The Structure and Pathologies of Spacetime

In our models, space emerges as the large-scale limit of our spatial hypergraph, while spacetime effectively emerges as the large-scale limit of the causal graph that represents causal relationships between updating events in the spatial hypergraph. An important result is that (subject to various assumptions) there is a continuum limit in which the emergent spacetime follows Einstein’s equations from general relativity.

And given this, it is natural to ask what happens in our models with some of the notable phenomena from general relativity, such as black holes, event horizons and spacetime singularities. I already discussed this to some extent in my technical introduction to our models. My purpose here is to go further, both in more completely understanding the correspondence with general relativity, and in seeing what additional or different phenomena arise in our models.

It should be said at the outset that even after more than 100 years the whole issue of what weird features of spacetime Einstein’s equations can imply remains a rather confusing subject, that is still very far from completely understood. And in some ways I think our models may help clarify what’s going on. Yes, there’s complexity in taking large-scale limits in our models. But unlike with the Einstein equations, which effectively just tell one to “find a spacetime that satisfies certain constraints defined by the equations”, our models give a direct computational way to determine the structure that is supposed to limit to spacetime.

In what follows, we’ll see that our models can reproduce known implications of Einstein’s equations such as black holes, event horizons and spacetime singularities. But they also suggest the possibility of other, in some ways yet more exotic, spacetime phenomena. Some of these may in fact be things that could be found in the Einstein equations; others require descriptions of spacetime more general than the mathematical structure of general relativity can provide (notably in connection with dimension and topology change).

There’s long been a view that at small enough length scales the Einstein equations will have to be supplemented by some lower-level description of spacetime, presumably connected to quantum mechanics. Our models immediately provide a lower-level representation for spacetime in terms of the discrete structure of the spatial hypergraph and the causal graph—with familiar, continuum spacetime emerging as a large-scale limit, much like continuum fluid behavior emerges as a large-scale limit of molecular dynamics.

And there is already a lot that can be said about the structure and behavior of spacetime in our models just on the basis of the spatial hypergraph and the causal graph. But the models also have the feature that they inexorably involve quantum mechanics as a result of the multiway systems obtained by following different possible sequences of updating events. And this means that we can expect to see how various extreme features of spacetime play out at a quantum level, in particular through the multiway causal graph, which includes not only connections associated with ordinary spacetime, but also with branchtime and the branchial space of quantum entanglements.

I won’t go far in this particular bulletin into the exploration of things like the quantum mechanics of black holes, but we’ll get at least an idea of how these kinds of things can be investigated in our models. In addition to seeing the correspondence with what are now standard questions in mathematical physics, we’ll begin to see some indications of more exotic ideas, like the possibility that in our models particles like electrons may correspond to a kind of generalization of black holes.

What General Relativity Says

Perhaps the most famous prediction of general relativity is the existence of black holes. At a mathematical level, a black hole is characterized by the presence of an event horizon that prevents events occurring inside it from affecting ones outside (though events occurring outside can affect what’s inside).

So how does this actually occur in general relativity? Let’s talk about the simplest nontrivial case: the Schwarzschild solution to Einstein’s equations. Physically, the Schwarzschild solution gives the gravitational field outside a spherically symmetric mass. But there’s a wild thing that happens if the mass is high enough and localized enough: one gets a black hole—with an event horizon. The physical story one can tell is that the escape velocity is larger than the speed of light, so nothing can escape. The mathematical story is more subtle and complicated, and took many years to untangle. But in the end there’s a clear description of the event horizon in terms of the causal structure of spacetime: of what events can causally affect what other ones.

But what about the actual Schwarzschild solution? An important simplifying feature is that it’s a solution to Einstein’s equation in the vacuum, not really in any place where there’s actually mass present. Yes, there’s mass that produces the gravitational field. But the Schwarzschild solution just describes the form of the gravitational field outside of the mass. What about in the black hole case?

In a physical black hole created by a collapsing star, there are presumably remnants of the star inside the event horizon. But that’s not what the Schwarzschild solution describes: it just says there’s vacuum everywhere, with one exception—that at the very center of the black hole there’s some kind of singularity. What can one say about this singularity? The equations imply that right at the singularity the curvature of spacetime is infinite. But there’s also something else. Once any geodesic (corresponding, for example, to the world line of a photon) goes inside the event horizon, it’ll only continue for a limited time, always in effect ending up “at the singularity”. There’s no necessary connection between this kind of geodesic incompleteness and infinite curvature; but in a Schwarzschild black hole both occur.

