What Is Spacetime, Really?

Space network

A hundred years ago today Albert Einstein published his General Theory of Relativity—a brilliant, elegant theory that has survived a century, and provides the only successful way we have of describing spacetime.

There are plenty of theoretical indications, though, that General Relativity isn’t the end of the story of spacetime. And in fact, much as I like General Relativity as an abstract theory, I’ve come to suspect it may actually have led us on a century-long detour in understanding the true nature of space and time.

I’ve been thinking about the physics of space and time for a little more than 40 years now. At the beginning, as a young theoretical physicist, I mostly just assumed Einstein’s whole mathematical setup of Special and General Relativity—and got on with my work in quantum field theory, cosmology, etc. on that basis.

But about 35 years ago, partly inspired by my experiences in creating technology, I began to think more deeply about fundamental issues in theoretical science—and started on my long journey to go beyond traditional mathematical equations and instead use computation and programs as basic models in science. Quite soon I made the basic discovery that even very simple programs can show immensely complex behavior—and over the years I discovered that all sorts of systems could finally be understood in terms of these kinds of programs.

Encouraged by this success, I then began to wonder if perhaps the things I’d found might be relevant to that ultimate of scientific questions: the fundamental theory of physics.

At first, it didn’t seem too promising, not least because the models that I’d particularly been studying (cellular automata) seemed to work in a way that was completely inconsistent with what I knew from physics. But sometime in 1988—around the time the first version of Mathematica was released—I began to realize that if I changed my basic way of thinking about space and time then I might actually be able to get somewhere.

A Simple Ultimate Theory?

In the abstract it’s far from obvious that there should be a simple, ultimate theory of our universe. Indeed, the history of physics so far might make us doubtful—because it seems as if whenever we learn more, things just get more complicated, at least in terms of the mathematical structures they involve. But—as noted, for example, by early theologians—one very obvious feature of our universe is that there is order in it. The particles in the universe don’t just all do their own thing; they follow a definite set of common laws.

But just how simple might the ultimate theory for the universe be? Let’s say we could represent it as a program, say in the Wolfram Language. How long would the program be? Would it be as long as the human genome, or as the code for an operating system? Or would it be much, much smaller?

Before my work on the computational universe of simple programs, I would have assumed that if there’s a program for the universe it must be at least somewhat complicated. But what I discovered is that in the computational universe even extremely simple programs can actually show behavior as complex as anything (a fact embodied in my general Principle of Computational Equivalence). So then the question arises: could one of these simple programs in the computational universe actually be the program for our physical universe?

Cellular automata evolution from simple rules

The Data Structure of the Universe

But what would such a program be like? One thing is clear: if the program is really going to be extremely simple, it’ll be too small to explicitly encode obvious features of our actual universe, like particle masses, or gauge symmetries, or even the number of dimensions of space. Somehow all these things have to emerge from something much lower level and more fundamental.

So if the behavior of the universe is determined by a simple program, what’s the basic “data structure” on which this program operates? At first, I’d assumed that it must be something simple for us to describe, like the lattice of cells that exists in a cellular automaton. But even though such a structure works well for models of many things, it seems at best incredibly implausible as a fundamental model of physics. Yes, one can find rules that give behavior which on a large scale doesn’t show obvious signs of the lattice. But if there’s really going to be a simple model of physics, it seems wrong that such a rigid structure for space should be burned in, while every other feature of physics just emerges.

So what’s the alternative? One needs something in a sense “underneath” space: something from which space as we know it can emerge. And one needs an underlying data structure that’s as flexible as possible. I thought about this for years, and looked at all sorts of computational and mathematical formalisms. But what I eventually realized was that basically everything I’d looked at could actually be represented in the same way: as a network.

A network—or graph—just consists of a bunch of nodes, joined by connections. And all that’s intrinsically defined in the graph is the pattern of these connections.

A graph

Space as a Network

So could this be what space is made of? In traditional physics—and General Relativity—one doesn’t think of space as being “made of” anything. One just thinks of space as a mathematical construct that serves as a kind of backdrop, in which there’s a continuous range of possible positions at which things can be placed.

But do we in fact know that space is continuous like this? In the early days of quantum mechanics, it was actually assumed that space would be quantized like everything else. But it wasn’t clear how this could fit in with Special Relativity, and there was no obvious evidence of discreteness. By the time I started doing physics in the 1970s, nobody really talked about discreteness of space anymore, and it was experimentally known that there wasn’t discreteness down to about 10-18 meters (1/1000 the radius of a proton, or 1 attometer). Forty years—and several tens of billions of dollars’ worth of particle accelerators—later there’s still no discreteness in space that’s been seen, and the limit is about 10-22 meters (or 100 yoctometers).

Still, there’s long been a suspicion that something has to be quantized about space down at the Planck length of about 10-34 meters. But when people have thought about this—and discussed spin networks or loop quantum gravity or whatever—they’ve tended to assume that whatever happens there has to be deeply connected to the formalism of quantum mechanics, and to the notion of quantum amplitudes for things.

But what if space—perhaps at something like the Planck scale—is just a plain old network, with no explicit quantum amplitudes or anything? It doesn’t sound so impressive or mysterious—but it certainly takes a lot less information to specify such a network: you just have to say which nodes are connected to which other ones.

But how could this be what space is made of? First of all, how could the apparent continuity of space on larger scales emerge? Actually, that’s not very difficult: it can just be a consequence of having lots of nodes and connections. It’s a bit like what happens in a fluid, like water. On a small scale, there are a bunch of discrete molecules bouncing around. But the large-scale effect of all these molecules is to produce what seems to us like a continuous fluid.

It so happens that I studied this phenomenon a lot in the mid-1980s—as part of my efforts to understand the origins of apparent randomness in fluid turbulence. And in particular I showed that even when the underlying “molecules” are cells in a simple cellular automaton, it’s possible to get large-scale behavior that exactly follows the standard differential equations of fluid flow.

Cellular automaton fluid

So when I started thinking about the possibility that underneath space there might be a network, I imagined that perhaps the same methods might be used—and that it might actually be possible to derive Einstein’s Equations of General Relativity from something much lower level.

