Difference Between Temporal and Spatial Dimensions?

Factual Questions
1

Can someone say, or point to a page somewhere that says, what the difference is, exactly, between a temporal and a spatial dimension?

I assume there’s an answer to this which is relatively noncontroversial, since I see physicists here and elsewhere defining spaces all the time in terms of the number of spatial and the number of temporal dimensions they are supposed to have.

Also, if possible, can you say something about what it would mean for a space to have two temporal dimensions or more? (Of course, the answer to this may just fall out of the definition you’ll be giving me of a temporal dimension.)

Thanks,

-FrL-

2

Does this help?

3

[QUOTE=aktep]
Does this help?
[/QUOTE]

While the text describing the jump from imagining a fourth dimension of time to a fifth dimension of (what I might call) “metatime” pleased me on an aesthetic level, (and because in some important ways it matched my own private speculations as to what it might mean for there to be two temporal dimensions,) nevertheless, the work you’ve linked to unfortunately does not appear to me to be a reputable source for information on this topic.

-FrL-

4

It depends on what you mean by “the difference”. If you mean the difference in how they’re used in relativity, then it’s a factor of i (i.e., the square root of -1). I.e., the distance formula for points in spacetime is:
d = sqrt(x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] + (it)[sup]2[/sup]) = sqrt(x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] - t[sup]2[/sup])

Note the extra minus sign, whereas the traditional distance formula has only plus signs: d = sqrt(x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup])

(Technical note: Normally, you also have to include the speed of light as a conversion factor from time to space, but you can choose units where it’s equal to 1 and can thus be omitted.)

See spacetime for more information.

However, there are other “differences” between space and time. E.g., the “cause” of an event always proceeds it in time. Also, the entropy of a closed system tends to increase over time. These are each examples of the arrow of time – i.e., they are processes that distinguish the future from the past. (Some subatomic particle interactions also exhibit time-asymmetry.) There’s no equivalent preferred direction for space.

Also, we remember the past but not the future. But this is probably a consequence of one of the above-mentioned arrows of time, which may themselves be related.

See arrow of time for more information.

5

[QUOTE=aktep]
Does this help?
[/QUOTE]

Concerning that site and it’s author… Does that in ANY way resemble what physics holds to be the current understanding of our universe? Because it puts me in HIGH skepto-mode. Anyone?

6

The wikipedia spacetime article also addresses the question of what the universe would be like with different numbers of dimensions, but I can’t really vouch for its accuracy.

I think it’s safe to say that our notion of causality would be pretty screwed up with multiple time dimensions. Assuming, that is, that the second time dimension was even relevant to causality. If instead we perceived it like a spatial dimension, but with a minus sign in the metric (i.e., the distance function), then we’d have some pretty weird effects involving rotation.

Normally, if you rotate a stick from a horizontal alignment to a vertical alignment, the horizontal distance between the endpoints decreases as the vertical distance increases. Otherwise, we’d say that the length of the stick had changed. However, with our new metric the horizontal distance would increase as the vertical distance increased. With our new metric, we’d say the length of the stick stayed the same, but by our old standards we’d be stretching or contracting the stick with each rotation. And it would be like that not just for sticks, but for everything, including our own bodies. Of course, this is only true if our plane of rotation includes both a space dimension and a “time” dimension.

Of course, if we added this extra dimension to our existing dimensions (as opposed to replacing one of the space dimensions), then we’d have more than four total dimensions – which would introduce a lot of weirdness in its own right.

7

[QUOTE=tim314]
It depends on what you mean by “the difference”. If you mean the difference in how they’re used in relativity, then it’s a factor of i (i.e., the square root of -1). I.e., the distance formula for points in spacetime is:
d = sqrt(x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] + (it)[sup]2[/sup]) = sqrt(x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] - t[sup]2[/sup])

Note the extra minus sign, whereas the traditional distance formula has only plus signs: d = sqrt(x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup])

(Technical note: Normally, you also have to include the speed of light as a conversion factor from time to space, but you can choose units where it’s equal to 1 and can thus be omitted.)

See spacetime for more information.

However, there are other “differences” between space and time. E.g., the “cause” of an event always proceeds it in time. Also, the entropy of a closed system tends to increase over time. These are each examples of the arrow of time – i.e., they are processes that distinguish the future from the past. (Some subatomic particle interactions also exhibit time-asymmetry.) There’s no equivalent preferred direction for space.

Also, we remember the past but not the future. But this is probably a consequence of one of the above-mentioned arrows of time, which may themselves be related.