What is really “at” the center of the Schwarzschild black hole? Is it a place where Einstein’s equations don’t apply? Is it even part of the manifold that corresponds to spacetime? The mathematical structure of general relativity says one starts by setting up a manifold, then one puts a metric on it which satisfies Einstein’s equations. There’s no way the dynamics of the system can change the fundamental structure of the manifold. Yes, its curvature can change, but its dimension cannot, and nor can its topology. So if one wants to say that the point at the center of a Schwarzschild black hole “isn’t part of the manifold”, that’s something one has to put in from the beginning, from outside the Einstein equations.

There’s a certain amount that has been done to classify the kinds of singularities that can occur in general relativity. The singularity in a Schwarzschild black hole is a so-called spacelike one—that essentially “cuts off time” for any geodesic, anywhere in space inside the event horizon.

Discovered 50 years after the Schwarzschild solution is the Kerr solution, representing a rotating black hole. Its structure is much wilder than the Schwarzschild solution. A notable feature is the presence of a timelike singularity—in which time can progress, but space is effectively cut off. (Later, we’ll see that spacelike and timelike singularities have quite simple interpretations in our models.) If the angular momentum is below a critical value, there’s an event horizon (or, actually, two) in a Kerr black hole. But above the critical value, there’s no longer an event horizon, and the singularity is “naked”, and potentially able to affect the rest of the universe.

Like the Schwarzschild solution, the Kerr solution is a mathematical construct that defines a static configuration that a gravitational field can have according to Einstein’s equations when no matter is present (and when the underlying manifold has holes cut out for singularities). There has been a long-running debate about what features of such solutions would survive in more realistic situations, such as matter collapsing to form a black hole.

An early conjecture was the weak cosmic censorship hypothesis which stated that in any realistic situation any singularity must be “hidden” behind an event horizon. This meant for example that Kerr-like black holes formed in practice must have angular momenta below the critical value. Numerical simulations did seem to support that supercritical Kerr-like black holes couldn’t form, but a class of constructions were nevertheless found that could in a certain limit generate at least infinitesimal naked singularities.

Ordinary general relativity is firmly rooted to (3+1)-dimensional spacetime. But it can be generalized to higher dimensions. And in this case one finds new and more complex black-hole-like phenomena—and it seems to be easier for naked singularities at least theoretically to arise.

There are lots of other results from general relativity. One example is Penrose’s singularity theorem, which (with various assumptions) implies that (so long as there is nothing with negative mass AKA gravitational repulsion) as soon as geodesics are trapped within a region (e.g. by an event horizon) they must eventually all converge, and thus form a singularity. (The Hawking singularity theorem is basically a time-reversed version, which implies that there must be a singularity at the beginning of the universe.)

Another idea from general relativity is the “no-hair theorem” (or conjecture), which states that, at least from the outside, a limited number of parameters (such as mass and angular momentum) completely characterize any black hole—at least if it’s static. In other words, black holes are in a sense “perfectly smooth” objects; there aren’t gravitational effects outside the event horizon that reveal “inner complexity”. (Note that this is a classical conjecture; it’s quite likely not to be true when quantum effects are included.)

The Einstein equations are nonlinear partial differential equations, and it’s interesting to compare them with other such equations, such as the Navier–Stokes equations for fluid flow. One feature of the Navier–Stokes equations is that for sufficiently high fluid velocities (larger than the speed of sound) shocks develop that involve discontinuities that cannot be directly described by the equations. And particularly when one gets to hypersonic flow what physically happens is that the approximation that the fluid is continuous—and can be modeled by a partial differential equation—breaks down, and the specific dynamics of the molecules in the fluid start to matter.

Does something similar happen in the Einstein equations? It’s not known whether there can be shocks per se. But it seems likely that to understand things like what appear to be singularities one will have to go “underneath” the Einstein equations—as our models do.

What else can one learn from the fluid dynamics analogy? A notable feature of fluid flow is the phenomenon of fluid turbulence, in which random patterns of flow are ubiquitous. It’s still not clear to what extent turbulence is a true feature of the Navier–Stokes equations, and to what extent it’s associated with sensitive dependence on molecular-scale details. I strongly suspect, though, that like in rule 30 (or in the digits of π) the primary effect is an intrinsic generation of randomness, associated with the phenomenon of computational irreducibility. I’d be amazed if something similar doesn’t happen in Einstein’s equations, but it’ll no doubt be mathematically difficult to establish. Plenty of “randomness” is seen in numerical simulations, but absent a precise underlying computational model it’s very difficult to know if the “randomness” is a true feature of the equations that one is trying to approximate, or is just a feature of the approximation scheme used.