Maybe There’s Nothing But Space

But, OK, if space is a network, what about all the stuff that’s in space? What about all the electrons, and quarks and photons, and so on? In the usual formulation of physics, space is a backdrop, on top of which all the particles, or strings, or whatever, exist. But that gets pretty complicated. And there’s a simpler possibility: maybe in some sense everything in the universe is just “made of space”.

As it happens, in his later years, Einstein was quite enamored of this idea. He thought that perhaps particles, like electrons, could be associated with something like black holes that contain nothing but space. But within the formalism of General Relativity, Einstein could never get this to work, and the idea was largely dropped.

As it happens, nearly 100 years earlier there’d been somewhat similar ideas. That was a time before Special Relativity, when people still thought that space was filled with a fluid-like ether. (Ironically enough, in modern times we’re back to thinking of space as filled with a background Higgs field, vacuum fluctuations in quantum fields, and so on.) Meanwhile, it had been understood that there were different types of discrete atoms, corresponding to the different chemical elements. And so it was suggested (notably by Kelvin) that perhaps these different types of atoms might all be associated with different types of knots in the ether.

Sequence of knots

It was an interesting idea. But it wasn’t right. But in thinking about space as a network, there’s a related idea: maybe particles just correspond to particular structures in the network. Maybe all that has to exist in the universe is the network, and then the matter in the universe just corresponds to particular features of this network. It’s easy to see similar things in cellular automata on a lattice. Even though every cell follows the same simple rules, there are definite structures that exist in the system—and that behave quite like particles, with a whole particle physics of interactions.

Particle in a cellular automata

There’s a whole discussion to be had about how this works in networks. But first, there’s something else that’s very important to talk about: time.

What Is Time?

Back in the 1800s, there was space and there was time. Both were described by coordinates, and in some mathematical formalisms, both appeared in related ways. But there was no notion that space and time were in any sense “the same thing”. But then along came Einstein’s Special Theory of Relativity—and people started talking about “spacetime”, in which space and time are somehow facets of the same thing.

It makes a lot of sense in the formalism of Special Relativity, in which, for example, traveling at a different velocity is like rotating in 4-dimensional spacetime. And for about a century, physics has pretty much just assumed that spacetime is a thing, and that space and time aren’t in any fundamental way different.

So how does that work in the context of a network model of space? It’s certainly possible to construct 4-dimensional networks in which time works just like space. And then one just has to say that the history of the universe corresponds to some particular spacetime network (or family of networks). Which network it is must be determined by some kind of constraint: our universe is the one which has such-and-such a property, or in effect satisfies such-and-such an equation. But this seems very non-constructive: it’s not telling one how the universe behaves, it’s just saying that if the behavior looks like this, then it can be the universe.

And, for example, in thinking about programs, space and time work very differently. In a cellular automaton, for example, the cells are laid out in space, but the behavior of the system occurs in a sequence of steps in time. But here’s the thing: just because the underlying rules treat space and time very differently, it doesn’t mean that on a large scale they can’t effectively behave similarly, just like in current physics.

Evolving the Network

OK, so let’s say that underneath space there’s a network. How does this network evolve? A simple hypothesis is to assume that there’s some kind of local rule, which says, in effect that if you see a piece of network that looks like this, replace it with one that looks like that.

Sample network rules

But now things get a bit complicated. Because there might be lots of places in the network where the rule could apply. So what determines in which order each piece is handled?

In effect, each possible ordering is like a different thread of time. And one could imagine a theory in which all threads are followed—and the universe in effect has many histories.

But that doesn’t need to be how it works. Instead, it’s perfectly possible for there to be just one thread of time—pretty much the way we experience it. And to understand this, we have to do something a bit similar to what Einstein did in formulating Special Relativity: we have to make a more realistic model of what an “observer” can be.

Needless to say, any realistic observer has to exist within our universe. So if the universe is a network, the observer must be just some part of that network. Now think about all those little network updatings that are happening. To “know” that a given update has happened, observers themselves must be updated.

If you trace this all the way through—as I did in my book, A New Kind of Science—you realize that the only thing observers can ever actually observe in the history of the universe is the causal network of what event causes what other event.

And then it turns out that there’s a definite class of underlying rules for which different orderings of underlying updates don’t affect that causal network. They’re what I call “causal invariant” rules.

Causal invariance is an interesting property, with analogs in a variety of computational and mathematical systems—for example in the fact that transformations in algebra can be applied in any order and still give the same final result. But in the context of the universe, its consequence is that it guarantees that there’s only one thread of time in the universe.

Deriving Special Relativity

So what about spacetime and Special Relativity? Here, as I figured out in the mid-1990s, something exciting happens: as soon as there’s causal invariance, it basically follows that there’ll be Special Relativity on a large scale. In other words, even though at the lowest level space and time are completely different kinds of things, on a larger scale they get mixed together in exactly the way prescribed by Special Relativity.

Roughly what happens is that different “reference frames” in Special Relativity—corresponding, for example, to traveling at different velocities—correspond to different detailed sequencings of the low-level updates in the network. But because of causal invariance, the overall behavior associated with these different detailed sequences is the same—so that the system follows the principles of Special Relativity.

At the beginning it might have looked hopeless: how could a network that treats space and time differently end up with Special Relativity? But it works out. And actually, I don’t know of any other model in which one can successfully derive Special Relativity from something lower level; in modern physics it’s always just inserted as a given.

Deriving General Relativity

OK, so one can derive Special Relativity from simple models based on networks. What about General Relativity—which, after all, is what we’re celebrating today? Here the news is very good too: subject to various assumptions, I managed in the late 1990s to derive Einstein’s Equations from the dynamics of networks.

The whole story is somewhat complicated. But here’s roughly how it goes. First, we have to think about how a network actually represents space. Now remember, the network is just a collection of nodes and connections. The nodes don’t say how they’re laid out in one-dimensional, two-dimensional, or any-dimensional space.

It’s easy to see that there are networks that on a large scale seem, say, two-dimensional, or three-dimensional. And actually, there’s a simple test for the effective dimension of a network. Just start from a node, then look at all nodes that are up to r connections away. If the network is behaving like it’s d-dimensional, then the number of nodes in that “ball” will be about rd.

Graphs with different effective dimensions

Here’s where things start to get really interesting. If the network behaves like flat d-dimensional space, then the number of nodes will always be close to rd. But if it behaves like curved space, as in General Relativity, then there’s a correction term, that’s proportional to a mathematical object called the Ricci scalar. And that’s interesting, because the Ricci scalar is precisely something that occurs in Einstein’s Equations.