See arrow of time for more information.
[/QUOTE]

Thanks for that.

Is there a connection to be made between the i factor you mention and the directionality of time? For example, does the time-assymetry of the subatomic particle interactions you mentioend somehow result from the fact that distance in spacetime is properly measured by multiplying the time distance by a factor of i?

-FrL-

8

[QUOTE=tim314]
…However, with our new metric the horizontal distance would increase as the vertical distance increased…
[/QUOTE]

Thanks again for all that.

Now, from the part I quoted above, it sounds to me like in such a situation, no sense could be made of the notion of actually completing the rotation–since horizontal distance would be increasing the whole time, then whatever vertical distance you’ve achieved at any point in the rotation, the horizontal distance continues to be non-zero. (This all assumes both distances started out positive.) If the horizontal distance is alays non-zero, then the stick is never actually fully vertical. So it seems like it’s never possible to take a horizontal stick and complete a ninety degree rotation to vertical.

But this, in turn, just seems to mean its never possible to rotate anything 90 degrees, at all.

Of course I’m sure I’m importing three dimensional imagery and intuitions into what you’ve said. And I’m not sure whether the i lurking in all the equations describing these movements makes it somehow false that continually increasing positive distances can’t ever reach zero distance. But still, do you think there’s anything to be said addressing the observations I’ve just tried to make?

-FrL-

9

[QUOTE=cmyk]
Concerning that site and it’s author… Does that in ANY way resemble what physics holds to be the current understanding of our universe? Because it puts me in HIGH skepto-mode. Anyone?
[/QUOTE]

I tried to have a look at the site. I got as far as him ripping off Flatland, and lost interest.

It’s true that string theory proposes that there are extra spatial dimensions addition to the usual three. But these extra dimensions are “curled up” in such a way that we can’t see them. They only matter on the level of subatomic particles.

As an analogy, consider a straw – the kind you drink through. It’s basically a two dimensional surface – specifically, a cylinder – with length and a circumference. But if the circumference of the straw is small enough, then the straw just looks like a one dimensional line segment. In a roughly similar way, the extra dimensions of string theory aren’t observable except on incredibly small scales.

This theory is a proposed quantum gravity theory – basically, a way to combine the two very successful theories of general relativity and quantum mechanics, which as it stands don’t work well together. We need a theory that reconciles the two if we want to understand what happens in situations where quantum and gravitational effects are both important (like, say, the birth of the universe, or the center of a black hole). For various reasons a lot of physicists (but not all) think string theory is the most promising such theory – but as of yet, there’s no direct experimental evidence for it. Worse, it’s not even clear how the theory could be tested experimentally.

So, suffice it to say these ideas are still controversial. Whether the above website has anything to do with any of this stuff (other than a vague appropriation of the idea of “ten dimensions”) I don’t know. Like I said, I quit listening with the ripoff of “flatlanders.”

10

[QUOTE=Frylock]
Thanks for that.

Is there a connection to be made between the i factor you mention and the directionality of time? For example, does the time-assymetry of the subatomic particle interactions you mentioend somehow result from the fact that distance in spacetime is properly measured by multiplying the time distance by a factor of i?

-FrL-
[/QUOTE]

Well, the time-reversal asymmetry in subatomic particle interactions is an experimentally-observed phenomenon, which so far as I know is still not fully understood from a theoretical standpoint.

(Incidentally, this effect is called CP-violation, where C and P are the names of two symmetries which together are equivalent to time reversal.)

But so far as I know, it’s not directly connected to the difference in the metric. At any rate, it’s really just the factor by which the space and time dimensions differ that matters. It’s just as legitimate to apply the factor of i to the spatial dimensions, writing:
d[sup]2[/sup] = - x[sup]2[/sup] - y[sup]2[/sup] - z[sup]2[/sup] + t[sup]2[/sup]

instead of:
d[sup]2[/sup] = x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] - t[sup]2[/sup]

This choice is sometimes called the signature of the metric. The important thing is to remember your choice of signature and be consistent with it throughout the calculation.

11

Scientific American had a special edition in June 06 dedicated to time. This probably will only help to add to the confusion, but it’s interesting.

These are some of the quotes from one of the articles:
“neither scientists nor philosophers really know what time is or why it exists”
“The passage of time is probably an illusion”
“According to relativity, simultaneity is relative. Two events that occur at the same moment if observed from one reference frame may occur at different moments if viewed from another.”

12

[QUOTE=Frylock]
Thanks again for all that.

