Multiple Time Dimensions

[Asymptote]

I have, for the past several months, been captivated by the idea of a universe with two separate dimensions of time. Despite having tried numerous methods of visualizing such a universe -- comparing it to a two-variable parametric equation seems to help a bit -- I remain unable to do so. Has anybody else thought about this before, and if so, have you been successful?

[Keiji]

The problem with counting time as a dimension is it makes everything have multiple positions, probably infinite. Suppose you have a particle moving in the positive x direction in a 3-dimensional space, then it exists at all the positions [x,y,z,t]:

[k,0,0,0], [k+1,0,0,1], [k+2,0,0,2], etc.

Now if you were to have two time dimensions and simply said that this particle existed at the positions [x,y,z,t1,t2]:

[k,0,0,0,0], [k+1,0,0,1,0], [k+2,0,0,2,0], etc

then it would exist only for a single moment of the other time dimension...

--------------------------------------------

Alternatively, one can give the movement of time a "velocity". In one-time-dimensional-space, time would have a velocity of k, where if k was positive time would be going forwards and if k was negative time would be going backwards. Due to time dilation, the velocity of time would be different for any given object.

In two-time-dimensional-space, time would have a velocity of ai + bj, where i is a "time vector" of k in the t₁ dimension and j is a "time vector" of k in the t₂ dimension.

If this was the case, there should be some way for time dilation to affect the velocity of time in such a way that not only its magnitude was changed. Assuming this was possible, it would be possible for objects to disappear and reappear from the viewpoint of another object.

*shrug* It doesn't really make much sense, but time dimensions are confusing anyway. >.<

[PWrong]

With one time dimension, an object traces out a 1D curve in (3+1)D spacetime. With two time dimensions, you would trace out a surface in (3+2)D spacetime.

x=x(s,t)
y=y(s,t)
z=z(s,t)
s=s
t=t

normally, velocity is a vector of three derivatives. In (3+2)D, velocity would be a 3x2 matrix of partial derivatives

v = (dx/ds , dy/ds , dz/ds)
......(dx/dt , dy/dt , dz/dt)

Acceleration would be a 3x3 matrix, since you need d²/ds², d²/dsdt and d²/dt²

[Asymptote]

Yeah, trajectories becoming parametric surfaces was about as far as I got. I hadn't thought about velocity and acceleration as matrices, though. Could they be tensors (I never got very far with tensor analysis, I just couldn't wrap my head around it)?

[quickfur]

I'm not sure if this is what you're looking for, but recently I came across the following idea, in which time does seem two-dimensional (however, not in the geometric sense---I'm not sure how to interpret it in that way).

The context is that of building an artificial timeline (say, an imaginary history in some fictional world or some such). The creator starts out with a sketch of what this timeline might look like, and fleshes it out as he goes, perhaps rearranging the order of events in the internal (fictional) timeline to make things flow better. This happens as external time ("real" time) is moving forward. So if you wish to refer to a particular event in this artificial timeline, you refer both to its internal time (the "point in time" of the fictional timeline) and the external time (which "version" of the fictional timeline you're referring to). In this sense, you have two dimensions of time with which to navigate: you could move forward or backward in fictional time (consider the sequence of events in a particular version of the timeline), or move forward or backward in real time (select which version of the timeline you're looking at), and they are mutually orthogonal.

One may then take a step back, and consider a scenario where an entity could simultaneously move forwards in fictional time and in "real" time (in the sense of moving forwards to a newer "version" of the internal time). So you can have "diagonal" movement through 2-dimensional time.

[houserichichi]

Ever heard of F-theory? I'm not sure if it's yet been discredited but I remember it being along the lines of a 10+2-dimensional universe.

A quick internet search reveals that it had over 200 arxiv posts in 1996 but well under 50 as of 2006...that leads me to believe it's either off on the backburner, fringe, or just esoteric at this point.

[Asymptote]

Well, I just read through the article, and it's certainly interesting as a theory. It might have possibilities as far as simplifying string theory. But it seems that one of the time dimensions in question is compactified when the dimensions are mapped onto the two-torus. I was primarily referring to uncompactified time dimensions.

