Energy, Momentum & Matter
If space is woven from activity, then energy might just be how much of that activity flows through a slice of it — and matter, a knot in the weave that refuses to dissolve.
In Curvature & Gravity we let the hypergraph be space and watched it bend. But Einstein’s picture has two halves: geometry on one side, and energy, momentum and matter on the other — the stuff that tells spacetime how to bend in the first place. So if the Wolfram model is going to be about physics and not just geometry, it owes us an answer to a blunt question: where, in a churning web of relations, is the energy?
Wolfram’s answer is disarmingly simple to state, and it’s worth being upfront that it is a proposal, not a proof. The idea: energy isn’t a substance sitting somewhere in the hypergraph. It’s a measure of how much is happening — how much rewriting activity flows through a given region. More activity, more energy. Matter, in the same spirit, would be a pattern of activity that holds together and persists instead of washing away.
Read this page as a sketch of an identification Wolfram says he only fairly recently arrived at — appealing, suggestive, and not yet shown to reduce to the energy and momentum of textbook physics with the right units. Nothing here is derived; it’s a definition being floated.
Point at the causal edges
The cleanest place to make this concrete isn’t the hypergraph of space — it’s the causal graph, the web of “what had to happen before what.” Each edge in that graph is one strand of dependency between events. Here is the same causal graph we’ve been using:
Try this: step to the end, then picture slicing the graph the way we did for foliations — sweep a “moment of now” through it. Now count the causal edges your slice cuts across. Wolfram’s proposal is that that count — the flux of causal edges through the slice — is what we should call energy and momentum, depending on how the slice is oriented. [conjecture]
The two ingredients you already have — causal edges, and slices through them — are exactly the two ingredients the identification needs. That’s the whole appeal: no new machinery, just a new reading of structure that was already there.
The proposed identification
Here it is, as plainly as it can be put. In the foliation picture a slice can be spacelike (a “moment of now,” cutting across the flow of time) or timelike (running along the flow). Wolfram proposes:
- Energy the flux of causal edges through spacelike hypersurfaces. [setup]
- Momentum the flux of causal edges through timelike hypersurfaces. [setup]
In words: energy counts the causal connections threading through a slice of space at an instant; momentum counts the causal connections threading through a slice oriented along the direction of motion through time. Loosely, then, energy is just “the amount of activity in space” — how much rewriting is going on in a region. [conjecture] (2020 announcement)
That loose phrasing is doing real work, and it’s also where the honesty has to come in: counting edges gives you a number, but a number is not yet a physical energy. Nobody has shown that this flux, in some continuum limit, turns into the energy that appears in the conservation laws — measured in joules, with the right factors of . It is a candidate definition awaiting that demonstration.
Spacelike vs timelike flux — what's actually being counted
In relativity a hypersurface is a slice through spacetime, and its orientation matters. A spacelike slice is a snapshot of all space at one moment; a timelike slice is a “wall” extended through time. The flux through a surface is, roughly, how many things pass through it.
The Wolfram-model translation: the causal graph stands in for spacetime, its edges stand in for the causal/light-cone structure, and a foliation supplies the slices. Counting how many causal edges pierce a spacelike slice gives the proposed energy; counting how many pierce a timelike slice gives the proposed momentum. Because energy and momentum are then the same kind of thing (edge-flux) seen through differently oriented slices, they would naturally combine the way the relativistic energy–momentum vector does — which is the structural reason the identification is tempting in the first place. [conjecture]
Two cautions kept deliberately:
- This is a proposed identification, framed by Wolfram as a recent realization. It is not shown to reduce to standard energy/momentum, and no units are pinned down. [setup]
- The relativistic scaffolding it leans on — that these slices behave like genuine spacelike/timelike hypersurfaces at all — is itself a model-internal claim resting on causal invariance, treated formally (and far more carefully than in the blog essays) in Gorard’s relativity paper.
Grounded in Wolfram’s 2020 announcement; relativistic framing per Gorard.
Matter as a persistent knot
If everything is made of the same hypergraph, then a particle can’t be a different kind of thing — there’s nothing else for it to be made of. So the model posits matter as a persistent local feature of the hypergraph: a tangle or pocket of structure that, as the rule fires over and over, keeps reconstituting itself and moves around rather than dissolving back into the background. A particle would be a knot in the weave that refuses to come undone. [conjecture] (2020 announcement)
This dovetails with the energy story. A region where such a knot sits is a region of extra, sustained activity — extra causal-edge flux — which is exactly what the proposal calls energy, and (via the curvature picture) exactly what would make that region curve space and so gravitate. It’s a pleasingly unified mental image: matter is concentrated activity, activity is energy, energy curves space. [conjecture]
The crucial caveat: this is a picture, not a result. No specific particle has been identified in the hypergraph — not the electron, not anything. By Wolfram’s own account a year on, “we haven’t ‘found the electron’ yet,” only a claim of getting “closer.” [conjecture] (2021 update) There is, as yet, no rule for which knots are stable, no calculation of a mass, no spectrum of particles falling out of the model.
What is not claimed here — especially
It would be easy to read “energy is activity, mass is a persistent knot of activity” and leap to “so the model derives .” It does not. The famous relation is not derived anywhere in these sources — neither the equivalence itself, nor the factor of , nor a route from edge-counting to a rest mass. [conjecture] (2020 announcement)
This is the same restraint we owe the rest of the project. Reviewing it in Scientific American, Scott Aaronson warned that the framework is flexible enough to accommodate almost any result after the fact, and Daniel Harlow judged the claimed successes “at best, qualitative.” [setup] (Scientific American critique) An identification as suggestive — and as unconstrained — as “energy = activity” is precisely the kind of move that warning is about. The responsible summary: Wolfram proposes that energy and momentum are causal-edge fluxes through spacelike and timelike slices, and that matter is persistent hypergraph structure — proposals that are not yet shown to reproduce the energy, momentum, mass, or of established physics.
That closes out Part II, where space, time, gravity and (tentatively) their sources all emerged from one rewriting rule. Part III turns to the quantum side of the story — starting with Branchial Space, the geometry hiding inside the multiway system that Wolfram proposes as the home of quantum states and entanglement.
Next: Branchial Space (coming soon, Part III).
Sources for this page: 2020 announcement · 2021 update · Gorard — relativity & gravity · Scientific American critique