Graphs with different effective Ricci curvatures

There’s lots of mathematical complexity here. One has to look at shortest paths—or geodesics—in the network. One has to see how to do everything not just in space, but in networks evolving in time. And one has to understand how the large-scale limits of networks work.

In deriving mathematical results, it’s important to be able to take certain kinds of averages. It’s actually very much the same kind of thing needed to derive fluid equations from dynamics of molecules: one needs to be able to assume a certain degree of effective randomness in low-level interactions to justify the taking of averages.

But the good news is that an incredible range of systems, even with extremely simple rules, work a bit like the digits of pi, and generate what seems for all practical purposes random. And the result is that even though the details of a causal network are completely determined once one knows the network one’s starting from, many of these details will appear effectively random.

So here’s the final result. If one assumes effective microscopic randomness, and one assumes that the behavior of the overall system does not lead to a change in overall limiting dimensions, then it follows that the large-scale behavior of the system satisfies Einstein’s Equations!

I think this is pretty exciting. From almost nothing, it’s possible to derive Einstein’s Equations. Which means that these simple networks reproduce the features of gravity that we know in current physics.

There are all sorts of technical things to say, not suitable for this general blog. Quite a few of them I already said long ago in A New Kind of Science—and particularly the notes at the back.

A few things are perhaps worth mentioning here. First, it’s worth noting that my underlying networks not only have no embedding in ordinary space intrinsically defined, but also don’t intrinsically define topological notions like inside and outside. All these things have to emerge.

When it comes to deriving the Einstein Equations, one creates Ricci tensors by looking at geodesics in the network, and looking at the growth rates of balls that start from each point on the geodesic.

The Einstein Equations one gets are the vacuum Einstein Equations. But just like with gravitational waves, one can effectively separate off features of space considered to be associated with “matter”, and then get Einstein’s full Equations, complete with “matter” energy-momentum terms.

As I write this, I realize how easily I still fall into technical “physics speak”. (I think it must be that I learned physics when I was so young…) But suffice it to say that at a high level the exciting thing is that from the simple idea of networks and causal invariant replacement rules, it’s possible to derive the Equations of General Relativity. One puts remarkably little in, yet one gets out that remarkable beacon of 20th-century physics: General Relativity.

Particles, Quantum Mechanics, Etc.

It’s wonderful to be able to derive General Relativity. But that’s not all of physics. Another very important part is quantum mechanics. It’s going to get me too far afield to talk about this in detail here, but presumably particles—like electrons or quarks or Higgs bosons—must exist as certain special regions in the network. In qualitative terms, they might not be that different from Kelvin’s “knots in the ether”.

But then their behavior must follow the rules we know from quantum mechanics—or more particularly, quantum field theory. A key feature of quantum mechanics is that it can be formulated in terms of multiple paths of behavior, each associated with a certain quantum amplitude. I haven’t figured it all out, but there’s definitely a hint of something like this going on when one looks at the evolution of a network with many possible underlying sequences of replacements.

My network-based model doesn’t have official quantum amplitudes in it. It’s more like (but not precisely like) a classical, if effectively probabilistic, model. And for 50 years people have almost universally assumed that there’s a crippling problem with models like that. Because there’s a theorem (Bell’s Theorem) that says that unless there’s instantaneous non-local propagation of information, no such “hidden variables” model can reproduce the quantum mechanical results that are observed experimentally.

But there’s an important footnote. It’s pretty clear what “non-locality” means in ordinary space with a definite dimension. But what about in a network? Here it’s a different story. Because everything is just defined by connections. And even though the network may mostly correspond on a large scale to 3D space, it’s perfectly possible for there to be “threads” that join what would otherwise be quite separated regions. And the tantalizing thing is that there are indications that exactly such threads can be generated by particle-like structures propagating in the network.

Searching for the Universe

OK, so it’s conceivable that some network-based model might be able to reproduce things from current physics. How might we set about finding such a model that actually reproduces our exact universe?

The traditional instinct would be to start from existing physics, and try to reverse engineer rules that could reproduce it. But is that the only way? What about just starting to enumerate possible rules, and seeing if any of them turn out to be our universe?

Before studying the computational universe of simple programs I would have assumed that this would be crazy: that there’s no way the rules for our universe could be simple enough to find by this kind of enumeration. But after seeing what’s out there in the computational universe—and seeing some other examples where amazing things were found just by a search—I’ve changed my mind.

So what happens if one actually starts doing such a search? Here’s the zoo of networks one gets after a fairly small number of steps by using all possible underlying rules of a certain very simple type:

Networks from different evolution rules

Some of these networks very obviously aren’t our universe. They just freeze after a few steps, so time effectively stops. Or they have far too simple a structure for space. Or they effectively have an infinite number of dimensions. Or other pathologies.

But the exciting thing is that remarkably quickly one finds rules that aren’t obviously not our universe. Telling if they actually are our universe is a difficult matter. Because even if one simulates lots of steps, it can be arbitrarily difficult to know whether the behavior they’re showing is what one would expect in the early moments of a universe that follows the laws of physics as we know them.

There are plenty of encouraging features, though. For example, these universes can start from effectively infinite numbers of dimensions, then gradually settle to a finite number of dimensions—potentially removing the need for explicit inflation in the early universe.

And at a higher level, it’s worth remembering that if the models one’s using are simple enough, there’s a big distance between “neighboring models”, so it’s likely one will either reproduce known physics exactly, or be very wide of the mark.

In the end, though, one needs to reproduce not just the rule, but also the initial condition for the universe. But once one has that, one will in principle know the exact evolution of the universe. So does that mean one would immediately be able to figure out everything about the universe? Absolutely not. Because of the phenomenon I call “computational irreducibility”—which implies that even though one may know the rule and initial condition for a system, it can still require an irreducible amount of computational work to trace through every step in the behavior of the system to find out what it does.

Still, the possibility exists that one could just find a simple rule—and initial condition—that one could hold up and say, “This is our universe!” We’d have found our universe in the computational universe of all possible universes.

Of course this would be an exciting day for science.

But it would raise plenty of other questions. Like: why this rule, and not another? And why should our particular universe have a rule that shows up early enough in our list of all possible universes that we could actually find it just by enumeration?

