What Can Replace Space-Time?

Nima Arkani-Hamed is famous for believing that space-time is doomed, that as physicists we will have to abandon the concepts of space and time if we want to find the ultimate theory of the universe. He’s joked that this is what motivates him to get up in the morning. He tends to bring it up often in talks, both for physicists and for the general public.

The latter especially tend to be baffled by this idea. I’ve heard a lot of questions like “if space-time is doomed, what could replace it?”

In the past, Nima and I both tended to answer this question with a shrug. (Though a more elaborate shrug in his case.) This is the honest answer: we don’t know what replaces space-time, we’re still looking for a good solution. Nima’s Amplituhedron may eventually provide an answer, but it’s still not clear what that answer will look like. I’ve recently realized, though, that this way of responding to the question misses its real thrust.

When people ask me “what could replace space-time?” they’re not asking “what will replace space-time?” Rather, they’re asking “what could possibly replace space-time?” It’s not that they want to know the answer before we’ve found it, it’s that they don’t understand how any reasonable answer could possibly exist.

I don’t think this concern has been addressed much by physicists, and it’s a pity, because it’s not very hard to answer. You don’t even need advanced physics. All you need is some fairly old philosophy. Specifically we’ll use concepts from metaphysics, the branch of philosophy that deals with categories of being.

Think about your day yesterday. Maybe you had breakfast at home, drove to work, had a meeting, then went home and watched TV.

Each of those steps can be thought of as an event. Each event is something that happened that we want to pay attention to. You having breakfast was an event, as was you arriving at work.

These events are connected by relations. Here, each relation specifies the connection between two events. There might be a relation of cause-and-effect, for example, between you arriving at work late and meeting with your boss later in the day.

Space and time, then, can be seen as additional types of relations. Your breakfast is related to you arriving at work: it is before it in time, and some distance from it in space. Before and after, distant in one direction or another, these are all relations between the two events.

Using these relations, we can infer other relations between the events. For example, if we know the distance relating your breakfast and arriving at work, we can make a decent guess at another relation, the difference in amount of gas in your car.

This way of viewing the world, events connected by relations, is already quite common in physics. With Einstein’s theory of relativity, it’s hard to say exactly when or where an event happened, but the overall relationship between two events (distance in space and time taken together) can be thought of much more precisely. As I’ve mentioned before, the curved space-time necessary for Einstein’s theory of gravity can be thought of equally well as a change in the way you measure distances between two points.

So if space and time are relations between events, what would it mean for space-time to be doomed?

The key thing to realize here is that space and time are very specific relations between events, with very specific properties. Some of those properties are what cause problems for quantum gravity, problems which prompt people to suggest that space-time is doomed.

One of those properties is the fact that, when you multiply two distances together, it doesn’t matter which order you do it in. This probably sounds obvious, because you’re used to multiplying normal numbers, for which this is always true anyway. But even slightly more complicated mathematical objects, like matrices, don’t always obey this rule. If distances were this sort of mathematical object, then multiplying them in different orders could give slightly different results. If the difference were small enough, we wouldn’t be able to tell that it was happening in everyday life: distance would have given way to some more complicated concept, but it would still act like distance for us.

That specific idea isn’t generally suggested as a solution to the problems of space and time, but it’s a useful toy model that physicists have used to solve other problems.

It’s the general principle I want to get across: if you want to replace space and time, you need a relation between events. That relation should behave like space and time on the scales we’re used to, but it can be different on very small scales (Big Bang, inside of Black Holes) and on very large scales (long-term fate of the universe).

Space-time is doomed, and we don’t know yet what’s going to replace it. But whatever it is, whatever form it takes, we do know one thing: it’s going to be a relation between events.

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3 thoughts on “What Can Replace Space-Time?

  1. Jan Reimers

    I think a softer version of what Nima is saying (which he sometimes uses) is space-time must be an emergent phenomena. As you know there are lots of examples of emergent phenomena and Robert Laughlin is very good describing this in lay terms https://www.youtube.com/watch?v=yF869nAYlfQ.

    Of course the next question is “emergent from what” and that is the part that is hard to imagine right now, and impossible to measure (even in principle ?).

    Other provocative statements from Nima that many physicists may struggle with are things like: “There are no fields just particles. The field is construct that allows us to define local interactions and operators … but because of gravity there are no truly local observables”.

    Anyway I think the direction people like you and Nima are headed is fascinating. I look forward to seeing a formulation of known theories like QED or QCD with no Lagrangian where locality and unitarity are emergent.

    Cheers
    JR

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