General Relativity & Gravity

Put special relativity and hypergraph curvature side by side and a tantalising road opens up — one that is argued to lead all the way to Einstein's equations.

This page is a meeting point. Two threads we’ve been pulling on separately turn out to be the two halves of the same picture.

From the causal graph we got Special Relativity: slice the web of events into “moments of now,” and the freedom in how you slice becomes relative motion, time dilation, and a maximum signal speed. From the hypergraph-as-space we got Curvature & Gravity: space bends, and that bending shows up as a tiny correction in how fast a ball fills out as it grows.

General relativity is what you get when you insist on both at once — a single curved spacetime in which the bending of space and the relativity of time are inseparable. The claim of the Wolfram model is that the same hypergraph that gave us each piece separately also stitches them together, with Einstein’s equations emerging as a large-scale, statistically-averaged feature of the rewriting — not as an assumption baked in from the start. [derived-in-model, partial]

Recall where curvature lives

Curvature isn’t painted onto the hypergraph; it’s read off from how quickly a neighbourhood grows. Here is the triangle rule again — every edge closes into a triangle, so neighbourhoods fill in fast and the structure curls into a denser web. That “fills up faster than flat space would” is, in miniature, what positive curvature means.

Rule: {x, y} → {x, y}, {y, z}, {z, x} — each edge becomes a triangle with a new node

gen 0 / 5nodes 2edges 1

Try this: play it to the end and watch how quickly the region around any one node packs in new neighbours. Now imagine reading off the order- $r^2$ shortfall between that count and a flat $r^d$ — that number is the model’s stand-in for the Ricci scalar, the very curvature quantity that sits inside Einstein’s equations. [setup] The full counting argument is on the Curvature & Gravity page.

Special relativity + curvature = the road toward GR

The logic of the synthesis is short. Special relativity already follows, within the model, from causal invariance plus the freedom to foliate the causal graph in different ways. [derived-in-model] Curvature already follows from the order- $r^2$ correction to ball growth. [derived-in-model] General relativity is the statement that matter and energy bend spacetime, and that bending is gravity — so the model’s route to GR is to show that its curvature, averaged over large regions, obeys the same equations Einstein wrote down, while respecting the relativistic structure the causal graph supplies.

Wolfram’s own summary is carefully hedged: “in various limits, and subject to various assumptions, our models do indeed reproduce Einstein’s equations.” [derived-in-model, partial] (2020 announcement) The honest, peer-reviewed treatment of how the discrete causal structure is argued to approach Lorentzian spacetime is Gorard’s relativity paper — the source to trust over the blog essays.

What is actually claimed — and what isn’t

This is the place to be most careful, because it is exactly the kind of result that is easy to overstate.

The concrete claim is for the vacuum Einstein equations — empty space, no matter term — recovered in certain limits, subject to assumptions. [derived-in-model, partial]
The full equations with matter are described as promised, not shown; that case is an aspiration. [conjecture]
A fully rigorous derivation is explicitly acknowledged as unfinished, even for the vacuum case.

So the responsible one-line summary is: the model is claimed to recover the vacuum Einstein equations in certain limits, with the rigour openly incomplete — not that “Wolfram derived general relativity.” The matter side of the theory is taken up next, in Energy, Momentum & Matter.

The precise version (and exactly what is and isn't claimed)

The synthesis rests on two model-internal results, each developed elsewhere:

Relativistic structure. For a causally invariant rule, causality-respecting foliations of the causal graph behave like inertial reference frames; boosting corresponds to tilting the slicing, and special relativity — time dilation, a maximum signal speed — is argued to follow. [derived-in-model] (See Time, Foliations & Reference Frames.)
Curvature. The volume of a small geodesic ball expands as
$V(r) \;=\; c_d\, r^{d}\left(1 \;-\; \frac{R}{6(d+2)}\,r^{2} \;+\; \cdots\right)$
and the model estimates the same order- $r^2$ shortfall combinatorially, identifying it with the Ricci scalar $R$ . [derived-in-model] (See Curvature & Gravity.)

Putting the two together, Wolfram argues that the large-scale, statistically-averaged behaviour of these hypergraphs reproduces Einstein’s equations “in various limits, and subject to various assumptions.” [derived-in-model, partial] The qualifications stressed in the source itself are not decoration:

The demonstrated case is the vacuum equations; the with-matter equations are promised, not derived. [conjecture]
The rigorous derivation is openly unfinished.

Grounded in Wolfram’s 2020 announcement; the formal, peer-reviewed grounding for the relativistic and gravitational properties — the continuum limit toward Lorentzian spacetime, and the role of causal invariance — is Gorard’s relativity paper.

A note of caution

It is worth holding two things at once. The ingredients are real: the counting definition of curvature is genuine differential geometry, and the causal-graph route to special relativity is a clean idea. But the leap from those ingredients to “our models reproduce Einstein’s equations” is precisely the limit- and assumption-laden claim that skeptics have flagged. Reviewing the project in Scientific American, Scott Aaronson argued the framework is flexible enough to accommodate almost any result after the fact, and Daniel Harlow called the claimed successes “at best, qualitative.” [setup] The project also bypassed normal peer review while drawing heavy media attention. (Scientific American critique)

That is why this page says claimed in certain limits and points you to Gorard’s formal paper rather than asserting a finished derivation. The road from special relativity plus curvature toward general relativity is real and worth walking — but it is a road, not a destination already reached.

Previous: Special Relativity · Next: Energy, Momentum & Matter

Sources for this page: 2020 announcement · Gorard — relativity & gravity · Scientific American critique

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