Q re: the nature of dimensions

Factual Questions
1

Can someone with a physics background explain the difference between the dimensions that we perceive (i.e. 3 of space, 1 of time)? From the reading I’ve done on the subject, I understand that the space-time construct is best explained in terms of either ten or twenty-six dimensions, and that degeneration is responsible for the four dimensional space-time in which we construct our own physical laws, but I’m having difficulty with the concept of time and space being distinct orthogonal entities - if that is the case, are higher (n) dimensional spaces also combinations of n-1 space dimensions and 1 time dimension, or some other combination incorporating two or more orthogonal time dimensions?

Does this question make any sense?

2

The difference between the space dimensions and the time dimension is basically a minus sign. To illustrate:

Suppose we have two points in a Cartesian coordinate system. If I ask you how far apart they are, you might (for instance) tell me the difference in their X coordinate. But that’s not a very useful measure: If I set up a different coordinate system, the difference in X would have a different value. Much more meaningful would be the quantity (x[sub]2[/sub] - x[sub]1[/sub])[sup]2[/sup] + (y[sub]2[/sub] - y[sub]1[/sub])[sup]2[/sup] + (z[sub]2[/sub] - z[sub]1[/sub])[sup]2[/sup] : You may recognize this as the square of the distance between the two points, and that will be the same regardless of what coordinate system you use.

In three dimensions, at least. In relativity, we find that that 3-d distance measure can still vary in different reference frames. Specifically, if one reference frame is moving relative to another, that 3-d distance will be different in those two frames, just like the difference in X was different in different coordinate systems. However, if you instead take (x[sub]2[/sub] - x[sub]1[/sub])[sup]2[/sup] + (y[sub]2[/sub] - y[sub]1[/sub])[sup]2[/sup] + (z[sub]2[/sub] - z[sub]1[/sub])[sup]2[/sup] - (t[sub]2[/sub] - t[sub]1[/sub])[sup]2[/sup], where t is time, you’ll find that for that distance measure, the distance between two points in spacetime again doesn’t depend on your coordinate system. Notice that the time part is subtracted, rather than added: This is the reason why time seems so different than space.

As for your second question, in all of the various extensions to relativity with ten or more dimensions, there is still only a single timelike dimension. One can discuss physics in any number of dimensions if exactly one of them is timelike, or if all but one is timelike, but for any other number of timelike dimensions, things start getting really weird.

3

Thanks for the reply.

Yes, I recognize the distance formula - via Pythagoras’ theorem, this can be extrapolated to give the distance between two points in any number of spatial dimensions. I did not, however, realize that the time component is negative.

Is there an explanation for the time term being negative, other than that being an arbitrary convention? The further apart two events are in time, the smaller the linear distance between them would be - this does not intuitively make sense to me.

4

On the question of more than one dimension -

In 1919 it was theorized that we have more than just 3 space dimensions. Klein (what is the other “K” one?) stated that it could be possible to have curled up dimensions at every point of space time. Jumping ahead a bit, superstring theory requires there to be 10 or so space dimensions because, mathmatically, the probabilty that events occur is infinity if we do not allow for these extra dimensions. In this case, there are 9 space dimensions and 1 time dimension. As far as I am aware, there is only one time dimension regardless of the number of space dimensions.
And, for what it is worth, our perception of the dimensions allows us for three indepent movements in space: left-right, back-front, up-down. Contrast this with time, which only allows for one movement (that we are aware of), and you can see why there is a difference between space and time dimensions. Time is still a medium for which things change, but it moves in one direction.

5

Well, I suppose it’s arbitrary to say that the time dimension has a negative sign and the space dimensions have a positive sign, as opposed to the other way around. However, what is not arbitrary is that time has the opposite sign as the space dimensions. There is a reason for this, which has to do with relativity.

Let me see if I can give a simplified explanation, without too much math. Hopefully I won’t simplify it to the point where it becomes wrong, but if I screw up, I’m sure someone will correct me.

