space-based gravitational wave detectors. By analyzing the emitted gravitational wave signals, we can probe the nature of compact objects in the strong-field region. To achieve this, we examine the effects of gravitational radiation. In this work, we base our calculations on the general relativistic Schwarzschild background and calculate the energy and angular momentum fluxes of gravitational waves. We perform a theoretical analysis of the equations of motion and the orbital evolution equations for EMRIs. The gravitational waveforms generated by the different periodic orbits of timelike test particles around scale-dependent Planck stars or renormalization group improved Schwarzschild black holes are investigated using both time-domain and frequency-domain methods. The time-domain method employs the ``analytic kludge" (AK) approach, while the frequency-domain method utilizes the discrete Fourier transform. We calculate the characteristic strain of the corresponding gravitational waves and compare them with the sensitivity curves of both ground-based and space-based detectors. These gravitational wave sensitivity curves can be experimentally tested for both spacetimes considered. Additionally, we use two approximate methods--the large eccentricity (EL) method and the small eccentricity (ES) method--to study the orbital evolution of EMRIs and compare the results with equatorial orbits derived from geodesic equations. Our findings will contribute to a deeper understanding of the nature of spacetime.

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