Gravitational Wave Extraction

📡 Gravitational Wave Extraction

GRANITE v0.6.8 | ← FAQ | HPC Deployment →

---

1. Method: Newman-Penrose Ψ₄

Gravitational waves are extracted via the Newman-Penrose Weyl scalar Ψ₄, which in the wave zone reduces to:

Ψ₄ = −(ḧ₊ − i ḧ×) / 2r

where h₊, h× are the two GW polarizations.

GRANITE computes Ψ₄ via the NP formalism applied to the evolved CCZ4 metric on coordinate spheres.

---

2. Spherical Harmonic Decomposition

Ψ₄(t, r) = Σ_{ℓ,m} Ψ₄^{ℓm}(t, r) · _{-2}Y_{ℓm}(θ, φ)

The spin-weighted spherical harmonics _{-2}Y_{ℓm} are implemented via Wigner d-matrices (Goldberg et al. 1967) for all modes, including ℓ=3,4 sub-dominant contributions.

Dominant mode for non-precessing equal-mass BBH: ℓ=2, m=±2

---

3. Extraction Radii

Ψ₄ is evaluated at multiple extraction radii to enable extrapolation to null infinity (Nakano et al. 2015):

Radius [M]	Purpose
50	Near-zone check
100	Standard extraction
150	Convergence test
200	Far-field
300	Near-null
500	Null-infinity proxy

Recommended minimum: 3 extraction radii for Richardson extrapolation. All 6 radii for publication-quality results.

---

4. Strain Recovery

The GW strain is obtained by double time-integration of Ψ₄:

h₊ − i h× = ∫∫ Ψ₄ dt dt

Fixed-frequency integration (Reisswig & Pollney 2011) is used to suppress integration drift (low-frequency noise from integration constants):

h̃_{ℓm}(ω) = Ψ̃₄^{ℓm}(ω) / max(ω, ω₀)²

where ω₀ is the cutoff frequency (typically ω₀ = 0.5/M for BBH).

---

5. Radiated Energy and Momentum

Energy:

dE/dt = r² / (16π) · Σ_{ℓ,m} |Ψ₄^{ℓm}|²

Linear momentum (recoil kick):
Implemented via Ruiz, Campanelli & Zlochower (2008) adjacent-mode coupling formula:

dP^z/dt = (r²/16π) · Σ_{ℓ,m} Im[Ψ̄₄^{ℓm} · ∫Ψ₄^{ℓ,m+1}] · a^{ℓm}
a^{ℓm} = √[(ℓ−m)(ℓ+m+1)] / [ℓ(ℓ+1)]

---

6. Current Status in v0.6.8

GW extraction infrastructure is implemented in src/postprocess/postprocess.cpp and python/granite_analysis/gw.py. Full activation in production runs is targeted for v0.7.

---

See also: Physics Formulations | Benchmarks & Validation

---