One might think that it’d just be something about us being in this universe, and that causing us to choose an enumeration which makes it come up early. But my current guess is that it’d be something much more bizarre, such as that with respect to observers in a universe, all of a large class of nontrivial possible universe rules are actually equivalent, so one could pick any of them and get the exact same results, just in a different way.

OK, Show Me the Universe

But these are all speculations. And until we actually find a serious candidate rule for our universe, it’s probably not worth discussing these things much.

So, OK. Where are we at with all this right now? Most of what I’ve said here I had actually figured out by around 1999—several years before I finished A New Kind of Science. And though it was described in simple language rather than physics-speak, I managed to cover the highlights of it in Chapter 9 of the book—giving some of the technical details in the notes at the back.

A New Kind of Science, Chapter 9

But after the book was finished in 2002, I started working on the problem of physics again. I found it a bit amusing to say I had a computer in my basement that was searching for the fundamental theory of physics. But that really was what it was doing: enumerating possible rules of certain types, and trying to see if their behavior satisfied certain criteria that could make them plausible as models of physics.

I was pretty organized in what I did, getting intuition from simplified cases, then systematically going through more realistic cases. There were lots of technical issues. Like being able to visualize large evolving sequences of graphs. Or being able to quickly recognize subtle regularities that revealed that something couldn’t be our actual universe.

I accumulated the equivalent of thousands of pages of results, and was gradually beginning to get an understanding of the basic science of what systems based on networks can do.

Notebooks from 2004

In a sense, though, this was always just a hobby, done alongside my “day job” of leading our company and its technology development. And there was another “distraction”. For many years I had been interested in the problem of computational knowledge, and in building an engine that could comprehensively embody it. And as a result of my work on A New Kind of Science, I became convinced that this might be actually be possible—and that this might be the right decade to do it.

By 2005 it was clear that it was indeed possible, and so I decided to devote myself to actually doing it. The result was Wolfram|Alpha. And once Wolfram|Alpha was launched it became clear that even more could be done—and I have spent what I think has probably been my most productive decade ever building a huge tower of ideas and technology, which has now made possible the Wolfram Language and much more.

To Do Physics, or Not to Do Physics?

But over the course of that decade, I haven’t been doing physics. And when I now look at my filesystem, I see a large number of notebooks about physics, all nicely laid out with the things I figured out—and all left abandoned and untouched since the beginning of 2005.

Should I get back to the physics project? I definitely want to. Though there are also other things I want to do.

I’ve spent most of my life working on very large projects. And I work hard to plan what I’m going to do, usually starting to think about projects decades ahead of actually doing them. Sometimes I’ll avoid a project because the ambient technology or infrastructure to do it just isn’t ready yet. But once I embark on a project, I commit myself to finding a way make it succeed, even if it takes many years of hard work to do so.

Finding the fundamental theory of physics, though, is a project of a rather different character than I’ve done before. In a sense its definition of success is much harsher: one either solves the problem and finds the theory, or one doesn’t. Yes, one could explore lots of interesting abstract features of the type of theory one’s constructing (as string theory has done). And quite likely such an investigation will have interesting spinoffs.

But unlike building a piece of technology, or exploring an area of science, the definition of the project isn’t under one’s control. It’s defined by our universe. And it could be that I’m simply wrong about how our universe works. Or it could be that I’m right, but there’s too deep a barrier of computational irreducibility for us to know.

One might also worry that one would find what one thinks is the universe, but never be sure. I’m actually not too worried about this. I think there are enough clues from existing physics—as well as from anomalies attributed to things like dark matter—that one will be able to tell quite definitively if one has found the correct theory. It’ll be neat if one can make an immediate prediction that can be verified. But by the time one’s reproducing all the seemingly arbitrary masses of particles, and other known features of physics, one will be pretty sure one has the correct theory.

It’s been interesting over the years to ask my friends whether I should work on fundamental physics. I get three dramatically different kinds of responses.

The first is simply, “You’ve got to do it!” They say that the project is the most exciting and important thing one can imagine, and they can’t see why I’d wait another day before starting on it.

The second class of responses is basically, “Why would you do it?” Then they say something like, “Why don’t you solve the problem of artificial intelligence, or molecular construction, or biological immortality, or at least build a giant multibillion-dollar company? Why do something abstract and theoretical when you can do something practical to change the world?”

There’s also a third class of responses, which I suppose my knowledge of the history of science should make me expect. It’s typically from physicist friends, and typically it’s some combination of, “Don’t waste your time working on that!” and, “Please don’t work on that.”

The fact is that the current approach to fundamental physics—through quantum field theory—is nearly 90 years old. It’s had its share of successes, but it hasn’t brought us the fundamental theory of physics. But for most physicists today, the current approach is almost the definition of physics. So when they think about what I’ve been working on, it seems quite alien—like it isn’t really physics.

And some of my friends will come right out and say, “I hope you don’t succeed, because then all that work we’ve done is wasted.” Well, yes, some work will be wasted. But that’s a risk you take when you do a project where in effect nature decides what’s right. But I have to say that even if one can find a truly fundamental theory of physics, there’s still plenty of use for what’s been done with standard quantum field theory, for example in figuring out phenomena at the scale where we can do experiments with particle accelerators today.

What Will It Take?

So, OK, if I mounted a project to try to find the fundamental theory of physics, what would I actually do? It’s a complex project, that’ll need not just me, but a diverse team of talented other people too.

Whether or not it ultimately works, I think it’ll be quite interesting to watch—and I’d plan to do it as “spectator science”, making it as educational and accessible as possible. (Certainly that would be a pleasant change from the distraction-avoiding hermit mode in which I worked on A New Kind of Science for a decade.)

Of course I don’t know how difficult the project is, or whether it will even work at all. Ultimately that depends on what’s true about our universe. But based on what I did a decade ago, I have a clear plan for how to get started, and what kind of team I have to put together.

It’s going to need both good scientists and good technologists. There’s going to be lots of algorithm development for things like network evolution, and for analysis. I’m sure it’ll need abstract graph theory, modern geometry and probably group theory and other kinds of abstract algebra too. And I won’t be surprised if it needs lots of other areas of math and theoretical computer science as well.