[Eric B]

Wow; two years, I have missed this one. But here's my take on it (which I have added to my page of essays of stuff like this: http://www.erictb.info/math&science.html#space&time)

MULTIDIMENSIONAL TIME?
The easiest way to conceive of two-dimensional time is to picture a bunch of people living and moving about in an area filled with statues. But these statues are really the inhabitants of a perpendicular (2nd) time dimension. Our whole time line lies only in an instant of theirs, so we see them frozen in an instant of time; and their whole time line lies in an instant of ours. So we are frozen in time to them. Now how could we appear frozen to each other? Well, if you could use a stopwatch (like in Twilight Zone story "A Kind of Stopwatch", or the made for syndicated TV movie The Girl, the Gold Watch and Everything), and leave our time, and enter theirs, then everything in this world would stop, and the frozen statues of that world would start up, moving around. If you stopped the watch, then that world would stop, and this one would start again, all the objects in this time dimension picking up exactly where you left off. But now all the people in the 2nd time dimension have changed position from where you left off. So you're really not back in our time dimension, but in a parallel time dimension to ours, in a different position in the 2nd time dimension. From this dimension, you would disappear; going into hypertime is pretty much like disappearing into hyperspace. You could travel through both dimensions at the same time, in which you would see both times running in slow motion. With the right trajectory, you could reach the same point described above by stopping one time, living completely in another, and entering the parallel-to-the-first time. If you change your path to a smaller angle to one of the time lines, that one speeds up, and the other one slows down.
Now all of this could be taking place amidst other statues which are inhabitants of a third time dimension, and you could carry it to any number of dimensions, getting more complex as you increase the number, just like with space dimensions.

Where this might fail, is that the most popular definition of time, is in terms of the causality of events. Events only have one cause. Two dimensional time would be defined with events having two causes. And 3D time would have events with 3 causes. Multidimensional events are not clearly defined in the scenario I've given. Instead, it is more like our time, where if you approach the speed of light relative to earth, then the other frame of reference's proper time will stop. One Scientific American article on the subject a few years ago ("Parallel Universes", Max Tegmark, 5-2003, p.45 http://www.scribd.com/doc/17662852/Parallel-Universes), had a table with several space and time dimension number combinations, and only 3D space/1D time could sustain reality as we know it. Less dimensions of space, "complex structures cannot exist". 0-D space and/or time, a red backgrounded "events are completely unpredictable". More dimensions of space, "atoms are unstable". 2 or more dimensions of space with more dimensions of time together, had a green-backgrounded "events are completely unpredictable". I wonder if the different color meant it was not as bad as the 0-D scenarios (which obviously could not have any events at all). The flipside of our universe: 3D time/1D space, had "fields are unstable". I don’t know what that means, but it is funny that that does not seem to be as restricted as the other combinations. So complex structures can hypothetically exist in that 1D space, unlike ones with less time dimensions? Likewise, 1D space with 4 or 5 dimensions of time also had "atoms are unstable", like the reverse. I wonder what would those universes be like. Why would three or more dimensions of time only be viable with one dimension of space, and not more? Who knows. And who says such "universes" would even have the same laws ("the standard Model"), which governs things such as atomic stability? With different laws; atoms might be stable with more dimensions! (And Michio Kaku and others have theorized, apart of string theory, on other 3D space/1D time universes like this one that might have other than Standard Model laws, including a cataclysmic condition where the laws in this universe could change from the standard model, and matter would all break down and reform!)

[wintersolstice]

I have a few ideas on this aswell.

Time is three seperate things

a dimension: the point in time that something is

a quantity: the period between two events

something else: the progession of events


Space has three dimensions and therefore three quanties

distance: one dimension

area: two dimensions

volume: three dimensions

so if time had two dimensions it would have seperate quantities plus you could say where it is in each time dimension

I've thinking of using this concept to see what "two time dimensions" would be like

When pwoplw say "time isn't a dimension" I think they mean "it isn't just a dimension"

[Eric B]

What they basically mean is that it is not a space dimension.

Then, there's the notion of a third kind of dimensional continuum I toss out here: viewtopic.php?f=4&t=1318

[PWrong]

Basically what we want is a universe with a Lorentzian inner product with two minuses in it. Like (-1,-1,1,1) would give a universe with two space and two time dimensions. The first thing we'd have to do is work out how light behaves in this universe. There'd have to be some equivalent of Maxwell's equations for light to make sense.

Any object would trace out a world-sheet instead of a world line. There'd have to be some kind of entropy law, which would be important to talk about any interesting real-world stuff going on the in the universe. So we'd have to work out how entropy is related to the Lorentzian inner product.

Then we could talk about defining events and draw a bunch of light cones, eventually try to work out some system like causality

Sounds like a lot of work .

