Black Hole Ruler

A scale-invariant framework and reproducible dataset for comparing black holes across the mass spectrum (Nagy 2025).

Stephen L. Nagy · Independent Researcher · September 2025
A Scale-Invariant Ruler for Black Holes: From Stellar-Mass to Ultra-Massive with Unified Uncertainties (V1)
License: MIT

BHRuler implements the Black Hole Ruler framework for scale-invariant, reproducible comparison of black holes from stellar to ultra-massive regimes. It provides code and data to compute gravitational units and ISCO landmarks, spin-aware Kerr corrections, dual accretion prescriptions (η-bridge and ADAF/RIAF), Blandford–Znajek jet-power estimates, environmental metrics (σ, (r_{\rm infl}), (r_{\rm infl}/R_e)), and a TDE module with a logistic capture boundary—plus a versioned catalog schema for cross-scale studies.

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Table of Contents

• Why BHRuler?
• Core Equations
• Features
• Repository Layout
• Quick Start
• Data Products
• Schema (V1)
• Reproducing the Trio & Sensitivity Results
• Quality Control & Provenance
• Contributing
• Citation
• License

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Why BHRuler?

• Scale-invariant: Everything is expressed in gravitational units so times/lengths/frequencies map across (10^+) orders of mass.
• Universal invariant: (f_{\rm ISCO},t_g = 1/(6^{3/2}2\pi)\approx 0.01083) provides a built-in sanity check for Schwarzschild baselines.
• Model-transparent: Dual accretion prescriptions (η-bridge vs ADAF/RIAF) make model dependence explicit for jet-power predictions.
• Cross-scale validation: Unifies X-ray binaries, AGN, EHT imaging, and GW remnants on the same ruler.
• Reproducible: Versioned catalog schema with per-row provenance and QC flags.

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Core Equations

Gravitational units [ t_g=\frac{GM}{c^3},\quad r_g=\frac{GM}{c^2},\quad r_s=2r_g ]

Schwarzschild ISCO [ r_{\rm ISCO}=6r_g,\qquad f_{\rm ISCO}=\frac{c^3}{6^{3/2}2\pi GM},\qquad f_{\rm ISCO},t_g=\frac{1}{6^{3/2}2\pi} ]

Kerr ISCO (equatorial, prograde/retrograde)
Using Bardeen–Press–Teukolsky: [ f_{\rm ISCO}(a_)=\frac{c^3}{2\pi GM},\frac{1}{r_{\rm ISCO}^{3/2}(a_)\pm a_*} ]

Accretion prescriptions

• η-bridge: (\dot m=\lambda_{\rm Edd}/\eta_{\rm eff}), (B_H\propto \dot m^{1/2}M^{-1/2}), (P_{\rm BZ}\propto a_*^2(\lambda_{\rm Edd}/\eta_{\rm eff})M).
• ADAF/RIAF: (\lambda_{\rm Edd}=\kappa\dot m^2\Rightarrow \dot m=(\lambda_{\rm Edd}/\kappa)^{1/2}), (B_H\propto(\lambda_{\rm Edd}/\kappa)^{1/4}M^{-1/2}), (P_{\rm BZ}\propto a_*^2(\lambda_{\rm Edd}/\kappa)^{1/2}M).

Blandford–Znajek (order-of-magnitude) [ P_{\rm BZ}\approx 10^{45}\ {\rm erg,s^{-1}}\left(\frac{a_*}{0.9}\right)^2\left(\frac{B_H}{10^4\ {\rm G}}\right)^2\left(\frac{M}{10^9M_\odot}\right)^2 ]

Environment & TDE [ r_{\rm infl}=\frac{GM}{\sigma^2},\qquad t_{\rm fb}\approx 41,{\rm d}\left(\frac{M}{10^6M_\odot}\right)^{1/2}!\left(\frac{R_}{R_\odot}\right)^{3/2}!\left(\frac{M_\odot}{m_}\right) ] TDE capture boundary handled with a logistic truncation (S(M;a_*)).

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Features

• Ruler core: (t_g, r_s, f_{\rm ISCO}) + invariant check.
• Spin-aware: (r_{\rm ISCO}(a_), f_{\rm ISCO}(a_)) (prograde/retrograde).
• Accretion toggle: η-bridge vs ADAF/RIAF (user-selectable (\eta_{\rm eff},\kappa)).
• Jet power: (B_H) estimates and (P_{\rm BZ}) ranges (MAD/SANE recommendation).
• Environment: (\sigma\rightarrow r_{\rm infl}), (r_{\rm infl}/R_e), morphology flags.
• TDE: Disrupt vs swallow flags; (t_{\rm fb}); per-galaxy rate model placeholders.
• Schema + QC: Versioned, with provenance fields and automated checks.

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Repository Layout

BHRuler/
├─ data/                  # CSVs used in the paper & examples
│  ├─ bh\_ruler\_check\_2025.csv
│  ├─ bh\_spin\_aware\_ISCO\_2025.csv
│  ├─ bh\_trio\_spin\_field\_jet\_2025.csv
│  ├─ M87\_RIAF\_sensitivity\_2025.csv
│  └─ bh\_ruler\_env\_SgrA\_M87\_V1.csv
├─ src/                   # Python implementations
│  ├─ ruler.py            # core units, ISCO (Schwarz/Kerr), invariants
│  ├─ accretion.py        # η-bridge & ADAF/RIAF, B\_H, P\_BZ
│  ├─ environment.py      # σ→r\_infl, ratios, QC checks
│  └─ tde.py              # capture boundary S(M;a\*), t\_fb, rate stubs
├─ notebooks/             # optional demos reproducing figures/tables
├─ paper/                 # LaTeX source for V1
│  └─ Black\_Hole\_Ruler\_V1\_Nagy\_2025.tex
├─ README.md
├─ LICENSE
└─ CITATION.cff