Basically, according to Einstein’s theory of Relativity, if one person is moving relative to someone else, they may measure the distance between two objects and get different answers. Likewise, they may measure the time elapsed between two events and get different answers. The reason these discrepancies aren’t usually noticable in our day-to-day life is because even though I may be moving relative to you, I’m moving slowly (meaning at much, much less than the speed of light), so the differences in our measurements are very small.

At any rate, there are somethings which we won’t disagree on. One of those is the speed of light (in vacuum.) If you see a beam of light, and I see a beam of light, we will both agree that it’s moving at about 300,000 km/s. This is surprising, because we may disagree on the speeds of a lot of other things. But that’s the way it is.

Another thing we should agree on is the space-time interval between two events. In other words, if a particle travels from point A to point B, we may disagree on the distance between A and B, and we may disagree on the amount of time it took the particle to travel that distance, but we will agree on the distance that the particle traveled in four-dimensional space-time.

For instance, let’s consider a case where we agree on the time but not on the distance travelled in each direction. Suppose I thought the particle traveled three kilometers in the x-direction and four kilomerters in the y-direction, and it took it an hour to do so. (I don’t mean 3 km in the x-direction and then 4 km in the y-direction, I mean both at once, like it’s traveling in a “diagonal” direction relative to my axes.) But you think the particle traveled four kilometers in the x-direction and only three kilometers in the y-direction, again taking an hour to get there. So we disagree on how far it went in the x-direction, and we disagree on how far it went in the y-direction, but we agree on the overall distance travelled in space-time. (In this case, I simplified things by saying we already agree on how far it travelled in the time dimension, so if we agree on how far it travelled in space then we agree on the space-time interval as well.) Note that since you measured a smaller y-distance, you had to measure a bigger x-distance to get the same total distance. So clearly x and y distances enter into the total distance in the same way, i.e. they have the same sign (which we’re calling positive).

Now, let’s say we want to measure how long it takes a beam of light to travel between two points. But let’s say we disagree on the distance between the points. You measure the distance to be twice as big as I measure it. But since we agree on the speed of light, this means you must also measure the time it takes to travel between the points to be twice as much. If it’s going at the same speed but going twice as far as I thought, then it takes twice as long. So in this case, you get a bigger distance in space and a bigger distance in time (whereas in the above example you got a bigger distance in one direction and a smaller distance in the other one.) So in order for us to agree on the distance the beam of light travelled in space-time, the time dimension has to enter into our distance formula with a negative sign. That way, the fact that you measured a bigger time interval can compensate for the fact that you measured a bigger space interval.

So, that’s why the time dimension gets a different sign than the space dimensions, more or less. Does that make sense?

6

Eloquent explanation. Thanks.

7

You’re welcome.

You might also wonder how we know that the “distance” we measure in space-time (i.e. the space-time interval) should be the same for each of us, even if I’m moving relative to you. Why not say “Let’s just square the distance in every direction, including time, then add them all together, take the square root, and call the result ‘distance’, even though we might measure different values for it.” The short answer is that if you defined the space-time interval in that way it wouldn’t be nearly as useful.

The long(er) answer, without going into a bunch of math and talking about vectors and the definition of a metric space and so forth, is that generally when we say “distance” we’re talking about a quantity that doesn’t change under translations or rotations in the space. If I carry my laptop computer across the room or I spin it around, the width of the screen should remain unchanged. Likewise, if you shift or rotate an event in space-time (or more accurately if you apply a “Lorentz transformation” to it), so it’s now a different length in the time dimension and a different length in one of the space dimensions, it should still have the same length overall, or else we haven’t made a very appropriate choice for what we’re calling “length” or “distance” in space-time.

But what about the other thing that I told you we should both agree on: the speed of light? Well, there are theoretical reasons for thinking the speed of light will be the same no matter who measures it, and there are experiments that confirm this. But it took the genius of Einstein to recognize that this is what the theory was saying, and to realize the consequences of it, despite the fact that it flies in the face of common sense. (Common sense says that if I chase after a beam of light, and I myself am travelling at half the speed of light, then the beam of light should only appear to move away from me at half the speed of light. But common sense is wrong. We’re used to seeing things move at relative speeds much less than the speed of light, but we can’t apply the intution we gain from those observations to things moving at light speed or near light speed and get answers that are anywhere close to correct.)