[anderscolingustafson]

Multiple dimensions of time would definitely be very weird weirder in fact than multiple spacial dimensions. Events could be in the past in the 1st dimension and past in the second dimension, past in the 1st dimension and present in the 2nd dimension, past in the 1st dimension and futer in the 2nd dimension, present in the 1st dimension and past in the 2nd dimension, present in the 1st dimension and present in the 2nd dimension, present in the 1st dimension and futter in the 2nd dimension, futter in the 1st dimension and past in the 2nd dimension, futer in the 1st dimension and present in the 2nd dimension, and futer in the 1st dimension and futer in the 2nd dimension.

The 2 possible scenarios for 2 time dimensions would be that you could only remember events that were in the past in both dimensions, or you could remember all events except those that were in the futer in both dimensions. If you could not remember any events accept those that were in the past in both time dimensions than you would have the tremendous problem of being blind to events which were in the past in 1 of the dimensions but in the futer in the other. You might start saying a word at 2 different points in time that were both in the futer relative to each other only to discover that both your words met a fraction of a second later and had them either interfair with each other or you had to decide which word would cut the other off. The scenario were you could only remember things if they were in the futer of both time dimensions would be very chaotic perhaps to chaotic for intelligent life. The other scenario were you could remember events unless they were in the futer in both time dimensions would probably work better but it would have the besar implication of remembering certain events without having any information about what those events could remember from the times that were in the futer in both time dimensions relative to you but in the past in 1 of the dimensions relative to those events. This would probably work because even if an event that you had no information about could affect an event that you could remember there would be an infinite number of other events from the time that would have no effect on your time that could have the axact same affect on the event you could remember.

If there were 2 time dimensions to find the number of organisms at any given generation assuming each organism had to offspring at the end of its life in each time dimension you would add the number of organisms that there were in the generation that was the previous generation from the 1st time dimension but same in the 2nd time dimension to the number of organisms in the generation that was the same generation in the 1st time dimension but previous in the second 2nd dimension.

Below I have a model of what population growth of what many organisms particularly micro organisms would look like if there were multiple time dimensions. In the model the numbers going from left to right represent one time dimension those from up to down represent another time dimension and the numbers themselves represent the number of individuals in each generation.

: 1 : 2 : 4 : 8 : 16 : 32 : 64 :
: 2 : 8 : 24 : 64 : 160 : 384 : 896 :
: 4 : 24 : 96 : 160 : 640 : 2048 : 5888 :
: 8 : 64 : 160 : 640 : 2560 : 9216 : 30208 :
: 16 : 160 : 640 : 2560 : 10240 : 38912 : 138240 :
: 32 : 384 : 2048 : 9216 : 38912 : 155648 : 587776 :
: 64 : 896 : 5888 : 30208 : 138240 : 587776 : 2351104 :

[PWrong]

This is very interesting stuff, and like you said, very weird and complicated.

The 2 possible scenarios for 2 time dimensions would be that you could only remember events that were in the past in both dimensions, or you could remember all events except those that were in the futer in both dimensions. If you could not remember any events accept those that were in the past in both time dimensions than you would have the tremendous problem of being blind to events which were in the past in 1 of the dimensions but in the futer in the other.

Memory is closely related to entropy. An important physical question is, "why do we remember the past but not the future?" Is it just a coincidence that computers also remember the past but not the future? It turns out that the "future" is defined by the direction in which entropy increases. So to talk about memory we have to talk about entropy. Presumably entropy would tend to decrease in all "temporal directions" from a "centre point" (0,0) when the big bang happened. That is, s=0,t=0. So if you happen to exist at time (0,1) then entropy is increasing as t changes, and stays constant as s changes. Thus you could remember events in the "t-past" but memory probably wouldn't work either way in the s direction. Assuming physics is invariant under rotation in s-t, you would define your axes based on the gradient of entropy. Perhaps this kind of universe could be indistinguishable from our own.

If there were 2 time dimensions to find the number of organisms at any given generation assuming each organism had to offspring at the end of its life in each time dimension you would add the number of organisms that there were in the generation that was the previous generation from the 1st time dimension but same in the 2nd time dimension to the number of organisms in the generation that was the same generation in the 1st time dimension but previous in the second 2nd dimension.

I don't think you would add them. Maybe you'd do a Pythagorean kind of thing. That would fit the invariance principle that I'm assuming. However this is all assuming it makes sense to talk about "organisms" or "offspring" in this kind of space. In particular what does it mean for an object to split in two if there are two time dimensions?

[Keiji]

Presumably entropy would tend to decrease in all "temporal directions" from a "centre point" (0,0) when the big bang happened. That is, s=0,t=0. So if you happen to exist at time (0,1) then entropy is increasing as t changes, and stays constant as s changes. Thus you could remember events in the "t-past" but memory probably wouldn't work either way in the s direction.

