CRR Gravitational Waves — Coherence · Rupture

Mathematical Foundation

The Coherence-Rupture-Regeneration (CRR) framework provides a process-theoretic description of how systems accumulate history, undergo irreversible transitions, and reconstitute themselves. Here we derive gravitational wave dynamics through CRR's three fundamental equations.

THE THREE CRR EQUATIONS

1. C(x,t) = ∫_-∞^t L(x,τ) dτ Coherence accumulates through the integral of local "learning" L(x,τ)

2. δ(now) Rupture as Dirac delta — the scale-invariant "choice moment" where past becomes future

3. R(t) = ∫ φ(x,τ) exp(C(x,τ)/Ω) Θ(t-τ) dτ Regeneration weighted by memory kernel exp(C/Ω); Θ is the Heaviside step function

Ω (Omega) is the precision parameter — it determines how sharply the system accesses its coherence history. From the Ω-symmetry hypothesis: Ω = 1/φ where φ is the phase (in radians) required to reach rupture. For Z₂ symmetric systems, φ = π, giving Ω = 1/π ≈ 0.318.

Gravitational Waves as CRR Process

A binary black hole merger naturally decomposes into the three CRR phases:

INSPIRAL → COHERENCE

The binary system accumulates orbital coherence as it spirals inward. Each completed orbit adds to the system's phase history.

L(τ) = dΦ/dτ = ω(τ) /* Angular velocity as coherence rate */

From post-Newtonian theory, the orbital frequency evolves as:

ω(τ) ∝ (t_merger - τ)^-3/8

Therefore coherence accumulates as:

C(t) = ∫ ω(τ) dτ ∝ (t_merger - t)^5/8

Key insight: C approaches the critical value C* = Ω⁻¹ as merger approaches. When C reaches this threshold, rupture becomes inevitable.

MERGER → RUPTURE

The merger is the δ-function rupture — instantaneous on the timescale of the inspiral, irreversible, marking the ontological transition from "two" to "one".

δ(t_merger) : Binary → Single Black Hole

At this moment:

dC/dt reaches maximum (coherence rate peaks)
The topology of spacetime changes irreversibly
Information about the binary's structure is "scrambled" into the final BH

Key insight: The rupture is scale-invariant — the same C→δ→R structure occurs whether the masses are 10 M☉ or 10⁹ M☉. This is CRR's process-theoretic universality.

RINGDOWN → REGENERATION

The newly-formed black hole "regenerates" toward its equilibrium Kerr state. The ringdown waveform is determined by quasi-normal modes (QNMs).

h(t) = A₀ exp(-t/τ_damp) cos(ω_QNM · t)

In CRR terms, this is the regeneration integral:

R(t) = ∫ φ(τ) exp(C(τ)/Ω) Θ(t-τ) dτ

The exp(C/Ω) memory kernel is crucial:

Large Ω → exp(C/Ω) ≈ 1 → all history weighted equally → slow regeneration
Small Ω → exp(C/Ω) peaked → only high-coherence moments accessible → fast regeneration

Key insight: The damping time τ_damp = Ω⁻¹ × (geometric factor). Symmetric mergers (lower Ω) ring down faster because exp(C/Ω) is more sharply peaked.

Spacetime Curvature Dynamics

INSPIRAL · COHERENCE

Coherence

Rupture

Regeneration

Gravitational Wave Strain h(t)

Primary Mass (M☉)

30 Solar Masses

Secondary Mass (M☉)

30 Solar Masses

Animation Speed

1.0 × realtime

Sound On

Mass Ratio (q) 1.000

CRR Ω 0.3183

Predicted CV 0.1592

Symmetry Class Z₂

f_ISCO 73 Hz

f_QNM 251 Hz

τ_QNM 4.0 ms

M_final 57.0 M☉

E_rad 5.0%

Complete Mathematical Derivations

1. Deriving Ω from Mass Ratio

The Ω-symmetry hypothesis states that Ω = 1/φ where φ is the effective phase (in radians) required to reach rupture. For gravitational wave systems, the symmetry class is determined by the mass ratio q = m₂/m₁ (where m₂ ≤ m₁).

