Initial Data

🔢 Initial Data

GRANITE v0.6.8 | ← HPC Deployment | Known Fixed Bugs →

Reference for all initial data types: physics background, YAML configuration, and known limitations.

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1. Two-Punctures / Bowen-York (Binary Black Holes)

1.1 Physics

The Two-Punctures method (Brandt & Brügmann 1997) solves the constraint equations for two compact objects using a spectral decomposition of the conformal factor. The Bowen-York (1980) extrinsic curvature provides analytic solutions with specified momenta and spins.

The conformal factor is written as:

ψ = ψ_BL + u

where ψ_BL is the Brill-Lindquist background and u is computed via a spectral expansion satisfying the Hamiltonian constraint.

1.2 Quasi-Circular Momentum — p_t Formula

For an equal-mass BBH at separation d with total mass M_total = 1:

d [M]	p_t (PN, leading order)	p_t (PN, 1.5PN)	Notes
6M	±0.133	±0.124	Close, strong-field
8M	±0.101	±0.096	Standard range
10M	±0.0840	±0.0812	Default GRANITE config
12M	±0.0724	±0.0706	Wide separation
15M	±0.0616	±0.0605	Far separation

Formula (leading-order PN):

p_t ≈ (M_reduced / 4) × √(M_total / d)
    = (m₁ m₂ / M_total) / 4 × √(M_total / d)
    
For equal-mass (m₁=m₂=0.5):
    p_t ≈ 0.25 × 0.25 × √(1/d) = 0.0625 / √d [M²·M^(-1/2)] = 0.0625/√10 ≈ 0.0198

[Full PN 1.5PN values require Pfeiffer et al. 2007 tables]

1.3 YAML Configuration

initial_data:
  type: two_punctures
  bh1:
    mass:      0.5           # m₁/M_total
    position:  [5.0, 0.0, 0.0]   # [M] (separation = 10M)
    momentum:  [0.0, 0.0840, 0.0] # [M] tangential ← REQUIRED FOR INSPIRAL
    spin:      [0.0, 0.0, 0.0]   # dimensionless spin a/M
  bh2:
    mass:      0.5
    position:  [-5.0, 0.0, 0.0]
    momentum:  [0.0, -0.0840, 0.0]  # ← equal and opposite
    spin:      [0.0, 0.0, 0.0]

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2. Brill-Lindquist (Single or Multiple BHs at Rest)

2.1 Physics

Brill-Lindquist data uses a simple superposition for N black holes at rest:

ψ_BL = 1 + Σᵢ (mᵢ / 2rᵢ)

This satisfies the Hamiltonian constraint analytically but has zero extrinsic curvature — all BHs are momentarily at rest.

2.2 YAML Configuration

initial_data:
  type: brill_lindquist
  bh1:
    mass:     1.0
    position: [0.0, 0.0, 0.0]

2.3 Boundary Condition Requirement

⚠️ MANDATORY: Use boundary.type: copy with Brill-Lindquist data.
boundary.type: sommerfeld produces ‖H‖₂ 8× worse from step 1.
This is a confirmed incompatibility — see Known Fixed Bugs: Sommerfeld+BL.

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3. TOV Neutron Star

3.1 Physics

The Tolman-Oppenheimer-Volkoff equations for a static perfect fluid sphere:

dP/dr = −(ρ + P)(m + 4πr³P) / [r(r − 2m)]
dm/dr = 4π r² ρ

Integrated outward from r=0 with boundary conditions P(0)=P_central, m(0)=0 until P=0 (surface).

3.2 Critical Unit Note

⚠️ BUG HISTORY: Early code used RSUN_CGS (~6.957e10 cm) for km→cm conversion.
   CORRECT: 1 km = 1.0e5 cm

This was fixed as bug "TOV" — never revert this conversion.

3.3 Expected Results (Standard Polytrope, Γ=2)

Quantity	Expected	Tolerance
ADM mass M	≈ 1.4 M☉	±5%
Radius R	≈ 10 km	±10%
Central density ρ_c	≈ 5×10¹⁴ g/cm³	±15%

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4. Compatibility Matrix

Initial Data	Sommerfeld BC	Copy BC	Notes
Brill-Lindquist	❌ FORBIDDEN	✅ Required	8× constraint violation with Sommerfeld
Two-Punctures	✅ Recommended	✅ Also OK	Sommerfeld preferred for long runs
Bowen-York	✅ OK	✅ OK	Either works
TOV	❌ Untested	✅ Use this

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See also: Parameter Reference | Benchmarks & Validation

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