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Note: This is a rewrite of the original question, which was titled What would time be for 2D beings?

In my current, non-physicist's understanding, every instant of our three‑dimensional world is just another 'slice' of a four‑dimensional body. I don't mean that as an analogy, but quite literally... Obviously, it would not be a straight 'slice', it would still be bent and curved by gravity, speed and other relativistic factors. Is this wrong?

Also, both 'spacial' and 'temporal' dimensions are — in my mind — fundamentally the same thing, given different names because we experience them differently because of our nature. I've had people explicitly say in the comments that this is wrong and that time and space are not the same 'type' of dimension. I'd like to understand what are the fundamental differences.

In my original question I used these two assumptions of mine (that space is a slice of time and that all dimensions are fundamentally the same) to make an analogy.

I noted that taking two-dimensional slices of a three-dimensional body — just as a slice of four‑dimensional time is three‑dimensional space — and displaying them is rapid succession looks like a bunch of matter that is changing over time (like in this brain scan below).

Brain scan

Assuming all dimensions, temporal or spatial, are fundamentally the same, would that mean that for a hypothetical two‑dimensional being time would be the thrid dimension, not that fourth?

The same question in a more general form: for any $n$‑dimensional being, would time for it be the $n+1$ dimension?

In particular, what would time be for a being living in a hypothetical fifth dimension?

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I did not know there was a Theoretical Physics board here on SE. If you think this question is better suited there, please move it. theoreticalphysics.stackexchange.com – Paul Manta Feb 4 '12 at 10:27
The theoretical physics board was designed to be research level. They wouldn't be happy about your question. – Nick Kidman Feb 4 '12 at 15:02
You might find this 1-minute video, Distance and Special Relativity: How far away is tomorrow?, interesting. – Warrick Feb 4 '12 at 18:00
You can perform rotations in spatial coordinates, but not in temporal directions. Or can you ... ? – ja72 Aug 29 '12 at 14:15

5 Answers

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I think you are correct for this hypothetical 2D being. However, you should be aware that time is not just a dimension. In space-dimensions, you can in principle move freely forward and backward, while in time, your motion is fixed.

With respect to the brain scan: this way of visualisation is chosen for simplicity. A regular 3D image, where you can look at any depth you want, will give a clearer image of what is happening in this third dimension. Some information is a bit lost for the observer: you clearly see structures in the x,y-plane, but for vertical coordinate, it is not that obvious.

Some bit off-topic reading material may be: http://en.wikipedia.org/wiki/Flatland

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"you can in principle move freely forward and backward, while in time, your motion is fixed" -- But is that a characteristic of the dimension, or a limitation of our technology and/ or dimensional nature? Afaik, there's no evidence that time travel in the past is impossible, so there's nothing that can be said for certain about that for now. – Paul Manta Feb 4 '12 at 9:30
"With respect to the brain scan: this way of visualisation is chosen for simplicity" -- I know, but that is besides the point for this question. :) I picked that image because it illustartes how 2D scans in rapid succession look like a substance that evolves over time. – Paul Manta Feb 4 '12 at 9:30
+1 "Flatland" seems very interesting. Thanks for pointing it out. – Paul Manta Feb 4 '12 at 9:33
You may be right, but moving backwards in time, will not be as convenient as moving backwards in space in the foreseeable future. I assumed the same constraints for the 2D being as we perceive in daily life. – Bernhard Feb 4 '12 at 9:37
@PaulManta: Space dimension is not the same as time dimension. It has a special significance in GR. A 2D person would experience the time as we are experiencing it! a monotonically increasing function. So, it will be only that there's won't be the "thickness" factor, but as far as time is concerned, it will be same as ours. – Vineet Menon Feb 4 '12 at 13:09
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My take is time is indeed fundamentally different from space, as, for example, time enters the invariant interval with a different sign: $(ds)^2=(dt)^2-(dx)^2-(dy)^2-(dz)^2$.

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That is a good argument, and one that I was hoping I would get. But how does the generalized form of that equation look like? Or, at least, how does it look if you introduce another dimension? – Paul Manta Feb 4 '12 at 16:51
@PaulManta: The way I read that formula is the time dimension is just like any spatial dimension, provided you measure it in imaginary units (multiplied by $i$). As far as adding dimensions, you can modify the formula in any way that seems like it might help you understand. – Mike Dunlavey Feb 4 '12 at 17:07
@PaulManta: generalized in what way? If you add another spatial dimension $w$, just add $-(dw)^2$ to the expression above. – akhmeteli Feb 4 '12 at 17:16
@PaulManta: For example if two events are separated by $dx=5$ light seconds, and $dt=4i$ seconds, then the distance between them is $\sqrt{dx^2+dt^2}=\sqrt{25-16}=\sqrt{9}=3$ light seconds. – Mike Dunlavey Feb 4 '12 at 17:28
I'm not saying this answer is useless but it only points out how time is modelled different than space in relativity theory and is not saying anything about the sense in which they are different. It's a statement about how the abstractions 'space' and 'time' and our observation gets translated into a theory and is therefore a statement like 'we seem to always experience only one time' itself. – Nick Kidman Feb 4 '12 at 18:01
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This is related in several ways to time traveler paradoxes. What if you were able to go back in time and kill your grandfather before he had a child? But this is a silly and undisciplined way to think about the problem.

A much better way is to use the principles that working physicists use in their descriptions of reality. Now, one of the fundamental principles is the conservation of matter. Matter is neither created nor destroyed, it is simply changed from one form to another while undergoing some kind of work.

A time traveler going back in time to kill his grandfather is changing the material content of the universe he is going back to. In accepting the paradox as a possibility, we are forced into a dubious proposition: give up this conservation principle as a way of describing objective reality.

If we allow that we can get something from nothing, then perpetual motion machines follow on as a consequence. Perpetual motion machines break the sense of proportionality which all the laws of physics as we know them depend upon.

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Time and space is a way of splitting the set of all space-time events into two sets:

Space is the family of sets of space-time events simultaneous with one another, with each element of a set paramaterised by three real numbers called the space coordinates; each set paramaterised by a real number t called time.

Alternatively, time is the family of sets of space-time events that aren't simutaneous with one another, each element of a set parameterised by a real number t; each family paramaterised by three real numbers called the space coordinates.

On the one hand space and time are identical in both being partitioning functions; on the other, they're different partitioning functions.

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Isn't it a circular definition to simply say that "time" is the dimension along which spacetime events aren't "simultaneous?" Also, "x" is family of sets of spacetime events whose x coordinates are not the same. That still doesn't really say anything about what makes t different from x. – Larry Gritz Feb 8 '12 at 19:40
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The sign is actually metric tensor $g_{\mu\nu}$,It takes the value $-1$,$1$,$1$,$1$ only in Minkowski space time.In here you may find what you need http://en.wikipedia.org/wiki/Metric_tensor

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