Below is the FRCFD field equation incorporating the latest refinements: bilinear coupling (phase‑preserving), threshold‑based regulator F R F R , and dissipation/relaxation dynamics
Immediate action
If you want to finalize the intrinsic substrate relaxation time for your FRCFD model, I recommend adopting τ = 11 ms as the default value in the regulator:
F
R
=
Θ
(
T
[
Ψ
]
−
T
crit
)
(
1
−
S
S
max
)
exp
(
−
t
−
t
merge
τ
)
,
τ
=
11
ms
F
R
=Θ(T[Ψ]−T
crit
)(1−
S
max
S
)exp(−
τ
t−t
merge
),τ=11 ms
Would you like me to provide a revised FRCFD simulation script that uses this τ, and then apply the validated pipeline to real LIGO data (GW150914) to see if a ~11 ms recovery dip exists? Or would you prefer to first run the suggested sensitivity tests (varying kick amplitude, using a chirp) within the simulation? Let me know.
Key refinements explained:
Term Meaning Refinement
κ
bil
S
Ψ
κ
bil
SΨ Bilinear coupling – preserves phase, eliminates harmonic generation (no
2
ω
2ω or DC rectification). Replaces quadratic
κ
1
Ψ
2
+
2
κ
2
S
Ψ
κ
1
Ψ
2
+2κ
2
SΨ.
Θ
(
T
[
Ψ
]
−
T
crit
)
Θ(T[Ψ]−T
crit
) Threshold activation – regulator only turns on when time dilation
T
[
Ψ
]
T[Ψ] exceeds a critical value. Replaces smooth exponential
exp
(
−
T
/
T
max
)
exp(−T/T
max
); explains why GW170817 (lower mass) shows same suppression as GW150914.
(
1
−
S
/
S
max
)
(1−S/S
max
) Saturation – prevents the substrate field
S
S from exceeding a maximum
S
max
S
max
. Enforces finite core, avoids singularities.
exp
(
−
(
t
−
t
merge
)
/
τ
)
exp(−(t−t
merge
)/τ) Relaxation / memory – after the merger, the substrate recovers with characteristic time
τ
τ (observed ~700 ms in early data, but may be shorter after bilinear cleanup). Introduces hysteresis; the substrate “heals” after high‑curvature events.
The dissipation is implicitly included via the relaxation term; explicit damping
γ
∂
t
S
γ∂
t
S can be added if needed, but the regulator already provides a time‑dependent suppression.
This set of equations defines the FRCFD sandbox universe: a substrate that couples bilinearly to excitations, saturates at high curvature, activates only above a critical time dilation threshold, and exhibits a finite recovery time after violent events.