Now, from the part I quoted above, it sounds to me like in such a situation, no sense could be made of the notion of actually completing the rotation–since horizontal distance would be increasing the whole time, then whatever vertical distance you’ve achieved at any point in the rotation, the horizontal distance continues to be non-zero. (This all assumes both distances started out positive.) If the horizontal distance is alays non-zero, then the stick is never actually fully vertical. So it seems like it’s never possible to take a horizontal stick and complete a ninety degree rotation to vertical.

But this, in turn, just seems to mean its never possible to rotate anything 90 degrees, at all.

Of course I’m sure I’m importing three dimensional imagery and intuitions into what you’ve said. And I’m not sure whether the i lurking in all the equations describing these movements makes it somehow false that continually increasing positive distances can’t ever reach zero distance. But still, do you think there’s anything to be said addressing the observations I’ve just tried to make?

-FrL-
[/QUOTE]

This is essentially correct. In fact, the same thing shows up in ordinary spacetime – e.g., you can’t continuously rotate a vector pointing forward in time so that it points backwards in time. (That’s actually a 180 degree rotation, but the 90 degree rotation is forbidden too – basically it would mean turning a vector pointed in the time direction so that it points in the space direction.) This is related to the restriction that nothing can be accelerated to speeds greater than the speed of light.

For those familiar with group theory, the point is that SO(1,3) – the group of (orientation-preserving) rotations in 4-dimensional spacetime – is disconnected. This differs from the the SO(3) which represents (orientation-preserving) rotations in 3-dimensional space, and is connected. This difference is a consequence of the metric.

One might think this explains the arrow of time, but things aren’t actually that simple. For one thing, just because we can’t change directions in time doesn’t mean we couldn’t have been going in the opposite direction to begin with. In fact, this is precisely what antimatter does. So the question then is “Why are we made of matter and not anti-matter?” The answer, or part of the answer, may very well be the aforementioned CP-Violation.

(Also, none of this addresses the thermodynamic arrow of time – i.e., the question of why entropy is increasing. Entropy increase makes statistical sense, but only if the universe began in a state of very low entropy, and it’s not at all clear why this should have been the case. See Loschmidt’s paradox.)

Perhaps I should also elaborate on what rotation through spacetime has to do with the “universal speedlimit.” Changes in speed can be treated as rotations in spacetime. Here’s a somewhat simplified explanation. Everyone travels through spacetime with the same speed, but in different directions. If the angle between your trajectory and the time axis is greater, then you will cover more distance per unit time – thus, this corresponds to a greater speed through space. (Note that speed through space is different from speed through spacetime.) I’m glossing over the fact that the orientation of the axes is different for observers in different reference frames. You’ll always be moving along the time axis in your own reference frame – which is just another way of saying that your speed is always zero relative to yourself.

13

[QUOTE=cmyk]
Concerning that site and it’s author… Does that in ANY way resemble what physics holds to be the current understanding of our universe? Because it puts me in HIGH skepto-mode. Anyone?
[/QUOTE]

Just to add to my previous comments, I forced myself to watch the rest of the video “explaining” ten dimensions, and I now feel I can definitively declare that site bullshit. First of all, it’s conflating string theory and the many worlds interpretation (a minority interpretation of quantum mechanics.) The two don’t have anything to do with each other. The extra dimensions in string theory are spatial dimensions, not alternate realities or timelines. They also are ultra-microscopic – you’re not going to find another version of you living in one of them. The “explanation” also mentions alternate initial conditions for the universe – while tangentially related to string theory (see string landscape), this also doesn’t have anything to do with the extra dimensions.

The claims that the extra dimensions of string theory represent alternate timelines, or divergent realities, or paths for somehow moving between these realities (which isn’t possible even in the many-worlds interpretation) are total sci-fi baloney. That’s not what string theory is talking about, at all.

The claim that the existence of ten dimensions has something to do with the total number of possible alternate universes is also complete nonsense. Extra dimensions in string theory are required for anomaly cancellation.

As far as I can tell, the author read the jacket of a book on string theory and then filled in the gaps in his knowledge by plagiarizing science fiction. He’s deliberately misleading people – no one who had read even one book on string theory could possibly misunderstand it so badly. They might not understand it at all, but they certainly wouldn’t end up believing this nonsense. Unless of course the one book they read was written by a fraud like this guy.

14

I’m a bit late to the party here, but if you want some further reading you can try Greg Egan’s excellent essays on relativity and other subjects:

http://gregegan.customer.netspace.net.au/FOUNDATIONS/index.html#CONTENTS

His section on special relativity in particular is a excellent primer on the geometry of space vs. time.