It’ll need serious, sophisticated physics—with understanding of the upper reaches of quantum field theory and perhaps string theory and things like spin networks. It’s also likely to need methods that come from statistical physics and the modern theoretical frameworks around it. It’ll need an understanding of General Relativity and cosmology. And—if things go well—it’ll need an understanding of a diverse range of physics experiments.

There’ll be technical challenges too—like figuring out how to actually run giant network computations, and collect and visualize their results. But I suspect the biggest challenges will be in building the tower of new theory and understanding that’s needed to study the kinds of network systems I want to investigate. There’ll be useful support from existing fields. But in the end, I suspect this is going to require building a substantial new intellectual structure that won’t look much like anything that’s been done before.

Is It the Right Time?

Is it the right time to actually try doing this project? Maybe one should wait until computers are bigger and faster. Or certain areas of mathematics have advanced further. Or some more issues in physics have been clarified.

I’m not sure. But nothing I have seen suggests that there are any immediate roadblocks—other than putting the effort and resources into trying to do it. And who knows: maybe it will be easier than we think, and we’ll look back and wonder why it wasn’t tried long ago.

One of the key realizations that led to General Relativity 100 years ago was that Euclid’s fifth postulate (“parallel lines never cross”) might not be true in our actual universe, so that curved space is possible. But if my suspicions about space and the universe are correct, then it means there’s actually an even more basic problem in Euclid—with his very first definitions. Because if there’s a discrete network “underneath” space, then Euclid’s assumptions about points and lines that can exist anywhere in space simply aren’t correct.

General Relativity is a great theory—but we already know that it cannot be the final theory. And now we have to wonder how long it will be before we actually know the final theory. I’m hoping it won’t be too long. And I’m hoping that before too many more anniversaries of General Relativity have gone by we’ll finally know what spacetime really is.



    One random comment while I’m part way through the article: Is there a special name or theory for graphs where a sub-graph can itself be treated as a node, and edges can be treated as nodes?

    Daniel Bigham

    Dear Steve, You are now given your missing matter:

    The internal process of protons and neutrons swaps dimensions. Space becomes time and matter grows time. Matter shrinks space and grows time. That is gravity and the expansion of the universe. 5 nanoseconds is from the proton volume swap. With those magnitudes, space becomes quantifiable for all aspects of physics.


    Great blog post, thanks. My gut reaction is to wonder whether or not people will be the best ones to do this job. Here’s the big irony: Your work on the Wolfram Language, then Alpha (NLU), and now increasingly AI, may result in the creation of a system that is much more capable of doing the investigation you’re outlining than your own human mind.

    It’s like a man trying to decide whether he should try walking across town to get to an event, or wait 10 more minutes so that he can hop in a taxi.

    For the last 25 years you’ve been building a taxi, and some would say the taxi is about to arrive.

    Daniel Bigham

    You created a highly successful company that can now provide the material support for you to pursue a profoundly important goal – the unification of physics. The physicists that you mentioned are not original thinkers, and have accepted a certain dogma as the status quo; they will make no fundamental advances.

    I’ve personally thought that it’s clear that space/time/matter/energy are all manifestations of a unifying simplicity. I’ve thought that it’s along the lines of particles being solitonic waves in the stuff of space. But a basic problem is to understand the properties of that stuff, such that such waves are both possible and correspond to observed physical measurement. Perhaps this can be understood as the emergent nature of something even more fundamental.

    I hope that you do focus on physics. Not to be gloomy, but we all have finite lives and we’re not getting younger (at least not until rejuvenation research pays off), including our brains. I think others can run your company; but nobody else can advance your ideas of physics as well as you, and unfortunately, time is running out.

    Phil Oliver

    Work on physics. Other people are already working on the other problems, you are (more!) uniquely enabled to work on this one. In a decade these other problems will have made good progress, and if not solved you can re-engage. That being said, if your primary concern is personal legacy, then don’t work on physics. To your point, it may be a dead end, while the others are almost certainly solvable in the same time frame. And if you get hit by a bus in the meantime, the machines will get around to your network physics eventually!


    I hope you give this quest another try! It will be worth the effort, even if it ends in failure.

    Give yourself a specific amount of time to delve into this again, and when the time’s up, evaluate whether it makes sense continuing with a larger effort (more capital).

    Thank you for everything!



    Not to take this great article in it’s complexity down to a simple question but: does this mean you’re going to be working on physics as a priority again?


    You know my opinion…


    I was reading something about tensor network and quantum entanglement at the boundary of the universe which seems shed light on our understanding of spacetime, as showed in the recent work of Brian Swingle and Van Raamsdonk. And I’ve always been wondering, if ‘Mathematica’ had been invented when you were 20 years old, would you continue to be a physicist?


    Hello Stephen Wolfram, I wrote down 5 basic points – not so much about spacetime but spacetime of the universe as a whole (that is mainly: “gravity of the universe does not exist”). Sorry, it’s not very polite to link to a text somewhere else


    Sounds like a great idea! As a Physics PhD – off to work in the software business for a while – I too am looking at this problem again.

    It seems like the current path is not the way there. More hours have been spent on theoretical physics since 1960 (or some other date close by) than were spent in all of human history before – with precious little to show for it.

    I do share your ideas that complex behaviours can arise from simple systems – one of my favourites is how K&R C could build essentially every piece of tech in existence – and its a really simple recipe!

    To put it in one sentence, I think that classical general relativity has enough freedom and strength to easily build an emergent quantum mechanics and electromagnetism from.


    Also my thoughts – https://knobsturner.files.wordpress.com/2015/08/emergent_qm_gr.pdf


    So is this the same idea? https://www.nature.com/news/the-quantum-source-of-space-time-1.18797

    If you think you can have any reasonable chance of success I think you should do it. AI would likely be a gradual evolution with better and better systems. But if you are correct then there is one true theory of the universe to be discovered once and one team will make the key discovery.


    For me it is obvious that every single scientist trying to find new things using old methods is perfectly wrong. If they keep on doing the same stuff, they’ll get find the same stuff too. We have to reset even math in order to find diferent kinds of things, for example fractals. Congrats for trying new things!! Its a risky business!!

    Rub teran

    Where do I sign up? ☺

    Andrew Wozniewicz

    Great stuff.