In our models, computational irreducibility and intrinsic randomness generation are ubiquitous. But exactly how such “microscopic” randomness will scale up to “gravitational turbulence” isn’t clear.

One further point, applicable to both fluid flow and gravitation, has to do with computational capability. The Principle of Computational Equivalence strongly suggests that in both cases, sophisticated computation will be ubiquitous. One consequence of this will be computation universality. And in fact it seems likely that this will already be manifest even in such simple cases as the interactions of a few fluid vortices, or just three point masses. The result is that highly complex behavior will be possible.

But that doesn’t mean that—for example in the spacetime case—one can’t summarize certain features of the behavior in simple terms, say by describing overall causal structure, or identifying the presence of an event horizon. However, the presence of underlying computational sophistication does mean that there may be computational limitations in coming up with such summaries. For example, to determine whether a certain system has a certain global causal structure valid for all time may in effect require determining the infinite-time behavior of an irreducible computation, which cannot in general be done by a finite computation, and so must be considered formally undecidable.

Identifying Causal Structure

An event horizon is a feature of the global causal structure of a spacetime—a boundary where events “inside” it can’t affect events (or “observers”) “outside” it. In traditional general relativity, it can be mathematically complicated to establish where there’s an event horizon; it effectively requires proving a theorem about how all possible geodesics will be trapped. (In numerical relativity, one can try to just identify “apparent horizons” where geodesics at least seem to be going in directions where they’ll be trapped, even if one can’t be sure what will happen later.)

But in our models, everything is considerably more explicit. We’re dealing with discrete updating events, occurring at discrete points. And at least in principle we can construct causal graphs that fully describe the causal relationships between all possible events that can occur in spacetime.

Here’s an example of such a causal graph:

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Remember that the only thing that’s ultimately defined by this causal graph is the causal dependency of events—or in effect the partial ordering of events. Time emerges as an overall inexorable progression of events. Space is associated with the relationship of events in slices defined by foliations of the causal graph.

Any given event can affect any event which it can reach by following directed edges in the causal graph, or, in the language of physics, any event which is in its future light cone. But what happens in the infinite limit of the future light cone of a given event? It could be that that infinite limit effectively covers the whole (“spatial”) universe. Or it could be that it only reaches part of the universe. And in that latter case we can expect that there’ll be an event horizon: a boundary to where the effects of our event can reach.

But how can we map which events have how large an effect—or in a sense what the causal structure of spacetime is? Let’s define what we can call a “causal connection graph” that shows the relationships between the future light cones of events. Given two events, it could be that the limits of their future light cones are exactly the same (both covering the whole universe, or both covering the same part of it). In that case, let’s consider these events “causally equivalent”.

In the causal connection graph, the nodes are collections of causally equivalent events. The edges in the graph are determined by the relationships between the future light cones of these collections of events. Consider two collections A and B. If the limit of the future light cone of A contains the limit of the future light cone of B, then we draw a directed edge in the graph from A to B. And if the limit of the future light cone of A intersects the limit of the future light cone of B (but doesn’t contain it), we draw an undirected edge.

There’ll be a causal connection graph defined for events on any spacelike hypersurface that can be specified as a slice through the causal graph. To find the ultimate causal connection graph, we’d then have to look at the infinite limits of future light cones from events in this spacelike hypersurface. But in practice, we can at least try to approximate this by looking not at the infinite limit of the future light cones, but instead at their limit on some spacelike hypersurface “far enough” in the future. (By the way, this is similar to issues that arise in thinking about “absolute” vs. “apparent” horizons in general relativity.)

OK, so let’s look at the causal graph we had above, and consider four events starting at the top:

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The future light cones for events 1 and 2 contain the whole causal graph. But the future light cones for events 3 and 4 are respectively:

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These future light cones are distinct. So if we construct the causal connection graph for the spacelike hypersurface containing events 3 and 4, it’ll just consist of two separated nodes. In effect the universe here has broken into two non-communicating subuniverses. In the language of general relativity, we can say that there’s a cosmological event horizon that separates these two subuniverses, and no information can go in either direction between them.

The signature of this in the causal connection graph is quite straightforward. Looking at the first few levels in the causal graph, one has:

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Computing the causal connection graph for successive levels, one gets:

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At the first and second levels, no “event horizon” has yet formed, and there’s just a single set of causally equivalent events. But at level 3, there start to be two separate sets of causally equivalent events; an “event horizon” has formed.