Surely time (0,1) would have less entropy than any other (s,1). Therefore you couldn't remember any events in (s,t>1). But perhaps you could remember all events lying in the circle centred at (0,0) with radius 1, since they have less entropy than time (0,1).

[PWrong]

You're right, I should have said that entropy increases in the radius direction, and not in the angle direction. So you're right that

Surely time (0,1) would have less entropy than any other (s,1).

but not very much less, if s is small.

But perhaps you could remember all events lying in the circle centred at (0,0) with radius 1, since they have less entropy than time (0,1).

Those events would all have about the same entropy. The expected entropy at time (s,t) is an increasing function of the radius s^2 + t^2. This is assuming that there is a single low entropy event which we call a big bang, and that the universe has temporal rotational symmetry.

Btw the reason (which I've just remembered) that memory depends on entropy, is that to create and store memory you have to do work, which makes heat and increases entropy.

[anderscolingustafson]

If there were 2 time dimensions to find the number of organisms at any given generation assuming each organism had to offspring at the end of its life in each time dimension you would add the number of organisms that there were in the generation that was the previous generation from the 1st time dimension but same in the 2nd time dimension to the number of organisms in the generation that was the same generation in the 1st time dimension but previous in the second 2nd dimension.

I just realized I forgot to mention that after adding them up you multiply by 2 because the organisms from each generation combine whith each other and each organism has 2 offspring.

I appolagize for any confusion to anyone who red my post before reading this.

[PWrong]

Thanks, I did wonder about that. However you should have a think about whether this is right.

Suppose the times in your table are given in "days" i.e. each organism splits in two after exactly one day. Notice that at time (3,4) you have 2560 organisms. Assuming a Pythagoras type rule, this is equivalent to waiting 5 days in some other "temporal direction", a combination of s and t. However on day 5 in the s direction, you only have 16 organisms.

Just by rotating the axes we use to measure time, you would completely change the population. Do you see the problem with this?

[anderscolingustafson]

So if I understand you correctly your saying that there would be a problem of how to determine exactly were to put each line from which each time dimension is measured from.

I see your point here but I don't realy understand what the alternative to having an angle at which time would be measured from would be.

[PWrong]

Well the alternative is to treat the time dimensions the way we treat our own spatial dimensions i.e. with no unique coordinate system. Look around you, do you see an x-axis? Could you point in a direction parallel to the universe's y axis? In the 2T universe, you could have clocks with fixed s,t axes, but these would be arbitrary.

For your formula, you could try something like 2^sqrt(s^2+t^2). This is rotationally invariant, so it makes no difference if you choose your axes differently.

However all this kind of assumes that entropy remains constant over time. With total entropy as a function of s and t, you will always get a well defined pair of axes, one equal to the gradient and one perpendicular to it. You could almost call them "real time" and "extra time". Since nothing interesting ever happens without increasing entropy, everything would happen in real time, including birth. So your formula would be simply 2^R where R is the amount of real time elapsed, which is also the change in entropy. If the entropy function is smooth, then locally you would in fact have a fixed s axis where things actually happen, and a fixed t-axis where "change" means something different or doesn't even exist. Very confusing.

[anderscolingustafson]

If entropy remained constant throughout time then what would make the 2 time dimensions time dimensions rather than spacial dimensions. In our universe the fact that entropy allways increases from past to futer is the thing that seperates time from the spacial dimensions. time also runes perpendicular to the three spacial dimensions so that only one of our four dimensions can be measured as time. I still dont see why living in a 2t universe would change this.

[PWrong]

Maybe that's the reason there is only one time dimension. If there was another time dimension it might be somehow equivalent to having an extra a spacial dimension. However I wouldn't jump to that conclusion immediately. The other difference between time and space crops up in relativity. 2+2D is a different kind of space to 3+1D, which is Minkowski space. However talking about relativity in 2+2D will be difficult since we can't really talk about the speed of light.

The problem with the entropy thing, is that entropy can't increase in all directions at once at all possible times. It might increase in both the (1,0) direction and the (0,1) direction. But then it would remain constant in the (-1,1) direction, or in some other direction. Then we would call (1,1) the real time axis, and (-1,1) the extra time axis.

