Numerical Relativity

ADM, BSSN and GHG formulations of the Einstein equations, with a 1D ADM diagnostic toolkit.

Status: exploratory educational demonstration, not validated physics. The simulations are simplified 1D teaching examples; production numerical relativity requires extensive validation and peer review. For the toolkit's tested mathematical results, see the Spectral-Gap Boundary audit.

What Is Numerical Relativity?

When two black holes spiral together and merge, they create ripples in spacetime called gravitational waves. Einstein's equations predict these waves, but the math is so complex that we can only solve it numerically on supercomputers. This is numerical relativity: simulating the most violent events in the universe using computer algorithms.

Just like the scattering amplitudes above, there are multiple mathematical formulations of Einstein's equations - ADM (the original approach), BSSN (better for black hole stability), and GHG (cleaner wave extraction). Each breaks spacetime into space plus time differently, but all three must predict the same gravitational wave signal that LIGO and VIRGO actually detect.

This page explains the formulations and shows a 1D ADM toolkit. It does not run BSSN or GHG and does not vary the calculus inside an Einstein solve; the interactive panel below is an illustrative cartoon.

3+1 Decomposition of Spacetime

Numerical relativity evolves the gravitational field by splitting spacetime into space + time. The same physical spacetime can be foliated in infinitely many ways, leading to different formulations:

ADM Formalism

Original Arnowitt-Deser-Misner formulation. Variables: metric gamma_ij and extrinsic curvature K_ij. Clear geometric meaning.

BSSN

Baumgarte-Shapiro-Shibata-Nakamura conformal decomposition: gamma_ij = phi^4 gamma_tilde_ij. Better numerical stability for black hole simulations.

GHG

Generalized Harmonic Gauge. Gauge source functions as dynamical variables. Principal hyperbolicity, clean wave extraction.

ADM Evolution Equations

Metric Decomposition

The spacetime metric decomposes into lapse alpha, shift beta^i, and spatial metric gamma_ij:

ds^2 = -\alpha^2 dt^2 + \gamma_{ij} (dx^i + \beta^i dt)(dx^j + \beta^j dt)

Evolution of Spatial Metric

How the 3-metric changes in time:

\partial_t \gamma_{ij} = -2\alpha K_{ij} + \mathcal{L}_\beta \gamma_{ij}

where $K_{ij}$ is the extrinsic curvature and $\mathcal{L}_\beta$ is the Lie derivative along the shift vector.

Evolution of Extrinsic Curvature

How the bending of spatial slices changes:

\partial_t K_{ij} = -D_i D_j \alpha + \alpha(R_{ij} + K K_{ij} - 2 K_{ik} K^k_j) + \mathcal{L}_\beta K_{ij}

where $D_i$ is the covariant derivative compatible with gamma_ij, and $R_{ij}$ is the 3-Ricci tensor.

Constraint Equations

Einstein equations split into evolution equations (above) and constraint equations (below). The constraints must be satisfied on every spatial slice:

Hamiltonian Constraint

\mathcal{H} = R + K^2 - K_{ij} K^{ij} - 16\pi \rho = 0

where $R$ is the 3-Ricci scalar and rho is the energy density.

Momentum Constraints

\mathcal{M}_i = D_j K^j_i - D_i K - 8\pi j_i = 0

where $j_i$ is the momentum density. There are 3 independent momentum constraints.

Scheme-Invariance Across Formulations

All three formulations (ADM, BSSN, GHG) are mathematically equivalent representations of Einstein equations. Physical observables must be independent of which formulation we use:

I_{NR} = \sum_{t} \left( \frac{|\mathcal{H}_{ADM}(t) - \mathcal{H}_{BSSN}(t)|}{|\mathcal{H}_{ADM}(t)| + 10^{-10}} + \frac{|\mathcal{M}_{ADM}(t) - \mathcal{M}_{BSSN}(t)|}{|\mathcal{M}_{ADM}(t)| + 10^{-10}} \right)

We monitor constraint violation over the evolution. Scheme-robustness requires that physical violations (not gauge noise) agree across formulations within numerical tolerance.

Key Insight: Different formulations have different numerical stability properties (BSSN is more stable for binary black holes), but they must produce the same constraint violations when evaluated at the same physical state. Divergence indicates either (a) numerical error accumulation, or (b) gauge artifact.

Interactive Demo: Constraint Evolution

Illustrative cartoon, not an Einstein solve. The curves below are scripted per-formulation growth/decay rates plus seeded noise, written to illustrate the textbook folklore that BSSN damps constraint violations better than ADM. No evolution equations are integrated in the browser, so the curves and the agreement score are not evidence about ADM, BSSN or GHG.

Python Module Usage

The meta_calculus.nr module provides short-duration spherical (1D) ADM diagnostics. There is no BSSN or GHG evolver in the toolkit.

from meta_calculus.nr import (
    quick_start_flat_space_test,
    quick_start_schwarzschild_evolution,
)

# Flat space should stay flat (constraint sanity test)
flat = quick_start_flat_space_test(N=100, t_final=1.0)

# Short 1D ADM evolution of Schwarzschild data, 1+log slicing, RK4
bh = quick_start_schwarzschild_evolution(M=1.0, N=100, t_final=1.0)

Implementation Status

ADM Evolution: short-duration 1D spherical ADM evolution in meta_calculus.nr, with Hamiltonian and momentum constraint monitoring, tested on flat space and Schwarzschild data.

Scheme Comparison: not implemented. The stored ADM/BSSN/GHG comparison (results/phase2_nr_results.json) is flagged by its own producer as a placeholder simulation (measured convergence order 0.13 against an expected 2.0), so it is not shown as evidence.

Future Work: BSSN/GHG evolvers, 3D evolution, adaptive mesh refinement and binary black hole simulations are research-level extensions that do not exist here.