So what does the spatial hypergraph “underneath” these look like? Here’s how it evolves (using our standard updating order) for the first few steps:

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It’s very simple compared to the spatial hypergraph for any “real universe”. But it’s still useful as an example for understanding ideas about causal structure. And what we see is that in this tiny “toy universe”, two “lobes” develop, which evolve like two separate subuniverses, with no causal interdependence.

But while there’s no causal interdependence between the “lobes”, the spatial hypergraph is still connected. In our models, however, there’s nothing to say that the spatial hypergraph can’t actually become disconnected—and here’s an example where it does:

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What’s the significance of this? If the spatial hypergraph stays connected, then even if two regions are at some point causally disconnected, it still remains possible for them to reconnect (in effect revealing that one did not have a true event horizon). But if the hypergraph is actually disconnected, no such reconnection is ever possible (unless one has a fundamentally nonlocal rule whose left-hand side is itself a disconnected hypergraph, so that it can effectively pick up elements “anywhere in any universe”, regardless of whether there are any existing relations between the elements).

In traditional general relativity, it is certainly possible to have multiple universes, each with their own disconnected manifold representing space. But while the Einstein equations support the formation of event horizons and causal disconnection, they do not support “forming a new universe” with its own topologically distinct disconnected manifold. To get that requires something like our model, with its much greater flexibility in the fundamental description of space. (In the usual mathematical analysis of general relativity, one defines up front the manifold on which one’s operating, though it’s perhaps conceivable that at least in some limit a metric could develop that’s compatible only with a different topology.)

Note that at the level of the causal connection graph, one just sees progressive formation of subuniverses, with cosmological event horizons; there’s no direct trace of disconnection in the spatial hypergraph:

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Black-Hole-Like Event Horizons

The type of causal disconnection that we’ve discussed so far is bidirectional: information can’t propagate in either direction across the event horizon. But for black holes, the story is different, basically with effects able to flow into the black hole, but not out.

Here’s a very simple example of something like this in our models:

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In the “universe at large”, represented rather trivially here by the outermost vertical edges, information can propagate back and forth. But if it goes into the “black hole” at the center, it never comes out again.

Here’s the rule that’s being used

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and here’s what’s going on in the spatial hypergraph:

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It’s a bit easier to see what’s happening if we look at every single event in the spatial hypergraph:

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And basically what we see is that events propagate causal effects in both directions on the “ring” that represents the “universe at large”, but any causal effect on the “prong” only goes in one direction, making it correspond to a very simple analog of a black hole.

It is important to note that the black hole “doesn’t form” immediately. The “universe” has to “expand” for a couple of steps before one can distinguish a “black hole” event horizon.

Given the causal graph

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one can construct a causal connection graph. Looking at this on successive steps one gets:

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For the first three steps, there’s only one set of causally equivalent events. But by step 4, a second set forms. And there’s now a one-way causal connection between the first set of causally equivalent events and the second: in other words, a “black-hole-like” event horizon has formed.

Let’s look at a slightly more complicated case, though still with a very simple rule:

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The causal graph in this case is:

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Looking at the causal connection graph for successive steps, we see that after a little while a “black hole event horizon” forms:

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If we look at forward light cones of different events, we see that some “fill the whole universe”, but others stay localized to the region on the left. One can think of the first set as being the light cones of events that are in the “universe at large”; the second set are the light cones of events that are “trapped inside a black hole”:

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Events trapped inside the black hole cannot affect events outside. But events outside can affect events inside, with events on the boundary in effect representing the process of things falling into the black hole. Note that since energy (and mass) are measured by the flux of causal edges through spacelike hypersurfaces, the black hole shown here effectively gains mass through events that deliver things into the black hole. (Our complete universe is also expanding, so there’s no overall energy conservation to be seen here.)

What does the underlying spatial hypergraph look like in this case? Once again, it’s quite simple:

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So where is the black hole? We can number the events in the causal graph:

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Then we can match these up with events in the spatial hypergraph:

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The basic conclusion is that the “main circle” acts like a black hole, with the smaller “handle” being like the “rest of the universe”.

Here’s another, slightly less trivial example:

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The causal graph in this case is:

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Some initial events have light cones which cover the whole space; others always avoid the “spine” on the left, which behaves like a black hole:

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Here’s the underlying spatial hypergraph:

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Comparing the list of events

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with the annotated causal graph

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we see that again the “black hole” is associated with a definite part of the spatial hypergraph.

Spacelike and Timelike Singularities

Disconnection is one kind of “pathology” that can occur in our models. Another is termination: a configuration can be reached in which the rules no longer apply, and “time stops”. (In the language of term rewriting systems, one can say in such a case that a “normal form” has been reached.)

