EDCA Gravity: How an Energy-Driven Cellular Automaton Naturally Produces Newtonian Gravity and a GR-like Geometry

Author: Raul Sanchez Perez · January 2026

1) Why talk about gravity in a cellular automaton?

Gravity in physics has two famous descriptions: Newtonian gravity (a force) and General Relativity (curved geometry). EDCA (Energy Driven Cellular Automaton) is not designed as a physics theory, yet two of its foundational properties naturally generate a gravity-like behavior:

Energy-gated dynamics: a site transitions only when energy arrives as a spot.
Dynamic space instantiation: the set of spatial sites is not fully instantiated; a cell site is allocated when needed.

Once those two principles exist, a feedback loop appears: spots drive activity → activity allocates space → allocated space is preferred → future spot collapses bias toward it. That feedback produces attraction, an emergent potential, and a geometric analog.

2) The foundational EDCA principle: spots gate all transitions

In EDCA, readiness alone does not cause a state update. A site may be “ready” for birth or death, but:

A ready-for-transition site performs its transition only if it is covered by an energy spot.

Spots are mobile energy quanta with polarity: positive (+) or negative (−). Spots propagate as wavefronts, collapse onto candidate sites, and can trigger birth/death transitions depending on polarity.

3) The spot cycle (intuitive summary)

Wavefront expansion: the spot expands outward like a wavefront.
Candidate set: when the wavefront touches ready sites, those sites become candidates.
Collapse: if candidates exist, the wavefront collapses onto one site via an incidence rule.
Polarity-conditioned transition:
- positive spot + birth readiness → birth
- negative spot + death readiness → death
- polarity mismatch → no transition
Polarity flip: after a successful collapse/transition, polarity flips before the next expansion cycle.

4) Dynamic space: “cell site is allocated when needed”

EDCA can operate without fully instantiating the lattice. Instead, cell sites are allocated/instantiated only when needed, for example:

to evaluate readiness in a region,
to represent wavefront coverage,
to support a transition at a site.

Important terminology note: “allocated site” means a computational location exists. This is not the same as a “living cell” (occupied/active matter state).

5) Allocation density ρ(x,t): the memory of cumulative activity

Because sites are allocated only when needed, a coarse-grained field arises: allocation density ρ(x,t), measuring how instantiated a region of space is.

Regions repeatedly traversed and evaluated by spots become more allocated. Therefore ρ(x,t) becomes a record of cumulative activity / energy usage in a region.

6) The key EDCA preference: high allocation density is favored

EDCA’s collapse incidence includes an allocation-density preference: wρ(x,t) = ρ(x,t)^γ. High allocation density regions are preferred because they require fewer resources (already-instantiated sites).

This yields a direct analogy: the system is biased toward regions where space is already “thicker” (more allocated), which creates a drift toward those regions.

7) Emergent potential and “gravity field”

Because preference includes ρ^γ, we define an effective potential:

    Φ(x,t) = −γ ln ρ(x,t)
  

The drift field that pulls dynamics toward high allocation density is:

    g(x,t) = −∇Φ(x,t)
  

8) Newtonian limit: recovering the inverse-square law

If allocation density is close to a background value and responds smoothly to activity, one obtains a Poisson-like equation:

    ∇²Φ = 4π GEDCA ε(x)
  

For a point activity source ε(x)=M δ(x), the solution is:

    Φ(r) = −GEDCA M / r   →   g(r) = −GEDCA M / r²
  

This reproduces the full Newtonian picture inside EDCA’s weak-field limit.

9) Geometry analog (GR intuition)

Newton treats gravity as a force. GR treats gravity as geometry. In EDCA, allocation density defines how “thick” the instantiated substrate is. A simple geometric analog is a conformal metric based on ρ(x,t), meaning distances and propagation are rescaled by allocation density.

10) Full tensor analogy (intuitive explanation)

GR uses a tensor equation: curvature = coupling × stress-energy. EDCA admits the same structure if we define an EDCA stress-energy tensor from spot events:

T₀₀ = ε: local activity/energy density (collapse + transitions + wavefront traversal)
T_0i = J_i: directional spot flux (momentum-like)
T_ij = Π_ij: stress/anisotropy (pressure/shear analog from propagation directionality)

The full EDCA–GR analog equation becomes: G_μν[g(ρ)] = κ_EDCA T_μν^EDCA.

11) Why conservation matters: the invariant 2σ_t + E_t

EDCA has a conserved quantity of the form: 2σ_t + E_t = constant, tying matter count (σ) and algebraic energy (E). This invariant provides conservation grounding for treating ε(x,t) as the curvature source (analog of conserved mass-energy in physics).

Interactive Toy Example: Allocation Density → Emergent “Gravity”

The following interactive demo illustrates the core EDCA feedback loop: collapse prefers high ρ → high ρ grows → collapse becomes even more concentrated.

Interactive Toy Demo

This demo runs on GitHub Pages. Click below to open it in a new tab.

Open Interactive Demo

Try this: increase γ to strengthen attraction toward high ρ. Increase λ to enforce more local collapse behavior. Switch to “Show Φ” to see how the potential emerges from ρ.

Reference

Foundational EDCA Paper: Energy Driven Cellular Automaton (EDCA): Fundaments, EDCAWorld, January 2026.

Click to access fundaments.pdf

Gravity and GR analogy formally: EDCA Gravitation and Curved Geometry: Newtonian and Tensorial GR Analogs from Energy-Spot-Driven Cellular Automaton Dynamics, EDCAWorld, January 2026.

Click to access gravity-expanded-1.pdf