8

Is the discrepancy between the apparent behaviour of light and the “common sense” that you speak of an artifact of our attempt to describe this behaviour with physical laws in four dimensional space-time, when in fact this could be a higher dimensional phenomenon with governing laws which are much simpler (and which correspond to the common-sense criteria) when expressed in higher dimensions?

9

One other thing – and here I’m going into an area that I don’t know much about , so if I’m wrong then someone please correct me – I think the “10 or 26 dimensions” thing might be out of date. There were a bunch of versions of String theory that called for the universe to have 10 dimensions, but there is now a theory (known as M-Theory) that requires an 11 dimensional universe. This theory incorporates five different versions of 10-dimensional string theory, as well as an eleven-dimensional theory called “Supergravity”. Basically, as I understand it, M-Theory is equivalent to these theories under certain choices of whatever parameters the theory depends on. So, as far as I know, the most commonly held view currently is that the universe is actually eleven dimensional.

I’m not sure where the idea that the universe has 26 dimensions came from, although I have heard that number as well. I also don’t know if whatever theory proposed that is still supported by many people or not.

–

I just saw your question. I’ll answer it in my next post, since it’s not really related to what I just posted above.

10

Kaluza.

11

The problem isn’t with how many dimensions we’re seeing the universe in – basically, in thinking that Gallilean relativity (I’ll explain this below) applies to all cases, we’re mistaking a limiting case for a general rule. Don’t worry if that’s confusing, I’ll ellaborate. I made a brief reference to this when I mentioned “chasing a beam of light at half the speed of light”, but I can go into a bit more detail.

Basically, according to Gallilean relativity (named for Gallileo Gallilei), if you are moving in the x-direction relative to me at a speed v, and we both measure the x-component of a velocity, the value that you measure will be v less than the value that I measure.

If that’s not clear, let me give an example:

Suppose I’m sitting next to the road, and I see a car go by at 45 mph. (I used kilometers in my previous examples, but since I’m an American I’m used to seeing road speeds in miles per hour, so that’s what I’ll use here. Anyway, the units don’t really matter for the purposes of this example.) Then I see another car go by at 30 mph. Suppose you are in the second car, and are trying to judge how fast the first car is moving. To you, it will only appear to be moving at 45 mph, because it is only going 45 mph faster than you.

You might say, “Hold on, I can just look down at my speedometer and see that I’m traveling at 30 mph, so I know the first car is really going 45 mph!” And yes, it’s true that you can easily compute that the first car is traveling at 45 mph relative to the road (or relative to me, the bystander standing next to the road). But it is meaningless to talk about how fast it’s really going. It’s going at 45 mph in my frame of reference and at 15 mph in your frame of reference. In other words, it’s going at 45 mph relative to me and 15 mph relative to you. Neither frame of reference is inherrently “more correct” than the other. You might think the “person who’s standing still” has the more correct measurement, but there really is no “person who’s standing still.” I’m moving relative to some things, and you’re moving relative to some things. For instance, we’re both moving relative to the Sun, since the Earth is orbiting it.

At any rate, the formula for converting the speeds I measure to the speeds you measure (assuming, for simplicity, that everything is moving along the same line, say the x-axis) is:

v[sub]2[/sub] = v[sub]1[/sub] - v.

Here v[sub]1[/sub] is the speed of the first car in my frame of reference, v[sub]2[/sub] is the speed of the first car in your frame of reference, and v is your speed in my frame of reference. (Obviously, your speed in your own frame of reference is zero, since you aren’t moving relative to yourself.) In the above example, v[sub]1[/sub] is 45 mph, v is 30 mph, and thus v[sub]2[/sub] is 15 mph.

So that’s the formula we use in Gallilean relativity. For a long time, it was basically considered “common sense” that this formula always works. (Again, it’s a little more complicated if we aren’t all moving along the same line, but I’m trying to keep things simple.)

But let’s say we replace the first car with a beam of light. And we’ll say that the beam of light is moving relative to me at 300,000 km/s, a value typically represented by the letter c. Let’s try to figure out how fast the beam of light is moving relative to you. If Gallilean relativity is to be believed,

v[sub]2[/sub] = c - v.

