Finite Coupled Monad Field Dynamics: A Geometry‑Free Substrate Model of Gravitational Relaxation
Finite Coupled Monad Field Dynamics
Abstract
We present a numerical implementation of Finite Coupled Monad Field Dynamics (FCMFD), a model in which gravitational‑wave‑like signals arise from a non‑abelian matrix network with fractional memory and amplitude saturation. No background spacetime geometry is assumed. The system is defined by a set of Hermitian operators Mi(t) coupled via an adjacency matrix Wij. Stress propagates through commutator terms [Mi, Mj], and saturation (“phase locking”) is enforced by spectral truncation. Fractional time evolution is approximated using Grünwald–Letnikov weights.
We simulate three configurations: a single saturated node, a binary inspiral with time‑varying coupling, and the same inspiral with stochastic noise added to the saturated core. From each we extract an effective stretched‑exponent parameter βeff by passing the network’s scalar observable through a phenomenological magnetar‑lag mapping and fitting the tail of the resulting rate to A·exp(−(t/τ)β).
The recovered values are:
- Single saturated core: βeff ≈ 0.29
- Binary inspiral (global mean eigenvalue): βeff ≈ 0.21
- Binary inspiral with stochastic core noise: βeff ≈ 0.19
- Binary inspiral using commutator norm: βeff ≈ 0.30
As a control, the same pipeline applied to real LIGO data from GW150914 yields βeff = 1.00, consistent with exponential ringdown predicted by General Relativity. The contrast shows that the fractional‑memory model produces distinct signatures that, while not present in this event, could in principle be tested against future observations.
1. Introduction
General Relativity describes gravity as curvature of spacetime. Its predictions have been confirmed to high precision, including the ringdown of black hole mergers. However, the theory contains singularities and assumes a geometric background. Finite Coupled Monad Field Dynamics (FCMFD) explores an alternative: a discrete, algebraically defined substrate with finite capacity and memory. In this approach, spacetime is not fundamental; rather, effective gravitational phenomena emerge from the dynamics of a network of matrix operators.
This paper presents a numerical implementation of FCMFD. The goal is to determine whether such a system can produce relaxation signatures reminiscent of stretched exponentials and chirp‑like waveforms without invoking curvature. We compare the model’s output to a phenomenological magnetar‑timing pipeline and to a control analysis of real LIGO data.
2. Model Definition
2.1 State representation
We consider N nodes (Monads). Each node i at time t is represented by a Hermitian 2×2 matrix Mi(t):
Mi = ai·I + bi·σx + ci·σy + di·σz
where σx, σy, σz are Pauli matrices and ai, bi, ci, di ∈ ℝ. The eigenvalues of Mi are interpreted as local “field strength” or stress.
2.2 Topology and coupling
A symmetric adjacency matrix Wij defines which nodes interact. For our simulations we use a ring topology: Wi,i±1=0.2 (background) plus a time‑varying binary coupling between nodes 0 and 1 to simulate an inspiral.
2.3 Fractional time evolution
We approximate a Caputo fractional derivative of order α (0<α≤1) using the Grünwald–Letnikov definition with alternating signs:
w0 = 1, wj = −wj−1·(α − j + 1)/j (j ≥ 1)
The update for node i at time step n is:
Mi(n) = − Σj=1n wj·Mi(n−j) + (Δt)α·Fi(n−1)
where Fi contains the local dynamics (see below). For α = 1 the method reduces to ordinary Euler integration.
2.4 Dynamical terms
The force term Fi consists of three contributions:
- Commutator torque from neighbours:
Ti = Σj Wij [Mi, Mj] - Nonlinear damping (cubic, to ensure boundedness):
Di = −γ·Mi3 - External driving Ji(t) (used only for the binary inspiral, applied symmetrically to nodes 0 and 1).
Thus Fi = Ti + Di + Ji.
2.5 Saturation (phase locking)
After each update, we compute the eigenvalues of Mi. If any eigenvalue exceeds a prescribed ceiling Smax in absolute value, we scale the whole matrix so that the maximum eigenvalue becomes exactly Smax. This ensures that node states cannot exceed the finite capacity of the substrate.
2.6 Observational mapping
To connect the matrix state to an observable akin to a pulsar timing residual, we define a scalar Seff(t). For single‑node studies we take the largest eigenvalue of that node. For binary inspiral we examine either the mean of the largest eigenvalues over all nodes (“global strain”) or the Frobenius norm of the commutator between the two binary cores:
Seff(t) = ||[M0, M1]||F = √( Tr( [M0,M1]† [M0,M1] ) )
This scalar is then fed into a phenomenological model of a magnetar glitch:
dθeff/dt = Ω / (1 + k·Seff(t)), lag(t) = (θeff(t) − Ω t)/(2π), rate(t) = |d(lag)/dt|.
We take the peak of the rate, discard the first 0.5 s thereafter to avoid the initial transient, and fit the remaining tail to a stretched exponential:
rate(t) = A·exp(−(t/τ)βeff).
The fitted exponent βeff is our diagnostic.
3. Numerical Simulations
All simulations use N=4 nodes, Δt = 0.002 s, duration 5–15 s, Smax=2.0, γ=0.4–0.5, and fractional order α=0.35 (chosen to match earlier scalar studies). The binary inspiral uses a time‑dependent coupling W01(t) driven by an accelerating phase Φ(t) = 2π( fstart t + (framp/3) t3 ).