Mass ratio: q = min(m₁, m₂) / max(m₁, m₂) ∈ [0, 1]

Symmetric mass ratio: η = (m₁ · m₂) / (m₁ + m₂)² ∈ [0, 0.25]

For equal masses (q = 1), we have perfect Z₂ exchange symmetry — swapping the black holes leaves the system invariant. This corresponds to reaching rupture in half a cycle:

q = 1 (Z₂ symmetric): φ_eff = π → Ω = 1/π ≈ 0.318

As symmetry breaks (q → 0), more phase accumulation is required before rupture:

φ_eff = π · (1 + (1 - η/η_max)) where η_max = 0.25

Ω(q) = 1 / φ_eff = 1 / [π · (2 - 4η)]

This predicts a coefficient of variation CV = Ω/2, giving CV_Z₂ ≈ 0.159 — testable against ringdown amplitude statistics.

2. Inspiral Waveform from Coherence Dynamics

The gravitational wave strain h(t) during inspiral comes from the quadrupole formula. In CRR terms, we express this through coherence accumulation:

Chirp mass: M_c = (m₁ · m₂)^3/5 / (m₁ + m₂)^1/5

Time to merger: τ = t_merger - t

The phase evolution (accumulated coherence) is:

Φ(τ) = -2 · (τ / 5M)^5/8 / η^3/8 /* This IS the coherence integral C(t) */

The amplitude evolution (coherence rate) is:

A(τ) ∝ M_c^5/4 · τ^-1/4 /* Amplitude grows as dC/dt increases */

The complete inspiral waveform:

h_inspiral(t) = A(τ) · cos(Φ(τ))

3. Ringdown from Regeneration Integral

After merger, the final black hole rings at its quasi-normal mode (QNM) frequencies. The dominant mode (l=2, m=2, n=0) has:

ω_QNM ≈ 1.2 / M_final /* Natural frequency of regeneration */

τ_damp ≈ 0.3 · M_final · (1 + 0.5(1 - q)) /* Damping time — depends on symmetry via Ω */

The CRR regeneration integral gives the ringdown waveform:

R(t) = ∫₀^t φ(τ) · exp(C(τ)/Ω) · Θ(t - τ) dτ

For a system where C(τ) decays from its peak at merger:

C(τ) = C_peak · exp(-τ / τ_damp)

The memory kernel exp(C/Ω) weights contributions from different times. Evaluating the integral yields the standard ringdown form:

h_ringdown(t) = A₀ · exp(-t / τ_damp) · cos(ω_QNM · t)

CRR Prediction: The damping time scales with Ω⁻¹. Since Ω_Z₂ = 1/π is smaller than Ω for broken symmetry, equal-mass mergers ring down faster. This is consistent with numerical relativity results showing cleaner ringdowns for symmetric binaries.

4. The CRR–Black Hole Correspondence

A remarkable correspondence emerges between CRR quantities and black hole thermodynamics:

C = S = 4πM² /* Coherence = Bekenstein-Hawking entropy */

Ω = T = 1/(8πM) /* CRR precision = Hawking temperature */

This gives the scrambling time:

t_scramble = Ω⁻¹ · log(C) = 8πM · log(4πM²)

This exactly matches the Sekino-Susskind scrambling time formula. The ringdown is literally the scrambling process — the merger's coherence being thermalized into the final black hole state.

Physical interpretation: The gravitational wave ringdown we hear is the audible signature of information scrambling. The exp(C/Ω) memory kernel determines how quickly the binary's coherence history becomes inaccessible — encoded holographically on the final horizon.

Inspiral → Coherence

As the binary spirals inward, each orbit adds to the system's phase coherence. The frequency chirps up as C(t) approaches the critical threshold Ω⁻¹.

f(τ) ∝ τ^-3/8
A(τ) ∝ τ^-1/4

Merger → Rupture

The merger is a true δ-function: instantaneous, irreversible, marking the ontological moment where two become one. Maximum dC/dt occurs here.

δ(now)
Two → One

Ringdown → Regeneration

The final black hole "regenerates" toward equilibrium. The exp(C/Ω) kernel weights memory access—symmetric mergers ring down faster.

h(t) ∝ e^-t/τcos(ω_qnmt)

Ω-Symmetry in Gravitational Waves

The mass ratio determines the system's effective symmetry, which in turn sets Ω—the precision parameter governing how sharply the system accesses its coherence history.

Equal Mass (q = 1)

Ω = 1/π

Z₂ symmetry: half-cycle (π radians) to rupture
CV ≈ 0.159 · Fast, clean ringdown

Unequal Mass (q → 0)

Ω → 1/2π

Broken symmetry: full cycle to rupture
Higher Ω · Slower, messier ringdown

Diegetic Sound — Hearing CRR

The sound you hear is a direct sonification of the CRR dynamics. Real gravitational waves are far too low frequency for human hearing (millihertz range), so we shift them up ~10 octaves while preserving the mathematical structure.