Here is an example of the spatial hypergraph for a rule in which this happens:

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The corresponding causal graph is:

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And given this causal graph, we can see that the future light cone of any event winds up “compressing down” to just the single final event—after which “time stops”.

Is there an analog of this in ordinary general relativity? Yes, it’s just a spacelike singularity—like what occurs in the Schwarzschild solution. In the Schwarzschild solution the spacelike singularity is in the future of events inside the event horizon; here it’s in the future of the “whole universe”. But it’s exactly the same idea. The future of anything that happens—or any geodesic—inevitably winds up in the “dead end” of the causal graph, at which “time stops”.

Here’s a slightly more complicated example that again involves termination, but now with two distinct “final events”:

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Like in a cosmological event horizon, different collections of events in effect lead to different branches of future light cones—though in this case whichever branch is followed, “time eventually stops”.

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The causal connection graphs in this case basically just indicate the formation of a cosmological event horizon:

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But now each branch does not lead to a “full subuniverse”; instead they lead to two subuniverses that each end up in a spacelike singularity at which time stops.

By the way, consider a “typical” causal graph like:

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Starting from any event in this causal graph, its future light cone will presumably eventually “cover the whole universe”. But what about its past light cone? Wherever we start, the past light cone will eventually “compress down” to the single initialization event at the top of our causal graph. In other words—just like in standard general relativity—the beginning of the universe also behaves like a spacelike singularity.

What about timelike singularities? What is their analog in our models? Remember that a spacelike singularity effectively corresponds to “time stopping”. Well, a timelike singularity presumably corresponds to “space stopping”, but time continuing. Here’s an example:

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In this example, the future light cones of all events concentrate down onto two separated “tracks”, in each of which there is an inexorable progression from one event to the next, with “no room for any space”.

Here’s a minimal version of the same kind of behavior, for the rule:

{{x, x}} → {{x, x}, {x, y}}

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Unsurprisingly, its spatial hypergraph does not succeed in “growing space”:

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If one starts looking at other rules, one quickly sees all sorts of strange behavior. The almost trivial unary rule

{{x}} → {{x}, {x}}

effectively gives a whole tree of timelike singularities:

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Here are a few other ways that collections of timelike singularities can be produced:

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Notice that for example in the last case only some events don’t lead to timelike singularities. A bit like in a Kerr black hole, it’s both possible to avoid the singularity, and be trapped in it.

But let’s go back to spacelike singularities. We’ve seen examples where “time stops” for the whole universe. But what about for just some events? Here’s a simple example:

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For most events, the future light cone continues forever. But that’s not true for the highlighted events. Each of these events is a “dead end”, for which “time stops”.

In this case, the dead ends are associated with spatial disconnections, but this does not need to be true:

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Here is a slightly more complicated case:

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Again, though, these are “single-event” dead ends, in the sense that after those particular events, time stops. But predecessors of these events still “have a choice”: their future light cones include both the spacelike singularity, and other parts of the causal graph.

It is perfectly possible to have a combination of spacelike and timelike singularities. On the left here is a series of spacelike singularities, while on the right are timelike singularities, effectively all separated by cosmological event horizons:

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Recognizing when there’s actually a genuine spacelike singularity can be difficult. Consider the case:

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At first it seems as if some of the events in the last few steps we have generated have no successors. But continuing for a few more steps, we find out that these ones do—though now there are new events that seem not to:

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Plotting the event numbers for “potential spacelike singularities” against the number of steps of evolution generated, we can see that most events only remain “candidates” for a few steps:

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In the limit of arbitrarily many steps, will any spacelike-singularity events survive? It doesn’t look likely, but it is hard to tell for sure. And this is exactly one of those cases where the general problem is likely to be undecidable: there is no finite computation which can guarantee to give the result.

One feature of all these examples is that they all involve spacelike singularities that affect just single events. At the beginning we also saw examples where “all the events in the universe” eventually wind up at a spacelike singularity. But what about a case that’s more analogous to a Schwarzschild black hole, where a collection of events inside an event horizon wind up at a spacelike singularity, but ones “outside” do not?

I haven’t explicitly found an example where this happens, but I’m confident that one exists, perhaps with slightly more complicated initial conditions than I’ve been using here.

In ordinary general relativity, Penrose’s singularity theorem implies that as soon as there is a trapped surface from which geodesics cannot escape, the geodesics will inevitably reach a singularity. This theorem is known to apply in any number of dimensions (with codimension-2 trapped hypersurfaces). And in our models, it might make one think that as soon as there is a black-hole-like event horizon, there must be a singularity of the kind we have discussed.

But what about a case like the following one we discussed above?