[Eric B]

I discuss what distinguishes one kind of dimension from another, here: viewtopic.php?f=4&t=1318&p=15144&sid=6cdee67a8ecbbcb0b4d4cf42cd4e17d8#p15144

The universe consists of different possible events, which are arrangements of matter and energy which change from event to event.
Time is the dimension in which particular arrangements are actualized through chains of causation. We are involuntarily dragged through different arrangements of matter and events.
Space is a realm in which different arrangements of matter exist simultaeneously, and you can freely travel from one event or configuration to another and back.

2D time would mean two dimensions of actualization of different universal arrangements of matter. I imagine if dragged through both at the same time and rate, it would look like one dimension. So I'm not compleely sure about that. But then space and time dimensions are said to be interchangeable; so it might look like a space dimension, as was said, if we weren;t being dragged through it.

[anderscolingustafson]

I also think planets and other objects that have orbits might have spherical orbits if time were 2d but its tough to visualize a spherical orbit though. It might also be possible that organisms might have planar digestive systems if time were 2d and rivers could potentially be planar if time were 2d. Living in three spacial dimensions with one time dimension would probably be like living in two spacial dimensions with one time dimension. I think adding a time dimension would have a surprisingly similar effect to removing a spacial dimension.

[PWrong]

Why?

[anderscolingustafson]

Why?

Which part of my post are you asking about with this question?

[PWrong]

Most of it. I'm just wondering how you conclude these things.

I can see how you got the 2D digestion, by naive addition. But it's hard to imagine what "digestion" could possibly mean in a world with two time dimensions. The idea of digestion is that you have some food going in at one place and time, you extract something useful from it, and it comes out in another place at a later time. Digestion is completely tied up with ideas like "cause and effect", "change" and "start and finish", which are vastly more complicated in a 2T universe.

As for rivers, these are kind of defined as being approximately 1D. I suppose you could define it as a body of moving water, in which case you'd need two directions for it to move in. Is that what you were thinking of?

Living in three spacial dimensions with one time dimension would probably be like living in two spacial dimensions with one time dimension. I think adding a time dimension would have a surprisingly similar effect to removing a spacial dimension.

This last bit is perhaps the most confusing

[Keiji]

Living in three spacial dimensions with one time dimension would probably be like living in two spacial dimensions with one time dimension. I think adding a time dimension would have a surprisingly similar effect to removing a spacial dimension.

This last bit is perhaps the most confusing

I think he meant that living in three spacial dimensions with two time dimensions would be like living in two spacial dimensions with one time dimension.

I can see what he's getting at - needing more dimensions for things to move in means you in some sense get less dimensions of freedom. But what is being forgotten here is the fact that the dimension being moved in isn't fixed - it can twist and turn in all sorts of ways. Put more simply, in (3+1)D, your spacial dimensions are 1 vertical, 1 frontal and 1 lateral. In (3+2)D, you more or less have 1 vertical and 2 frontal - so you have no lateral dimensions.

Perhaps, though, a (4+2)D universe would be more stable than a (4+1)D universe? I've come to the conclusion in the past that what makes 4D and up unstable is the existence of more than one lateral dimension, but in (4+2)D you'd have 1 vertical, 2 frontal and 1 lateral dimensions, avoiding that problem...

[PWrong]

In (3+2)D, you more or less have 1 vertical and 2 frontal - so you have no lateral dimensions.

Good point. That would be kind of restrictive in a way. There might be some similarity to 2+1T then, but not that much.

I think the problem is we have no way of talking about this. We're used to talking about time with words like "before", "start" and "then". We don't think of it as a parameter.
To talk about this properly, we'd have to invent and understand a number of new concepts. All of physics would be completely different. You have velocity as a 2x3 matrix instead of a vector, and acceleration as a 2x2x3 tensor. That's a huge problem for the orbit question. Newton's law of gravity no longer works, because you have a tensor on one side and a vector on the other.

I've come to the conclusion in the past that what makes 4D and up unstable is the existence of more than one lateral dimension

The distinction between vertical and lateral only really applies when there's some kind of constant field though. In the case of gravitational or atomic orbits, the instability comes from the inverse cube law. I suppose there might be problems for life having an extra lateral dimension, if we got past the orbit problem.

[Keiji]

You have velocity as a 2x3 matrix instead of a vector, and acceleration as a 2x2x3 tensor.

Where did this come from?

[PWrong]

In 3+1D, velocity is the vector (dx/dt, dy/dt, dz/dt). In 3+2D, velocity is
(dx/ds dy/ds dz/ds)
(dx/dt dy/dt dz/dt)

For acceleration you have to differentiate again. There's be a bit of redundancy since d²x/dsdt = d²y/dtds. So acceleration will have 9 elements.