SONIFICATION MATHEMATICS

We map gravitational wave frequencies to audio frequencies while preserving the CRR structure:

f_audio = f_base · (60 M☉ / M_total) /* Heavier masses → lower pitch (mass warps time) */

f_base = 80 Hz (inspiral start) → 400 Hz (pre-merger) /* Frequency chirps following τ^-3/8 */

f_QNM = 200 Hz · (60 M☉ / M_final) /* Ringdown frequency ∝ 1/M_final */

τ_decay ∝ Ω⁻¹ · (1 + 0.5(1 - q)) /* Symmetric mergers decay faster (lower Ω) */

Inspiral Sound

Two sine oscillators represent the orbiting masses. Frequency chirps up following f(τ) ∝ τ^-3/8 as coherence accumulates. Slight detuning creates beating — the "wobble" of the binary.

osc1.freq = f₀ → f₀ · (5 + 3q)
osc2.freq = osc1.freq · 1.01
gain ∝ τ^-1/4

Merger Sound

The δ-rupture manifests as noise burst + low "thump" — the irreversible crack of spacetime. Pitch drops as energy falls into the newly-formed horizon.

noise: 20ms attack, exp decay
thump: 80 → 20 Hz pitch drop
/* Two → One */

Ringdown Sound

The final black hole rings at its quasi-normal mode frequency. Exponential decay follows exp(−t/τ) where τ depends on Ω — symmetric mergers decay faster.

f = f_QNM (triangle wave)
amp = exp(-t / τ_damp)
τ_damp = Ω⁻¹ · geometric

TRY THIS — HEAR THE Ω DIFFERENCE

Adjust the masses and listen to how the sound changes:

• Equal masses (30+30 M☉): Pure chirp, clean ringdown, higher pitch — Ω = 1/π, maximum symmetry
• Unequal masses (30+10 M☉): More complex chirp, longer ringdown — Ω increases, symmetry broken
• Heavy total mass (80+80 M☉): Lower pitch throughout — mass warps time, f ∝ 1/M
• Light total mass (10+10 M☉): Higher pitch, faster evolution — less mass, less time dilation

Quick Reference — CRR Gravitational Wave Formulae

CORE CRR EQUATIONS

                        C(t) = ∫ L(τ) dτ

                        δ(now)

                        R(t) = ∫ φ exp(C/Ω) Θ dτ

                        Ω = 1/φeff

MASS PARAMETERS

                        q = m₂/m₁ ∈ [0,1]

                        η = m₁m₂/(m₁+m₂)² ∈ [0,0.25]

                        Mc = (m₁m₂)3/5/(m₁+m₂)1/5

                        Mfinal ≈ M(1 - εrad)

Ω-SYMMETRY VALUES

                        Z₂ (q=1): Ω = 1/π ≈ 0.318

                        CVZ₂ = Ω/2 ≈ 0.159

                        Broken: Ω → 1/2π ≈ 0.159

                        φeff = π(2 - 4η)

NUMERICAL RELATIVITY FITS

                        εrad ≈ 0.056η + 0.58η²

                        χfinal ≈ 2√3 η (non-spin)

                        Healy et al. (2014)

QNM FREQUENCIES (l=m=2)

                        MωR = 1.53 - 1.16(1-χ)0.13

                        M/τ = 0.09 + 0.34(1-χ)0.50

                        fQNM ≈ 32 kHz × (M☉/Mf)

                        Berti et al. (2006)

ISCO & INSPIRAL

                        fISCO ≈ 4400 Hz × (M☉/M)

                        f(τ) ∝ τ-3/8

                        h(τ) ∝ Mc5/4 τ-1/4

                        Post-Newtonian (3.5PN)

The Central Insight:
Gravitational wave ringdown is the audible signature of information scrambling. The CRR memory kernel exp(C/Ω) determines how the binary's coherence history becomes encoded on the final horizon — scrambled in time t_scr = Ω⁻¹ log(C), exactly matching the Sekino-Susskind formula.

PHYSICS REFERENCES

Post-Newtonian theory: Blanchet, Living Reviews in Relativity (2014)
Final state fits: Healy et al., PRD 90, 104004 (2014)
QNM frequencies: Berti, Cardoso & Will, PRD 73, 064030 (2006)
GW150914 detection: Abbott et al. (LIGO), PRL 116, 061102 (2016)