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There is a “trapped region” in the causal graph, but no sign of a singularity. There seem to be a couple of (not-mutually-exclusive) possibilities for what is going on here. The first is that there is in a sense so much “overall expansion in the universe” and lack of energy conservation (reflected in an increasing flux of causal edges) that the energy conditions in the theorem effectively do not apply. A second possibility is that what we’re seeing is “another universe” forming inside the event horizon. In the Kerr solution, the (probably unrealistic and fragile) mathematics implies that it is in principle possible to “go through” the singularity and emerge into “another universe” that is just as infinite as ours. Perhaps what we are seeing here is a similar phenomenon.

The Distribution of Causal Structures

What kinds of causal structures are possible in our systems? To start answering this, we can explicitly look at all 4702 possible 2₂ → 3₂ rules. We have to pick how many steps after the “beginning of the universe” we want to consider; we’ll start with 5. We also have to pick how many steps we want to use to check causal connectivity; we’ll start with 10. Then here are the possible causal connection graphs that occur, together with how many times they occur:

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Nearly 40% of these rules give causal graphs that terminate before 10 steps, corresponding to “universe-scale” spacelike singularities. Of the remainder, 76% produce no event horizons (at least at the steps we’re measuring). Then the next most common behavior is to generate “subuniverses” separated by cosmological event horizons. And then there are black holes. In aggregate these are produced in about 3% of cases. The most common is a single black hole, followed by a black hole combined with a subuniverse, two black holes, etc.

There are also more exotic cases, like the last case shown. The rule here is:

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And here is the causal graph:

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This is how the causal connection graph develops at successive steps:

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It takes a few steps for “black holes to form”. But in the end, we get what we can interpret as a nested set of black holes. The causally equivalent events corresponding to the node at the top have future light cones that eventually cover the universe. But events associated with “lower” nodes yield progressively more localized future light cones:

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Here’s what’s going on in the spatial hypergraph:

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Once again, it’s at first pretty obscure where the “black holes” are. It’s mildly helpful to see individual events. What seems to be happening is that the “exterior of the universe” is the main loop. But somehow information can get propagated into the “long hair”, but then can’t get out again.

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What is the analog of all this in ordinary continuum general relativity? Presumably it’s some kind of nested black hole phenomenon. Inside the event horizon of a black hole other black holes form. Perhaps all these black holes in some sense eventually merge (as the singularity theorem might seem to suggest), or perhaps there’s some structure with multiple levels of event horizons, as roughly happens in the Kerr solution.

Properties of Black Holes

When there’s a black hole in our models, what can one tell about what’s inside it? Causal edges go into the event horizon, but none come out. However, that does not mean that the structure of the spatial hypergraph doesn’t reflect what’s inside the event horizon. After all, every causal edge that crosses the event horizon originated in an event outside the event horizon. And that event represented some update in the spatial hypergraph. Or, in other words, while causal edges are “lost” into the event horizon, the “memory” of what they did is progressively imprinted in the spatial hypergraph outside the event horizon.

In ordinary general relativity, there are “no-hair” theorems that say that the gravitational effects of a black hole depend only on a few parameters, such as its overall mass and overall angular momentum. In our models, the overall mass is essentially just determined by the number of causal edges that end up crossing the event horizon. (Angular momentum is related to a kind of vorticity in the causal graph.) So the no-hair theorem for mass says that when there is an event horizon, none of the details of these causal edges matter; only their total number. It’s not clear why this would be true, but it seems conceivable that it could be an essentially purely graph theoretic result.

There’s another potential effect, however. Consider the following causal graph:

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The edges highlighted in red correspond in effect to the interior of an event horizon. The edges highlighted in purple form a pair. The one on the left will cross the event horizon, and “fall into the black hole”. The one on the right will escape to affect the rest of the universe.

Later on, we will see how similar phenomena occur in the multiway causal graph, and we will discuss how this relates to quantum phenomena such as Hawking radiation. But for now, here, everything we’re discussing is purely classical. But recall that while the flux of causal edges through spacelike surfaces corresponds to energy, the flux of causal edges through timelike surfaces corresponds to momentum. On average, the net momentum associated with a pair of causal edges should always cancel out. But if one of the edges crosses inside an event horizon, then whatever momentum it carries will just be aggregated into the total momentum of what’s inside the event horizon, effectively leaving the momentum associated with the other edge uncanceled.

What does this mean? It needs a more detailed analysis. But there seems to be some reason to think that these uncanceled causal edges should lead to a net momentum flux away from the black hole—a kind of “black hole wind”. The effect will be small, because it’ll essentially be proportional to the elementary length divided by the black hole radius (or, alternatively, the black hole surface area times the elementary length divided by the black hole volume). But conceivably in some circumstances—notably with small black holes—it could have measurable consequences.