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## Quick Start

**Requirements:** Python ≥ 3.10

```bash
# (optional) create a virtual environment
python -m venv .venv && source .venv/bin/activate  # Windows: .venv\Scripts\activate

pip install numpy pandas matplotlib

Minimal example (compute the ruler for a 10 M☉ Schwarzschild BH)

import numpy as np

G = 6.67430e-11
c = 2.99792458e8
M_sun = 1.98847e30

def tg_seconds(M_Msun): return G*(M_Msun*M_sun)/c**3
def rs_m(M_Msun):       return 2*G*(M_Msun*M_sun)/c**2
def fisco_hz(M_Msun):   return c**3/((6**1.5)*2*np.pi*G*(M_Msun*M_sun))

M = 10.0
print("t_g [s] =", tg_seconds(M))
print("r_s [km] =", rs_m(M)/1e3)
print("f_ISCO [Hz] =", fisco_hz(M))
print("Invariant f_ISCO * t_g =", fisco_hz(M)*tg_seconds(M))  # ~0.01083

Kerr ISCO (prograde) helper

def r_isco_over_M(a):
    Z1 = 1 + (1 - a*a)**(1/3)*((1+a)**(1/3) + (1-a)**(1/3))
    Z2 = np.sqrt(3*a*a + Z1*Z1)
    return 3 + Z2 - np.sqrt((3 - Z1)*(3 + Z1 + 2*Z2))

def fisco_kerr_hz(M_Msun, a):
    rM = r_isco_over_M(a)
    denom = (rM**1.5 + a)
    return c**3/(2*np.pi*G*(M_Msun*M_sun)) * (1.0/denom)

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Data Products

• bh_ruler_check_2025.csv — cross-scale sanity check table (Gaia BH3, GW231123 remnant, ΩCen IMBH, CEERS-1019, J0529-4351).
• bh_spin_aware_ISCO_2025.csv — spin-corrected ISCOs for credible spins (e.g., Cyg X-1, LMC X-1, GW150914 remnant).
• bh_trio_spin_field_jet_2025.csv — trio (Cyg X-1, Sgr A*, M87*) with η-bridge/RIAF branches: $r_{\rm ISCO}(a_), f_{\rm ISCO}(a_), B_H, P_{\rm BZ}$.
• M87_RIAF_sensitivity_2025.csv — M87* jet-power sensitivity vs. ADAF parameters (κ).
• bh_ruler_env_SgrA_M87_V1.csv — environment metrics and TDE flags for Sgr A* and M87*.

Each CSV includes a brief header and column units. Keep provenance in a companion README within /data.

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Schema (V1)

Core: name, type, z, distance, M, M_err, anchor_method, mass_ref, a_star, a_ref, quality_grade, use_in_fits

Ruler: t_g, r_s, f_ISCO_schw, fISCO_tg_invariant

Spin-aware: r_ISCO_a, f_ISCO_a, f_ratio

Accretion: L_Edd, L_band, L_bol, lambda_Edd, eta_eff, branch{eta_bridge|ADAF}, kappa

Magnetics/Jets: B_H, phi_BH, P_BZ_pred, P_BZ_low, P_BZ_high, branch_recommendation, jet_power_ref

Environment: sigma, sigma_err, sigma_source, sigma_aperture, sigma_ref, k_factor, k_source, Re, Re_err, morph_type, bar_flag, morphology_warning, rinfl, rinfl_err, rinfl_over_Re

TDE: tde_flag{disrupt|swallow}, Mcrit_est, tfb_days, Gamma_gal, Gamma_err, rate_method, rate_params_ref

Imaging (if any): theta_shadow, image_band, baseline_req, polarization_frac, EHT_ref

Versions: framework_version, catalog_version, row_version, last_updated, notes

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Reproducing the Trio & Sensitivity Results

import pandas as pd

trio = pd.read_csv("data/bh_trio_spin_field_jet_2025.csv")
print(trio.filter(["Object","a_* (assumed/est.)","r_ISCO [r_g] (Kerr)","f_Kerr / f_Schwarz","B_H [G]","P_BZ [erg/s]"]))

*M87 ADAF sensitivity:**

sens = pd.read_csv("data/M87_RIAF_sensitivity_2025.csv")
print(sens[["Model","mdot (dotM/dotM_Edd)","B_H [G]","P_BZ [erg/s]"]])

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Quality Control & Provenance

• M–σ outliers: flag when measured sigma deviates from M–σ prediction by >3σ.
• Aperture sanity: warn when sigma_aperture ≫ Re.
• Morphology: morphology_warning=true for pseudo-bulges/bars; k_source={orbit_model|virial} recorded for dynamical fallbacks.
• Every table row includes *_ref fields for traceability.

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Contributing

Issues and pull requests are welcome. Please:

1. Keep numeric changes reproducible (include code or notebook).
2. Update data/README and catalog_version when adding/replacing measurements.
3. Run simple invariant checks (e.g., $f_{\rm ISCO}t_g$) before submitting.

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Citation

If you use BHRuler or the accompanying datasets in your work, please cite:

Nagy, S. L. BHRuler

BibTeX

@misc{Nagy2025BHRuler,
  author       = {Stephen L. Nagy},
  title        = {A Scale-Invariant Ruler for Black Holes: From Stellar-Mass to Ultra-Massive with Unified Uncertainties (V1)},
  year         = {2025},
  eprint       = {XXXX.YYYYY},
  archivePrefix= n/a,
  primaryClass = n/a
  note         = {BHRuler software and data}
}

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License

This project is licensed under the MIT License. See LICENSE for details.