# Two time dimensions

I read an artilce a few months ago (I think in Discover) where they talked about the possibility of universes with two time dimensions. I would assume, like our one, that they would both be moving “forward” and picture a two dimensional graph but…

What the heck would it be like? It would mean you could go out and have your lunch hour, then go out and have it again. Excellent.

I don’t think so… I don’t see how having two means you can go back in ‘time’ I am quite comfortable with the possibility of each of our real dimensions…including time…as having an imaginary counterpart.

Each dimension…length, width, depth, time…instead of being
a real number line could be a complex plane.

My first thought was that the “sideways”, if you will, time dimension, might be analogous to moving among various parallel universes. But those are generally thought of as discrete and distinct, so I’m not sure how a sideways dimension would help there. Oh well…

The first question is how light behaves in two time dimensions. Suppose I send a light-speed signal from time [0,0]. When can you see my signal if you are 1 light year away from me? I suppose at time [ta,tb] where sqrt(ta^2+tb^2)=1 ly, and ta>0, and tb>0.

The next question is what determines my trajectory through time? Here in a 1-time-dimension universe everything has the same linear trajectory through time. But in a 2-t-d universe the time dimensions are orthogonal, so you might be advancing in them at different relative rates. So three days might pass in ‘a’-time during which only one day passes in ‘b’-time.

How much control can a being have over its trajectory in time? Can a person weave through time in order to age faster? How much energy does it take to change your relative progression through ‘a’ and ‘b’ time?

Instead of being able to perceive light from a source where distance/time=c we have something like distance/sqrt(atime^2+btime^2)=c.

A second time dimension is definitely another degree of freedom. We would have to think in terms of time arcs instead of time lines. In other words, there would be an angle in history. Historical events would lie somewhere on the spectrum between ‘a’-history and ‘b’-history. Similarly, we would be planning for an arc of future. We might be uncertain whether important events in our lives would happen in an ‘a’-ish or ‘b’-ish future.

The present would, however, always just be a point, like it is in 1-t-d.

There could be units of comparison between ‘a’-time and ‘b’-time, for example, our wedding is scheduled to be in three months, and .50 aishes, meaning the wedding is three months away, but at a doubly ‘b’-ish time. By aging in a different direction, (i.e. at 0.0001 aishes for 2.5 months and 1x10^5 aishes for 1.5 months) you might be able to attend that same wedding 4 months after receiving the invitation.

How could the present be a point if time were two dimensional? If, for example, a “time-line” is a vector originating from the origin (0,0), with an angle of “A”, then after time “t” had passed, the present would be an arc over the range of “A” with radius “t.” But then if a “time-line” were to change direction, it would no-longer be in the “present.” Um…I’ve just confused myself.

But yeah, if time had two full dimensions (not one complex dimension) then I don’t see how the “present” would be a single point. Myabe not even if time were a single, complex, dimension.

Not that I know what I’m talking about, or anything.

On a related note, how is a complex plane fundamentally different from a two dimensional real plane? Obviously certain maths are different, but are there any models that demand a complex plane over a two dimensional real plane?

Maybe having two time dimensions would result in something like the movie ‘Sliding Doors’.

Well, string theory has, as I recall, at least the potential for multiple time-like dimensions, but with any other than just the 1 we think of curled up. If my memory isn’t playing tricks with me, the answer then might be that a universe with two time dimensions would be quite familiar already…

askol, at least in standard physics I’m not aware of any models that require the use of complex numbers; a pair of real numbers will do. Using complex numbers is often a lot easier, but not normally required.

What is time anyway?

Keeve, your thought about parallel universes was exactly what I was thinking about but don’t know if it is valid since different each parallel universe would be discrete instead of a continuous breaks in the second time dimension.

I’ll have to digest Jawdirk’s post more.

Another thought I had is that if time flows at a ‘constant’ rate for both time dimensions (no personal control over movement like we have now) then it might appear as if there is only one? Is this what you are saying gr8guy?

Never saw the movie sliding doors.

Sort of a little bit. This is really hard for people (me included) to visualize, so let’s start with extra spatial dimensions first, and reason by analogy.

Suppose you’re an ant on the surface of a rope, walking merrily along. You’ve got two dimensions to move in: around the rope’s circumference, and along the rope’s length. Now pretend that the rope gets really super super thin, so it’s more like a fishing line or something like that, only far thinner still. You’ve still in principle got both options, but as far as you can tell, you can’t actually walk around it because it’s too skinny for you to see that there even IS such a thing as around it. Does that make sense? Basically, it looks to you as if it’s just a line, but it’s actually an incredibly thin tube.

This would be the idea of an extra spatial dimension: the ant in my little example thinks he’s walking along a line (1D) but he’s actually walking on a thin tube (so it’s actually 2D). Extra spatial dimensions in string theory would work the same sort of way, although it’s utterly impossible to visualize it. We think we’ve got 3 spatial dimensions, but really we might have far more that are just incredbly tiny and curled up, just like the tube is incredibly thin.

Okay, so the idea with two time-like dimensions that I was tossing out there is the analogous thing: we really could move anywhere in either one (except that we’re always going forward, of course). The trick is that we can’t tell that we’re moving in both, because one is too tiny for us to notice. That’s probably not really all that helpful, but I can’t think of a better analogy than that, since it’s how I think of it myself (at least, when I’m masochistically trying to understand some aspect of string theory).

gives whole new meaning to the term two-timing, don’t it

I’ve found a paper covering the dimensionality of spacetime which argues that life as we know it requires that there be exactly three space dimensions and one time dimension. I havn’t read the entire thing, but here’s the abstract:

(bolding mine) The paper is by Max Tegmark and can be found here.

