Hi EinsteinPy team,

Following up on #525 (adding new metric classes), I wanted to flag a recently derived
exact algebraic two-body metric that could potentially be added to einsteinpy.metric.

The metric:

For two masses M₁, M₂ at positions r₁, r₂:

# Exact two-body metric in QGD — pure algebra, no differential equations
import numpy as np

def sigma(M, r):
    G, c = 6.674e-11, 3e8
    return np.sqrt(2 * G * M / (c**2 * r))

def two_body_g00(M1, M2, r1, r2):
    """
    Exact g_tt component for two-body system.
    Cross-term 2*sigma1*sigma2 is gravitational binding energy (quantum interference).
    """
    s1, s2 = sigma(M1, r1), sigma(M2, r2)
    return -(1 - (s1 + s2)**2)

def two_body_grr(M1, M2, r1, r2):
    """Exact spatial component — no series expansion required."""
    s1, s2 = sigma(M1, r1), sigma(M2, r2)
    Sigma_sq = (s1 + s2)**2
    return 1.0 / (1 - Sigma_sq)

Key result: The cross-term 2*sigma1*sigma2 = 2*sqrt(M1*M2*rs1*rs2)/(r1*r2)^(1/2)
appears as exact gravitational binding energy — this is O(c⁻²) but contains all
higher-order interference corrections. In GR, equivalent cross-terms only appear at 2PN.

Binary merger condition follows algebraically from g₀₀ = 0:

Σ(x) = 1 → d_merger = 4 * r_schwarzschild

This is a prediction, not an assumption.

Why potentially relevant for EinsteinPy:

• Could be a new TwoBodyQGD class in einsteinpy.metric
• Zero numerical relativity required — pure Python, numpy-only
• Exact geodesics computable analytically

Preprint (Zenodo DOI): https://doi.org/10.5281/zenodo.18605058
Full code + notebooks: https://github.com/matshaba/Quantum-Gravity-Dynamics

Happy to contribute a PR with the metric class and test suite if there's interest.

Preprint: https://doi.org/10.5281/zenodo.18605058
Code: https://github.com/matshaba/Quantum-Gravity-Dynamics

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

Happy to discuss or contribute a comparison notebook.