    Perhaps it will be possible to ‘crowdsource’ the effort. Not everyone with the time, ability, and inclination to take part in this project will be in academia.

    george woodrow

    This sounds to me like a problem where more cycles is better, in possibly reaching a solution or at least a conclusion that much sooner. Could you examine the properties of more networks faster if you used a cycle sharing screen saver like SETI used to have to analyze data? Interested parties could contribute idle cycles to help complete the analysis in a world wide effort. I would certainly be interested in participating.

    John Romero

    I really like you work, always enjoining a lot this kind of posts!!!
    Your work is very bold in trying new methods outside mainstream.
    My current interest is in metaphysical assumptions in science, specially the subject-object division and the restriction to only “objective” phenomena. A subjective mind creating objective models of a system that involves it, I think that quantum physics has something more to say about that, or at least, it was the first time that physics touched that main issue.
    Your description of matter, particles as space structures is identical with the Buddhist idea of “emptiness” and “form”, but in that view, “awareness” is not an emerging propriety of the brain, “awareness” is fundamental, and i think that is analogous to “Computation”.
    I know that this may sound like crap religious nonsense to lot of people, but I’m pretty sure that there are some false assumptions in the metaphysics of science.

    Juan Manuel Jones Volonte

    Reading your post I immediately thought of the work by Robert Brady and Ross Anderson at Cambridge on modeling photons as phase vortices in a compressible inviscid fluid — that is, they’re able to generate Maxwell’s equations from a solution to the wave equation. Their solution also does indeed provide the required non-locality required by Bell’s inequality. https://www.lightbluetouchpaper.org/2015/02/23/maxwell/


    Are you familiar with Lee Smolin’s work?



    Consider if dark mass is external to our spacetime.

    If our universe is a black hole (in a nested Poplawski Multiverse – see Poplawski’s Metric, Einstein metric with torsion as quantum spin) then dark mass changing the curvature of our spacetime (likely holographically) can be explained by NO HAIR THEOREM. If it’s external to the black hole that is our universe, in the parent universe (see Smolin’s evolutionary multiverse theory), then there would be no force interaction other than graviational, the bending of our apparent horizon locally “beneath” the infalling mass/energy/information (specifically m;E;S – ie m, E, and Shannon Entropy).

    This nested Poplawski black hole multiverse with holographic principle, and Hawking thermodynamics, and with external dark matter subject to No Hair Theorem first of all explains why our visible universe has the exact mass considering only light matter (ie not dark matter) as a black hole of Schwarzschild radius equal to our internal light horizon, or that is to say equal to the radius of our visible universe, the visible horizon.
    It is unnecessary to presume anything beyond our light horizon. To us it doesn’t exist.

    And, what is beyond our universe, beyond that EVENT HORIZON, which is what a cosmic light horizon IS inherently. Beyond the cosmic event horizon, of our cosmic black hole, there is only the dark mass; which we can only detect gravitationally due to No Hair Theorem. This also explains the gamma emissions emanating from dark matter clusters, as particles converted from the external parent universe’s Planck scale to that our our ‘network’ as you would call it. Take a look at Hawking’s black hole thermodynamics, and you’ll see why such black hole universes MUST have relatively different Planck scales… otherwise the Boltzmann constant or the speed of light would have to be relative between them, and those are not the same in terms of measures: length for a length; ie Planck length for relative Schwarzschild radius. Consider a Poplawski set-up with complex black hole event horizons within parent universe black holes via Hamiltonian maths, and you’ll see why.

    IF there is such a larger Poplawski Multiverse, and the parent black hole universe also has a parent (eg Smolin’s evolution), then the entire system begins very quickly to look like a Hierarchical Temporal Memory, given Hawking Thermodynamics.

    Consider that information falls into each successive nested black hole universe, where it is processed by your cellular automaton-like structure. But, then the external parent universe cools, and the process reverses; and the processed information is then released back out to the external parent. But, that parent black hole universe will suffer the same fate, releasing that processed information, etc, etc.

    Consider the information FLOW with processing within such a Hawking-Poplawski Multiverse, and you’ll see it is an HTM.

    Evolution is scale invariant.

    Nick Albertini

    Two differing areas of space colliding form the Universe. The medium into which the energy generated by the colliding resists the expansion forming matter, first in trhe form of subatomic particles. The phenomenon of Time is the number of subatomic particles generated. As subatomic particles form they change arrangements. This change we perceive as Time.

    Steven E. Cooper

    The ultimate truth of the universe is much more simple than thought. The equation for everything is =

    Michael J Ahles

    Other people are coming around to your perspective, too: spacetime is weakly emergent over a discrete valued tensor network. Entanglement is not caused by proximity, instead, proximity results from entanglement. And it makes a lot of sense: if the universe is computational, then continuous-valued spacetime cannot exist (other than an approximate, emergent relationship). http://palmstroem.blogspot.com/2015/11/rethinking-quantum-mechanics-and.html

    Joscha Bach

    Dear Professor Wolfram, The biggest problem for mathematicians and physicists in regards to spacetime is what I refer to as the Ruby Slipper Conundrum.

    Once you consider that space-time could be made of something (up until recently it was thought that spacetime is made of nothing) and this something is in the form of a medium, you need to determine what the basic building block of that medium is; and herein lies a huge problem! Everything we know to exist; exists “in” space-time. Mathematicians and physicists had no other choice but to understand the properties of particles, the interactions of particles, and phenomena that result from particles as all particles exist “in” space-time.

    I use the word “entity” for the building block “of” space-time to distinguish it from particles that exists “in” space-time. The question to ask is; do physicists and mathematicians possess the cognitive capacity to conceive of a spacetime entity’s mathematical properties in the manner in which it actually exists? Such an entity cannot be mathematically expressed as a particle “in” space-time because it does not exist “in” space-time; it exists “as” space-time. Mathematically describing such an entity has never been done before in the history of physics!

    Let’s consider what would happen if we were to pluck one of these entities out of its structural position of the spacetime medium and place it where there is no space-time (isolated in the true void). Can we possibly conceive of its mathematical properties and describe them in mathematical terms that would make sense to us. I can assure you that no one, not even our greatest minds in physics and mathematics can think that far “outside the box”. Because this has never been done before, the mathematical description of this entity would be impossible to accept without first going through the “Ruby Slipper Conundrum”. This is why when reading “The GOD Entity: Gordon’s Theory of Everything”, you must commit to reading the entire book.

