Time, Foliations & Reference Frames
There's no master clock in the universe — only the order the causal graph forces; slice that order into "moments of now" and you've chosen a reference frame.
In Curvature & Gravity we treated the hypergraph as space and watched it bend. Now we ask the other question: where does time come from, if there’s no clock built into the model?
The answer is already sitting in the causal graph. Time isn’t a separate ingredient — it’s just the ordering forced by the arrows. When one event consumes what another produced, the first had to come before the second. That “had to come before” is the only sense in which anything happens “earlier” or “later.” [setup] No external stopwatch, no background timeline — just dependency.
Slicing the causal graph into “now”
Here’s the same causal graph from before. Notice again that events sitting side by side, with no arrow between them, are causally independent — nothing in the system says which one happened first.
Try this: step to the end and picture sweeping a line down through the graph, top to bottom. Each position of that line is a “moment of now” — everything above it has happened, everything below hasn’t yet. Because the side-by-side events have no arrows between them, you have real freedom in how you draw that line.
A complete sweep like this — a stack of “moments of now” that together cover the whole graph — is called a foliation. (“Foliation” as in foliage: you’re slicing the graph into leaves.) The only rule is that the slicing must respect causality: a slice can’t put an effect into an earlier moment than its cause. [setup]
A foliation is a reference frame
This is the move that turns a discrete bookkeeping graph into physics. Each valid foliation — each consistent way of stacking “nows” — is identified with a reference frame, the viewpoint of one particular observer. [setup] The sequence of slices is that observer’s experience of time passing.
And here’s the seed of relativity. Because independent events can be sliced in more than one order, two observers using different foliations can disagree about which distant events happened “at the same time” — and, within the model, neither is wrong. There’s no privileged slicing that the universe prefers. Tilt the slices relative to one another and that tilt corresponds to relative motion between the two frames. [setup→derived-in-model]
If that sounds like the relativity of simultaneity, that’s exactly the point — and it’s where the next note, Special Relativity, picks up (including an interactive foliation you can tilt yourself).
The precise version (and the catch)
A foliation is a sequence of slices through the causal graph — discrete analogues of spacelike hypersurfaces — chosen so that no slice ever separates a cause from its effect in the wrong order. Each such foliation is identified with an inertial reference frame; advancing from one slice to the next is the passage of time in that frame. Boosting to a frame in relative motion corresponds to tilting the slicing. [setup→derived-in-model]
The catch worth keeping honest:
- This only yields consistent, observer-independent physics if the rule is causally invariant — every updating order produces the same causal graph (see the causal graphs note). Causal invariance is what lets different observers slice differently yet still agree on the underlying web of cause and effect. The project ties it to Lorentz covariance. [setup]
- Calling these slices “spacelike hypersurfaces” and deriving special relativity — time dilation, a maximum signal speed — from foliation geometry is a stronger, model-internal claim. It is argued to hold “for any rule that has causal invariance,” [derived-in-model] but treat that as a claim within the model, not as established physics.
The formal, peer-reviewed grounding for all of this — the continuum limit in which the discrete causal structure is argued to approach Lorentzian spacetime — is in Gorard’s relativity paper, which is the source to trust over the popular essays.
Sources for this page: Gorard — relativity · 2020 announcement · Technical paper