By the way, it’s worth understanding the “physical origin” of this “wind”. Essentially it’s the result of a classical analog of vacuum polarization. Whereas in ordinary general relativity spacetime is a continuum, in our models it consists of discrete components, and this discreteness causes there to be “fluctuations” that are sensitive to the presence of the event horizon.

More Complicated Cases

If one starts looking at causal graphs for our models, there are plenty of complex things one sees, even with very simple underlying rules. Here are a few examples with various forms of overall causal structure:

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Other Exotic Phenomena in Spacetime, etc.?

Event horizons and singularities can be surprising and confusing in the Einstein equations, but the mathematical structure of the equations at least in principle allows them to be described there. But might there be other kinds of exotic phenomena in spacetime that just can’t be described using this mathematical structure? Our models definitely suggest so.

One example we already encountered is disconnection in the spatial hypergraph. Another major class of phenomena involve what amounts to dynamical change of dimension. In the timelike singularities we saw above, parts of the spatial hypergraph effectively degenerate to being zero-dimensional.

But in general, the whole or part of the spatial hypergraph can change its effective dimension. I’ve discussed elsewhere the possibility that the early universe might have had a larger effective dimension than the current universe. But what would happen if there was just a region of higher dimension in our current universe?

It’s not clear how to make a traditional continuum mathematical description of this, though presumably if one tried to describe it in terms of a manifold, it would show up as a region with infinite curvature. But given a finite—but perhaps very large—hypergraph, it’s much more straightforward to see how regions of different (approximate) dimension might exist together. Inevitably, though, there will be lots of intricate issues to unravel in seeing exactly how all the appropriate mathematical limits fit together.

Nevertheless, one can at least get some sense of what “dimension anomalies” might be like. A region of higher effective dimension will—by definition—have a denser pattern of connections than the surrounding parts of the spatial hypergraph. It’s not clear what the boundary of the structure will be like. But conceivably, to be stable, it will have to correspond to an event horizon of some sort.

One thing that’s fairly clear is that a region of higher dimension will tend to involve more causal edges than the surrounding hypergraph, and will therefore behave like a region of higher energy, so that it’ll show up as a tiny region of space in which energy (or mass) is concentrated.

But of course we’re already very familiar with one example of tiny regions of space in which energy (or mass) are concentrated: elementary particles, like electrons. I’ve always assumed that particles in our models are some kind of stable localized structures in the spatial hypergraph. Perhaps their stability has an essentially topological or graph theoretical origin. But perhaps instead it’s more connected to some kind of event-horizon-like phenomenon—that would be visible in the causal graph.

But what’s “inside” such a particle? It could be a collection of elements of the spatial hypergraph that are connected to form some higher (or effectively infinite) dimensional space. It could also simply be that at scales involving fairly few elements in the hypergraph, there are only a discrete set of possible forms for the local region of the hypergraph that support something like an event horizon.

As an example of the kinds of localization one can see in a causal graph, here are the forms of forward light cones one gets starting from different events in a particular causal graph:

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It’s not clear what the best signature to use in searching for particles in our models will be. But it seems like an intriguing—and appealing—possibility that in some sense particles like electrons are “generalized black holes”, or in effect that the apparent “perfection” of black holes is also manifest in the “perfection” of the smallest distinct constituents of matter.

(This reminds me of a personal anecdote. Back around 1975, when I was about 15 years old, I attended an evening physics talk in Oxford about black holes and their quantum features. After the talk, I went up to the speaker and asked if perhaps electrons could be black holes. “Absolutely not”, he said, rather dismissively. Well, we’ll see…)

What about other kinds of “anomalies in spacetime”? In our models, the structure of space is maintained by continual activity in the spatial hypergraph. Presumably in most situations such activity will on a large scale reach a certain statistically uniform “equilibrium” state. But it is conceivable that different regions of space could show different overall levels of activity—which would lead to different “vacuum energy densities”, effectively corresponding to different values of the cosmological constant.

It is also conceivable that there could be different domains in space, all with the same overall activity level (and thus the same energy density) but with different configurations of some large-scale (perhaps effectively topological) feature. And in such a case—as in many cosmological models—one can expect things like domain walls.

In ordinary general relativity, the basic topology of space remains fixed over the course of time. In our models, it can dynamically change, with the structure of space potentially being “re-knitted” in different topological configurations. I haven’t explicitly found examples of rules that generate something like a wormhole, but I’m sure they exist.