The existence of a four-dimensional geometry, with three dimensions of space and one of time as described by Einstein, has been verified experimentally in the sense that clock time measurments do indeed vary according to his equations. This can be interpreted in two ways, and the big question is which one is correct:

1. Is this the geometry of space itself? This would mean, for example, that if you blow up a balloon and then let it deflate, the shape of the balloon includes how long the balloon stays inflated the same way it includes how big the balloon got.

Taken literally, this implies Fatalism. To see why, consider replacing “inflated balloon” with “your life”… But the very definition of a law of physics is that motion A by itself causes motion B, so that if A happens and B doesn’t, that’s evidence that something else happened (e.g., The apple didn’t hit the ground because you caught it.) Fatalism, however, would mean that there could someday be an apple that would stop in midair for no reason, simply because that motion happens to be part of the shape of the universe. But in that case, everything that has ever been discovered about physics, from Einstein all the way back to cavemen learning how to throw stones, is just luck. The biggest reductio ad absurdum ever!!! 1. Is this the geometry of motion through space? This would be more like time as we know it, in the sense that the air in the balloon doesn’t form a closed four-dimensional shape until the balloon has deflated.

There’s a problem though: If everything is part of an unfinished shape of events, where is the open edge of events? It would seem that you could answer this just by looking at your watch and seeing what time it shows. But that would answer it only for the location of your watch. Even if you did this for every object in the universe, it still wouldn’t explain how relativity can work, because it would also have to apply to the space between them. But the space between them doing what? Moving through space?

There are two common ways of getting out of this dilemma:

A. You can assume that looking at your watch is as far as it can go, because that’s good enough to define the edge of what you observe. But this would be a different path through time than anyone else’s, so the only way to get a sensible physical theory out of this is to postulate an observer of everything. Some people like this idea, such as the people in the Templeton Foundation. :rolleyes:

B. You can assume that there actually is a sense in which space is moving through itself. But would this moving space be Aether? Would it be Dark Matter? Or what? Your guess is as good as mine or anyone else’s, so there’s no way to flesh this out into a good theory.

This is where Occam’s Razor comes in. There are two definitions of time, and no way to choose between them without making a mess. So why not just take both? If one of them is a dimension, then call the other a dimension too!

In some alternate, two-dimensional time path, this thread was just posted a week ago.

Here is a paper by a late colleague of mine, Joachim Lambek, that posits three dimensions of time. Look at http://www.math.mcgill.ca/barr/lambek/pdffiles/Quater2014.pdf.

The answer to the question of the difference between a 2 real dimensional space and a 1 complex dimensional space is that there is no real difference, but there is a complex difference in that you can multiply the elements of the second space by complex numbers but not those of the first. But there is a big difference between the real projective plane and the complex projective line in that the former has a line at infinity while the latter has only a point.

Three dimensions of time but only one of space is exactly equivalent to three of space and one of time. It’s just the convention that we call the odd-man-out dimension “time”.

What does this mean? If this comment makes any sense at all, could you explain to us what that sense is? What would a universe be like if it had one space dimension and three time dimensions?

On the surface, this post sounds like you’re just picking at the arbitrariness of the words we invent to label things.

In relativity (i.e. Einstein’s general and special theories) the metric tensor has the signature (3,1). In mathematical terms this means that at any event in spacetime if you define an orthonormal basis (i.e. a set of four spacetime vectors of unit length, such that each vector is at right angles to each other), of the four basis vectors, there will be three which we will label as a[sub]N[/sub] (N = 1,2,3) such that a[sub]N[/sub][sup]2[/sup] = 1 and one basis vector which we will label as a[sub]0[/sub] where a[sub]0[/sub][sup]2[/sup] = -1. Where ‘a[sup]2[/sup]’ just means taking the scalar product of the vector a with itself.

The physical interpretation of this is clear: the basis vector a[sub]0[/sub] represents the ‘time direction’ for a particular observer at that event in spacetime and basis vectors a[sub]N[/sub] represent 3 orthogonal spatial directions for that observer.

It might seem then that if we were to use a metric of signature (1,3) as opposed to signature (3,1) the natural physical interpretation would be 1 spatial dimension and 3 temporal dimensions, as the defining feature of the temporal dimensions when using a (3,1) signature is that minus sign when squaring orthonormal basis vectors and when using a (1,3) signature there are 3 of those minus signs. However there is a natural (and obvious) isomorphism between metrics of signature (3,1) and metrics of signature (1,3), so a metric of signature (1,3) can quite happily be used to represent 3 spatial and 1 temporal dimensions and indeed often are (with the two alternative signatures described as sign conventions).

The TL;DR (and can’t be bothered with a light bit of maths) version of this is that in spacetime spatial and temporal dimensions are distinguished by having different signs, the result of which means that there is no difference between a spacetime with n spatial dimensions and m temporal dimensions and a spacetime with m spatial dimensions and n temporal dimensions. As we usually assume there is only ever one temporal dimension and three or more spatial dimensions this isn’t a problem, but if you drop this assumption you may well have problems saying which set of dimensions are the temporal ones and which set are the spatial ones.