In this week's "Ask a Physicist," we're going to figure out why the universe we inhabit isn't Flatland, Tesseractland, or Doubletimeland. Or, at very least, we're going to figure out why those worlds would suck at harboring complex life.
This week's question was posed by Richard Henretta via my facebook fan page. He asks:
This could easily be classified a stoner question, but what would the universe look like if there were 2 dimensions of time?
It is a stoner question, but that won't stop me from answering it. Let me dispense with the obvious. From a macroscopic perspective, there are three dimensions in space. I don't care if you call them up-down,left-right,and forward-back, or x,y,z, or whatever you like. The fact is that we can only move in three-d.
There is also just a single dimension of time, and the main difference between time and space is that unlike with space, you only get to move through time one way, and at a rate of 1 second/second. It's this succession of events through time that define our experiences. Within the timeline of a single individual, it always makes sense to say, "A came before B." Once you start with multiple observers and relativity, all bets are off, but let's not get distracted with that today. It doesn't change the fact that in any local frame, there's just one way for time to go.
But why is the universe made this way? Everything we know about the standard model is built on the assumption that the universe has 3+1 dimensions, but it doesn't actually tell us why that has to be the case.
As hardcore geeks, many of you are probably familiar with M-theory, which says (among much else) that in reality, the universe has ten spatial dimensions and one for time. All but three of those spatial dimensions are presumably very small; they would essentially be a Pacman universe on scales not only much smaller than you and me, but also on scales smaller than atomic nuclei.
Suppose, for a moment, that M theory is right. Is there a place in the multiverse with more than 3 macro dimensions? Sure, there might be. But we couldn't live there, and as I'll try to argue below, neither could anything else.
I'm going to make a bunch of anthropic arguments, and somebody is going to get their panties in a twist by saying that we don't know that life has to be like it is here on earth. True enough, but I'm making some reasonable assumptions. That is, I'm assuming complex molecules and atoms heavier than hydrogen need to be able to form. Since we've never seen extra-terrestrial (let alone extra-universal) life, I could be wrong. That's a chance I'm willing to take.
So what's the (weak) anthropic argument? The basic idea is that there isn't just a single set of physical parameters that describe the universe, but that you'll only find intelligent (or any) creatures trying to be physicists in regions of the universe that are conducive to life. We talk a fair amount about this in the "User's Guide to the Universe", where we address, for example, why physical constants might actually vary in different patches of the multiverse.
So what's wrong with anything but 3+1?
Image via Flatland the movie.
Why not live in Flatland?
We'll start easy. I'm sure many of you have read Edwin Abott's beloved Flatland. If you haven't, the premise is that a (literal, geometric) square tells us all about the physics and civilization of his two-dimensional world. I assure you that it's more interesting than it sounds.
The problem with such a world is one of complexity. To pick a particularly gross example. Imagine yourself as a two-dimensional amoeba. A mouth-type opening takes in some food. How does your digestive system work? Well, presumably, there's a tube running through you, ending at your tuchus. The problem is that in 2-d, such a tube would split you in half. In other words, for your digestive system to work, your mouth would also have to serve double-duty as your butt.
Grossness aside, there's a general problem in two dimensions, let alone in one. Systems and organisms simply can't be complex enough to form anything approaching intelligence. For instance, because you can't cross things in 2-d, neurons wouldn't be able to cross one another, and brains (or anything like it) would be very, very limited.
Sculpture by Rob Millar
How about a Tesseractland with 4 dimensions?
It's easy to think about 2 dimensional universes, because we can draw them on paper or computer screens. It's far harder to visualize what life would be like in a universe with more than three dimensions. We have to at least consider the possibility, however. If M theory is right and there really are 10 dimensions, why are so many of them compact, and only three of them big? You've probably already seen Rob Bryanton's "Imagining the tenth dimension", but if you haven't, you should check it out.
Just because we can imagine a 10 dimensional universe doesn't mean that we (or anything like us) could ever live in one. I'm going to drop a little old-school classical mechanics on you. You may remember, dimly, something about gravity being an inverse square law. The idea is that if you double the distance between two objects, their force of gravity drops by a factor of four. The same rule holds for electromagnetism.
The inverse square law isn't an accident. It turns out that it's entirely a function of the fact that we live in a three-dimensional universe. If we lived a four-dimensional one then we'd have an inverse cube law.
It turns out, though, that an inverse square law is very special. Higher dimensional universes (with their inverse cube or inverse-fourth gravity laws or whatever) don't have any stable orbits. In other words, in a 4-dimensional universe, the earth would either spiral in toward the sun or fly away. We wouldn't get to enjoy the five billion or so years of nearly constant sunlight that we do in our universe.
This is true for all orbiting bodies (including planets, comets, stars in the galaxy and so on), but the situation gets even worse — at least from an anthropic point of view. Because electromagnetism also obeys an inverse square law, it turns out that atoms wouldn't be stable. They'd all spontaneously collapse. It's really hard to imagine complex life without atoms, and even tougher to imagine having this conversation without the existence of life.
A note to the experts. Somebody is likely to point out in the comments section that electrons don't "orbit" atoms in the same way that planets do the sun. True enough, but if you grind through the equations in quantum mechanics and do the problem correctly, you hit the same problem. No stable atoms. Sorry.
What's so weird about 2 dimensions of time?
So we're limited to three dimensions in space, but what about perhaps having more than one dimension of time?
To put it bluntly, the universe would be confusing. Even talking about it is confusing. MIT Physicist Max Tegmark has a very nice discussion of what life would be like in a universe with two dimensions of time, and I'm quite shamelessly borrowing from his work here. (Warning: the original paper is a bit on the mathematical side).
Every person (and particle, for that matter) would move through two different times, t1 and t2. But these two times can't proceed at exactly the same rate, because if they did, it would be exactly the same as if the universe had only one dimension of time. More generally, if you're alone in this freaky universe, you probably wouldn't even notice that anything is wrong.
But things get awkward if you have a friend. (Use your imagination if necessary.)
Normally, if you meet up with someone, it's because you're at (more or less) the same coordinates in space during an overlapping period of time. The problem is that if the two people are moving through the different time coordinates at different rates then even though they might remain in the same place, they won't remain in the same time.
Put more simply, while your personal clock may run normally, unless your loved ones are all on the same world-lines you're destined never to see them again.
But even supposing you're an angry loner, living in a two-spatial-dimensional universe is going to be tough. The big thing (and the one must likely to spawn debate in the discussion section) has to do with the fact that with 2 time dimensions (and at least two space dimensions), you can't say anything useful about the "future." One of the things that makes an intelligent observer is that I (assuming I am one) can look around and based on the state of things around me, determine with some probability what will happen elsewhere at some point in the future. With two dimensions of time, you simply can't do that. I'll skip the math, but the basic idea is that the "future" isn't very well defined if you've got two time dimensions. If there's no inference, there's no prediction and no science. It's very hard to imagine how creatures in those sorts of circumstances could make any decisions at all.
Life is unpredictable enough with one dimension of time; two would just be ridiculous.
Dave Goldberg is the author, with Jeff Blomquist, of "A User's Guide to the Universe: Surviving the Perils of Black Holes, Time Paradoxes, and Quantum Uncertainty." (follow us on twitter, facebook or our blog.) He is an Associate Professor of Physics at Drexel University. Feel free to send email to firstname.lastname@example.org with any questions about the universe.