3.1 Single saturated core (v6.3)
Node 0 is initialised with eigenvalues near 1.4; other nodes are near zero. The system evolves and the largest eigenvalue of node 0 is used as Seff. The resulting βeff is 0.2924 (mean of several runs). This serves as the baseline for a phase‑locked relaxation.
3.2 Binary inspiral – global strain (v6.5)
Nodes 0 and 1 are driven with the chirp coupling. Seff is the mean of the largest eigenvalues over all four nodes. The lag‑rate spectrogram shows a rising frequency ridge. The tail‑fit gives βeff = 0.2062.
3.3 Binary inspiral with stochastic core noise (v6.6, v6.7)
At each time step, independent Gaussian noise with standard deviation σ=0.03 is added to the eigenvalues of node 0 before spectral truncation. This simulates unresolved boundary fluctuations of a saturated core. The global strain yields βeff = 0.1883.
3.4 Binary inspiral – commutator norm (v6.8)
We use the same binary drive but set Seff to the Frobenius norm of [M0, M1]. This observable preserves phase information. The fitted βeff = 0.3048, which is close to the single‑core value.
3.5 Control: real LIGO data (GW150914)
We downloaded the whitened, bandpassed strain of GW150914 from the GWOSC archive. The same tail‑fitting pipeline was applied to the strain envelope (after peak alignment). The recovered exponent is βeff = 1.000 with a relaxation time τ ≈ 42 ms, consistent with the exponential ringdown predicted by General Relativity.
4. Results
| Configuration | Observable | βeff |
|---|---|---|
| Single core (v6.3) | Node 0 eigenvalue | 0.292 |
| Binary inspiral (v6.5) | Mean eigenvalue | 0.206 |
| Binary + stochastic core (v6.6) | Mean eigenvalue | 0.188 |
| Binary commutator (v6.8) | ||[M0,M1]|| | 0.305 |
| GW150914 (control) | Strain envelope | 1.000 |
All FCMFD simulations produce βeff significantly less than 1, indicating decay slower than exponential (stretched). The binary commutator case recovers a value nearly identical to the single‑core case, suggesting that the intrinsic memory kernel (α=0.35) dominates the direct torque interaction. The global strain, which averages over peripheral nodes, gives a lower exponent – a network‑level emergent effect. Stochastic core noise further lowers the exponent, implying that boundary fluctuations can enhance apparent memory.
The LIGO control yields β=1, the signature of a pure exponential decay. This matches the GR prediction for the fundamental quasinormal mode of a Kerr black hole. The pipeline therefore distinguishes between memory‑less exponential relaxation and fractional‑memory stretched relaxation.
5. Discussion
5.1 Geometry‑free dynamics
The model contains no spatial coordinates, no metric, and no partial differential equations. Propagation of stress occurs via the commutator coupling across the network. The time‑varying adjacency matrix encodes an inspiral without requiring a background manifold. Hence, the observed chirp‑like spectrogram arises purely from algebraic relations.
5.2 Comparison with General Relativity
| Feature | General Relativity | FCMFD (this work) |
|---|---|---|
| Foundational object | Spacetime metric | Matrix operator network |
| Singularities | Present | Removed by saturation |
| Gravitational wave | Metric perturbation | Commutator torque wave |
| Speed of light | Postulated constant | Emergent maximal commutator rate |
| Black hole horizon | Null surface | Phase‑locked core boundary |
| Ringdown tail | Exponential (β=1) | Stretched (β∼0.2‑0.3) for low‑stress regimes |
The key observational difference is the value of βeff in the late‑time tail. The FCMFD simulations produce β<1, while the only tested real event (GW150914) gives β=1. This suggests that if future gravitational wave events exhibit a stretched‑exponential tail, they would favour a substrate‑like memory model; otherwise, GR remains consistent.
5.3 Limitations
- The number of nodes (N=4) is small; larger networks may show different scaling.
- The fractional discretisation is heuristic; rigorous convergence has not been established.
- The magnetar‑lag mapping is phenomenological; a first‑principles derivation from the substrate equations is absent.
- Stochastic noise is added ad‑hoc; a physical model of boundary fluctuations is not derived.
- Only one real event (GW150914) has been tested; more events are needed.
6. Conclusion
We have implemented a coordinate‑free, algebraic substrate model based on a matrix network with fractional memory and amplitude saturation. The model produces stretched‑exponential relaxation (βeff ≈ 0.2–0.3) for single‑core and binary inspiral configurations. A control analysis of real LIGO data from GW150914 yields the exponential (β=1) signature of General Relativity. The framework is falsifiable: detection of a stretched‑exponential tail in a gravitational wave event would challenge the GR ringdown picture and support a finite‑capacity memory substrate.
Further work should include larger networks, more realistic driving terms, a systematic study of parameter sensitivity, and application to a broader set of observational data.
Acknowledgments
The numerical code and analysis were developed through iterative collaborative testing. All results are reproducible from the archived scripts (FCMFD v6.8). LIGO data were obtained from GWOSC.
References
- LIGO Scientific Collaboration, “Observation of Gravitational Waves from a Binary Black Hole Merger”, Phys. Rev. Lett. 116, 061102 (2016).
- Grünwald, A. “Über begrenzte Derivationen und deren Anwendung”, Z. Angew. Math. Phys. 12, 441 (1967).
- Oldham, K.B. and Spanier, J. “The Fractional Calculus”, Academic Press (1974).
- Hawking, S.W. “Particle Creation by Black Holes”, Commun. Math. Phys. 43, 199 (1975) – for the conceptual analogy of boundary radiation, not a derivation used here.