    Scott S Gordon

    A wonderful discussion of non-dimensional space and it’s possible structure base. I believe this is a bit like the thought game where one begins with “empty” space and proceeds to define mensuration and distance through the addition of points within the space. The “observer” is the first point and all other points must be added from an external set. With this idea of “information” space mixing with “connectivity” to create an interactive lattice there is the possibility to create mensuration points within the space it’s-self. This all fits neatly into my understanding that space is non-dimensional and that axis es of x,y,z, and t are all occurring in the same network of space. Most discussions of space use coordinate volume to define dimensions which is incorrect. A dimension must have a network with at least one degree of change in structure (a square, triangle, and circle could be three different space dimensions.). How can I become involved in your potential project? Please contact me with details. Thank you and Sincerely – William.

    William D. Jagger

    Count me in.

    Ashwin Rao

    Your certainly not alone in trying to figure out what’s space-time and how to represent it best. Definitely good equipped with a good understanding of numerical mathematics. The work of Lee Smolin, Carlo Rovelli and Martin Veldman are probably well known to you. So a suggestion from an applied physics guy ‘hobbying’ theoretical physics. Just because: hope that silence critics why things are done. If Galileo, Newton and Einstein were not stubbornly pursuing their idea, we would be still working with sticks and stones, figuring out how water boils.
    I’m (again) following is KK theory but now represented as a graph. Consider the cylindrical condition as the depth we can view into that 5th dimension. About Plank length. Interpreting charge as a 1/2 or 1/4 wave mode with it’s maximum at the borders and mass as phase shifted mode with it’s node on the border.
    Mass less photons are represented by a + and – wave canceling each other and being mass less KK gives an unification of Electromagnetism as stated by Maxwell and the GR quotations as stated by Einstein. Adding up the (almost) zero contributions of the phase shifted waves, the higher modes in the 5th dimension, coupled with either a charge wave would give a very small mass neutrino, a light mass for electron, positron and quite a heavy mass for the mix of electrons/positrons making up a proton.
    Probing deeper into 5D by squashing particles into each other, will show more (instable) modes (either as a projection in 3D or as ‘real’ particles, e.q. modified 4D) with heavier masses.

    Kaluza theory was rejected (by Einstein as well) on the grounds that the mass of an electron would be an staggering 10^30 more than it is measured. but that was a). based on the assumption that the size of the 5th dimension was about the Planck length (not the projection depth, where projected mass is expressed in inverse meter). b). that time is absolute and the same, whilst time is just as relative (and can be seen as negative for symmetrical reasons) giving a whole different result when Newton mechanics is applied.


    Fantastic blog post. It resonated with me based on stuff I’ve seen this morning to do with fluid mechanics. Where little balls of fluid can be ejected from a fluid and then waves on the surface of the fluid can force the ball to move around. You create a phenomena like wave particle duality. A thought I had at some point in your article was: what if the nodes can reconnect? Could you have disconnected parts of the network which some how moved through the network, perhaps reconnecting briefly here and there. Might be like the blob of fluid hovering over the surface. The whole idea of wormholes in spacetime being the same thing as entanglement really fits in with your ideas of networks. Personally I think you should try to finish the job! I’m currently trying to do the molecular construction problem to make our planet sustainable in the flux of energy from the sun. Biggest problem here is human politics! I wonder if you couldn’t imagine networks of people as an example of your kind of space and find the right order of reconnection you would need to set up a sensible global politics; the right order of exchanges between countries and companies in order to find a settlement on the move from fossil fuels to renewables, but without world war three!! Wilful reconnection as opposed to violently imposed! I’m also struggling with the direction I’m going to take. My choice is politics or technology… Any advice appreciated!


    Ooh, also alcubierre warp drive: could waves of reconnections in the network drive about disconnected self contained regions through space? Maybe you could reconstruct instantaneously elsewhere/move at arbitrary velocities a particular sub pattern??? Yeah! Ftl travel!!!! You should totally do this…..


    There is never really such a thing called right timing; it’s only possible to say so on hindsight.

    After all, “the only thing observers can ever actually observe in the history of the universe is the causal network of what event causes what other event.”

    Just do it.


    I am very impressed by your work as you have presented in this article. I haven’t studied your theory yet, but I will try. I have only two comments on this article:!) I get the impression that you assume that physics is computable. But we know that the most part of even discrete mathematics isn’t computable. And there has been a discussion, especially initiated by Roger Penrose, about the computability/non-computability of pysics. 2) Is there any connection between your theory about causality as basic and Russells ‘Causal Theory of Space-time’ from 1927 (as he presented it mainly in his “The Analysis of Matter”)?


    Although this is tremendously fascinating, I have to say that the fact that in Science there is no simpler concept than a graph doesn’t imply that the Universe should be necessarily made of graphs too. This merely tells us something more about the stuff our Mind is made of


    Coming from an outside observer, with no background in physics or mathematics, hence strictly intuitive:
    Do this.


    really cool perspective ….. more work must me done in this direction.

    arun ramanathan

    I wonder if you have ever encountered with the work of Christoph Schiller.
    It’s very similar to what you say in many key points. Like your points about Planck scales, and that space-time, matter and radiation are all made of the same entity.
    Although I must say that I think there are some problems in his theory.
    For example, I think knots must be avoided. and instead matter and radiation must be considered as movable tangles in vacuum strands.

    Some nice points that I find very interesting about his theory:
    1. space is 3 dimensional because knots are only possible in 3 dimensional space. in less than 3 dimensions knots are impossible, and in more than 3, knots will be opened.
    2. tangles are not necessarily localizable. they get localized when they interact with baths.
    3. some tangles are unstable and decay to stable ones.
    4. a tangle can be untangled with a mirror tangle (antimatter).
    5. mass is the amount of inter-tangledness, and thus it determines the inertial difficulty to move the tangle, as well as the curvature of nearby vacuum strands.

    here is a link to his website:

    Damoon Saghian

    If we are looking in electromagnetic spectrum we can see that it is not discrete and in the number system too you can see that its not discrete so the space can also be non discrete No knot in the string and all events are continuous Huygens for the explanation of wave theory introduced the ether medium but actually the universe is like a fluid, as the author told , is continuous and non discrete this is my humble thought…


    Blown away by the amazing work you’ve put in to giving a better explanation for the causal web. Your approaching the idea of networking instead of the manifold objective strings and branes so many mathematicians are locked in to is refreshing, and agreeably highlights some of the ‘coolest’ aspects of the study of spacetime!
    P.S. Can’t wait for the day we go back to saying aether, instead of Higgs Field. When something falls to the Earth, we don’t call it ‘Getting Newton’d’. This article in particular is a nice thumbing of the nose to dogmatic scientists just looking for steady checks.