I mentioned above the possibility of small regions of space having different effective dimensions. It’s also possible that there could be extended structures, such as tubes, with different effective dimensions. And this raises the intriguing possibility of what one might call “space tunnels” in which there is a tube of “higher-dimensional space” connecting two points in the spatial hypergraph.

No doubt there are many possibilities, and many kinds of exotic phenomena that can occur. Some may be visible directly in the large-scale structure of the spatial hypergraph, while others may be more obvious in the causal graph, or the causal connection graph. It is interesting to imagine classifying different possibilities, although it seems almost inevitable that there will ultimately be undecidable aspects to essentially any classification.

Some exotic phenomena may rely on the discrete structure of our models, and not have meaningful continuum limits. But my guess is that there will be plenty of exotic phenomena that can occur even in continuum models of spacetime, but which just haven’t been looked for before.

Cauchy Surfaces, Closed Timelike Curves, etc.

The questions of causal structure that we’re considering here are already quite complicated. But there are even more issues to consider. One of them is the question of whether it is possible to define a definite notion of “progression of time” in spacetime. The causal graph in effect specifies the partial ordering of events in a system. But now we can ask whether there is a valid foliation that respects this partial ordering, in the sense that all events in a given slice are strictly “before” those in subsequent slices. In the continuum limit, this is effectively asking whether the dynamics of our system exhibit strong hyperbolicity.

One case where this will definitely not be seen is when there are closed timelike curves, represented by loops in the causal graph. But there are also plenty of other cases where there is no strict way to make a “thickness-1” foliation in which no event occurring within a given slice has a successor in the same slice, as in:

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When it comes to taking a continuum limit, one should presumably “thicken” the foliations first. But there will inevitably still be cases where no limited amount of thickening will suffice to allow foliations to be created. And such cases one will presumably correspond to failures of Cauchy development in the continuum limit. And it seems likely that this phenomenon can be fairly directly connected to the singularities we have discussed above, but exactly how is not clear.

The Quantum Case

Everything I’ve said so far has involved “purely classical” evolution of the spatial hypergraph, and the causal graph that represents causal relationships within this evolution. But we can also consider multiway evolution and the multiway causal graph which in our models represent quantum behavior. So what happens to various forms of causal disconnection in this case?

The basic answer is that things rapidly become very complicated. Consider the very simple “cosmological horizon” causal graph:

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Here is the corresponding multiway system:

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And here is the multiway causal graph:

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And here is its transitive reduction:

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This multiway causal graph includes both spacelike causal edges, of the kind shown in the ordinary causal graph, and branchlike causal edges, which represent causal relationships between different branches in the multiway graph. If one looked only at spacelike edges, one would see the spacetime event horizon. But branchlike edges can in effect connect across the event horizon.

Another way to say this is that quantum entanglements can span the event horizon. And if we look at the branchial graph associated with the multiway system above, we see that all states are entangled, even across the event horizon:

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There’s lots to understand here. And it’s all quite complicated. But let’s look at a simpler case:

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The ordinary spacetime causal graph in this case is:

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The underlying spatial hypergraphs are

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and the multiway graph is just

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but the multiway causal graph is

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although its transitive reduction is just:

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As another example, consider

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whose ordinary causal graph is:

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The multiway causal graph is:

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But once again the transitive reduction of this graph is just:

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As a marginally more realistic example, consider the minimal “black-hole-like” case from above:

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The ordinary causal graph in this case is:

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The multiway causal graph in this case is:

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The transitive reduction of this is:

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Meanwhile, the branchial graph has the structure:

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And once again, even though in the spacetime causal graph there’s a (rather minimal) black-hole-like event horizon, things are considerably more complicated in the multiway (i.e. quantum) case, notably with quantum entanglements in the branchial graph apparently spanning the event horizon.

There is much more to study here. But it’s already clear that there are some complicated relationships between ordinary causal connectivity, and multiway (i.e. quantum) causal connectivity. Just as in ordinary causal graphs there can be event horizons that limit the spatial effects of events, so in multiway causal graphs there can be event horizons that limit the branchial effects of events. Or, said another way, there should be “entanglement event horizons” that limit the causal relationships between quantum states.

One way to view causal effects in spacetime is to say that a given event can affect events in its future light cone. But in the space of quantum states the causal effect of an event is also limited, but now to an entanglement cone rather than a light cone. And just as we looked at causal connection graphs on the basis of light cones, we can do the same thing for entanglement cones.

The result will hopefully be an elucidation of the quantum character of event horizons. But that will have to await another bulletin…