[Contribution idea] Algebraic two-body metric notebook — complement to numerical approach

[matshaba]

Hi,

The NRPy+ tutorial series is excellent for showing the numerical relativity pipeline.
I wanted to propose a complementary notebook: an algebraic two-body metric that can
reproduce key binary merger results without solving PDEs.

What the notebook would show:

import numpy as np
import matplotlib.pyplot as plt

G, c, M_sun = 6.674e-11, 3e8, 1.989e30

def qgd_two_body_metric(M1, M2, r1, r2):
    """Exact algebraic two-body metric (Matshaba 2026)."""
    rs1 = 2*G*M1/c**2
    rs2 = 2*G*M2/c**2
    s1 = np.sqrt(rs1/r1)
    s2 = np.sqrt(rs2/r2)
    Sigma = s1 + s2
    g00 = -(1 - Sigma**2)
    grr = 1/(1 - Sigma**2)
    return g00, grr, Sigma

# GW150914: predict merger separation
M1, M2 = 35.6*M_sun, 30.6*M_sun
rs1 = 2*G*M1/c**2
rs2 = 2*G*M2/c**2
# Merger: Sigma=1 at midpoint
# d_merger = 4*(rs1+rs2)/2 = 2*(rs1+rs2)
d_merger = 2*(rs1 + rs2)
print(f"Predicted merger separation: {d_merger/1e3:.1f} km")  # ~370 km

Side-by-side comparison:

• QGD algebraic waveform vs NRPy numerical waveform for GW150914
• Same physical parameters, two approaches

This wouldn't claim to replace NR — it would illuminate why NR is necessary for the
merger/ringdown phase while the algebraic approach works for inspiral.

Preprint (full derivation): https://doi.org/10.5281/zenodo.18605058
Complete notebooks already built: https://github.com/matshaba/Quantum-Gravity-Dynamics

Quick Reference:
https://github.com/matshaba/Quantum-Gravity-Dynamics/blob/main/core/two_and_three_body_solutions.py

Happy to submit a PR with a complete notebook if this seems like a good fit.