    Thanks for a very insightful post. Interestingly your ideas have a convergence with a book I am currently reading which basically advocates a view that life is an emergent property of networks at all levels (book: A System View of Life: http://www.amazon.co.uk/The-Systems-View-Life-Unifying/dp/1107011361). Your postulation that our universe and the effects we see at all levels are also an emergent property of an underlying network and connections is quite thought provoking in context of this view in other scientific disciplines.

    Grant McDonald

    Typing on a phone can be challengeing. Here is an edit item version of my comment belowhe Microscopic Connection between nodes can be seen as “outside of time,” not in a weird way but simply as in Bell’s non-locality of space. Imagine a starting point of a network as a zero dimensional point, is outside time. When a point is ‘moved’ in time it defines a 1 D line, movement on any other axis defines a 2 D Plane, then 3D, 4D, 5D etc, as long as there is a sufficiency of unfolding persistent nodes connecting in time, a timeline is established. Some matter is formed and some blinks out of existence, producing energetic wave/particles.
    The key in imagining this emergence is that time is the first dimension, and all dimensionality unfolds from a singularity “in-time. Outside of time, everything in that time line is still connected to every thing else that has arisen
    Now imagine the singularity at the center of a black hole, where mass accelerating toward it approaches the speed of light and time begins to slow down. As time slows to zero at the singularity near the center, matter breaks down from molecular nodes of attraction when the E and H fields are overcome, on and on all the way down to a near horizon of quark soup before merging “outside of time at the singularity.
    Now the tricky part to imagine the mass entering the black hole was still intimately connected to all the other mass in it’s timeline, so even though dimensionality has completely broken down and time is at zero, the Gravity remains.
    So, gravity is just the emergence of connection in time, Black holes can be imagined as nodes that connect to the time lines of everything in gravitational range, which drops of to zero connection at Planck limit of 10 to the minus 34th. This might be why some are working to show gravity is different on the very small scale than the typical square of the macro level distance apart.

    Daniel Washburn

    I vote for You’ve got to do it!” with many !!! added.

    It’s often said that the three pillars of modern physics are general relativity, quantum mechanics, the standard model, and string theory, but compared to the first two the standard model and string theory seem like Ptolemaic astronomy with all its ad-hoc tricks (make it more complex and perhaps it will work). I guess once a fundamental description of space-time, which includes both general relativity and quantum mechanics as limit cases for low energies and intermediate scales, is in place, the rest of physics will begin to fall in place.

    I guess what you have in mind is a cloud system based on Wolfram Language where participants run tentative models with automatic checks for desired features and a social karma system to rate the best proposals?

    Tell us more and let us know what we can do to help!

    Giulio Prisco

    Do it!


    As always, it’s a matter of whether our models are adequate to the thing itself, the phenomenon before us.

    If the universe is a network that has the capacity to inquire into itself, the way our universe appears to do through us, then what order of logical capacity is up to that task?

    For my part, I don’t think the common run of network models that we are seeing today have enough “logic” in them to do the job. Just to be cute about it, they need more nots in their ether.


    Interesting, does the expansion of the universe in this model mean that more nodes are added to the network or that the distance between them increases? I guess the former since we can perceive the expansion – if not will the increase distance between the nodes lead to that they de-link and the network collapses at some point?
    I like the concept – instead of messing with compressing and expanding space to travel long distances we can rip the nodes apart and re-link the local environment to the nodes of the destination – I shall name it the rip-drive… :)
    Cheers, Niclas


    Please do it. As you said, there is a risk that you’ll be wrong, but it’s a risk that not enough people are willing to take – for fear of being wrong – for a goal that is immeasurably significant if you do find it. Plus, if you do go ahead and do this out in the open, it’ll give the (interested) public a chance to learn from your investigative skills – whether or not you’re successful.


    Do you know the work of Thad Roberts?
    I’m a total layman… but I’ve read his book (Einstein’s Intuition: Quantum Space Theory) and loved it.
    It shares a lot with your theory.

    Duarte Cunha Leão

    Renate Loll has been doing something similar


    Dear Wolfram, my theory is similar to yours and indeed I mention your name early on in my write up. However, mine directly leads to QM with many interesting results. It is all about the fundamental entity which is random numbers where I derive space, time , matter, energy and interaction all in one coherent system.

    “Reality is nothing but a mathematical structure, literally”


    Essay Abstract

    “In This essay I shall derive the laws of nature from a simple mathematical system from a postulate that reality is indeed a mathematical structure. The system can be simulated by a computer program to generate many results that agree with Quantum mechanics. Also I will show that the system can be put in regular more familiar mathematical formalism. The postulate lead to assume particles are made of random lines were one end originates in a small region representing the particle and it extends to all other points in space and some ending on other particles. The points are really nothing but random numbers, hence reality is nothing but some relation between random numbers. Moreover, the lines are responsible for the interaction by a process of crossing or not crossing or meeting.The start point and the end point of these lines define space and the length of the line is interpreted as energy, time is just a change of state. The system unifies space, time, matter, energy and interaction, all in one coherent picture, so particles and the laws of nature governing them appear naturally. The simulations generate some basic Quantum Mechanics results and the 1/r law ….”


    Great work and an amazing perspective, would like to help as much as i can.

    Arun Ramanathan

    i’m afraid you’ll be working on just another theory (like so many other) in physics.
    maybe it will be elegant and beautiful, maybe it will match so many experiments … but …
    the problem in physics is not a lack of theories to explain this or that, the problem is that our
    experiments are so limited at the time … there’s a need of new experiments … we need to “see”
    more, way more, to better understand the universe.


    The Buddha worked this “universe as network” theory out a long time ago –classically expressed in the “Jeweled net of Indra” in Hua-Yen Buddhism

    Peter Timmerman