Einstein's field equations from three fluid equations
The 10 coupled nonlinear PDEs of general relativity simplify to three substrate equations: continuity, Euler, and P=ρc². The stiff equation of state makes pressure gravitate automatically, providing the factor of 4 that converts 4πG to 16πG.
Einstein's field equations
G_μν + Λg_μν = (8πG/c⁴) T_μν — 10 coupled nonlinear PDEs
in the substrate, become
∂ρ/∂t + ∇·(ρv) = 0
Continuity (mass conservation)
∂v/∂t + (v·∇)v = −∇P/ρ
Euler (momentum conservation)
P = ρc² (c_s = c)
Equation of state (stiff BEC)
ρ_eff = ρ + 3P/c² = 4ρ
Pressure gravitates!
4ρ × 4πG = 16πG·ρ
The factor of 4 → 16πG
□ h̄_μν = −(16πG/c⁴) T_μν
Linearized Einstein equations — all components reproduced
h₀₀ from Poisson, h₀ᵢ from drag, hᵢⱼ from compression
Three fluid equations replace ten coupled nonlinear PDEs
The key insight: P = ρc² makes pressure a gravitational source automatically