So v[sub]2[/sub] should be a bit less than c. In fact, in my previous post, where I talked about someone chasing a beam of light while traveling at half the speed of light, v would be c / 2, so v[sub]2[/sub] = c / 2.

However, this is wrong. According to Einstein, v[sub]2[/sub] should also be equal to c. The speed of light is the same in your reference frame as mine, even though this goes against Gallilean relativity! If my explanation of Gallilean relativity made complete sense to you, then the fact that it is wrong here should seem very odd indeed. This is what I meant when I said that Einstein’s theory of relativity defies common sense.

(Aside: I always find it amusing when people summarize Einstein’s theory of relativity by saying “Everything is relative.” It’s true that certain things that we thought were absolute, like distances and time intervals, actually depend on your frame of reference. But this is all based on the fact that the speed of light isn’t relative! It’s the same in all inertial reference frames. Basically, “inertial” means “unaccelerated” – to deal with accelerated reference frames, you need Einstein’s General Theory of Relativity.)

Anyway, according to Einstein’s relativity, the correct equation for calculating these speeds is:

v[sub]2[/sub] = (v[sub]1[/sub] - v) / (1 - v v[sub]1[/sub]/c[sup]2[/sup])

Note that if I plug in c for v[sub]1[/sub], then the equation becomes:
v[sub]2[/sub] = (c - v) / (1 - v/c)

Then I can multiply the right side by c / c (which is of course 1). I get:
v[sub]2[/sub] = c (c - v) / (c - v)

But (c - v) / (c - v) = 1, so I’m left with:
v[sub]2[/sub] = c

Thus, the value you measure for the speed of light is also c, just as I claimed. Note that I didn’t need to specify the value of v. In other words, you will still measure light to be traveling at speed c, regardless of how fast you are moving relative to me.

So, if this is the correct equation, how come everyone thought that Galilean relativity was right for so long? Well, look at what happens when the velocities involved are much smaller than c. In that case, v v[sub]1[/sub] / c[sup]2[/sup] is very small, in fact it’s basically zero. So if we set this to zero, our equation goes from:

v[sub]2[/sub] = (v[sub]1[/sub] - v) / (1 - v v[sub]1[/sub]/c[sup]2[/sup])

to

v[sub]2[/sub] = (v[sub]1[/sub] - v) / 1

Dividing by 1 doesn’t do anything, so we can just write this as:
v[sub]2[/sub] = v[sub]1[/sub] - v

which is the equation we had for Gallilean relativity. So Gallilean relativity works just fine for small velocitities. This is what I meant by saying that Gallilean relativity is a limiting case of Einstein’s relativity. The reason our common sense is wrong is because we’re only used to dealing with small velocities, so our intuition tells us Gallilean relativity should work all the time, which is wrong.

12

Now whether these higher dimensional theories give a “better” explanation of why physical constants like the speed of light are constant, I don’t really know. I don’t know much about such theories, other than that they involve a lot of complex math, so I doubt they’re exactly analogous to any common sense principle we have in our daily lives. Basically, our common sense is just wrong (or perhaps I should say "not applicable to all situations), because in most circumstances we only encounter a fairly limited range of sizes and energies. As far as the question of would it all seem obvious if we were eleven-dimensional beings? I haven’t got a clue. But we wouldn’t need to be eleven dimensional for relativity to seem natural – we’d just need to be traveling around at speeds close to that of light. Basically, common sense boils down to the assumption that the universe works however we’re used to it working.

13

But our perception allows no motion in the “higher” spatial dimensions, and AFAIK it’s not wholly settled that there might not be “higher” temporal dimensions. On the other hand, while we only consciously move into the forward light-cone, there is nothing physically preventing (again, as far as current physics goes) something from moving into the reverse light-cone. From a perceptual level, you’re pretty much right. From the physics, however, it’s a bit muddy.

14

The discrepancy arises directly from the fact that our common sense is based on everyday experience. All the objects with which we are familiar in everyday experience move so slowly in comparison to c that the correction factors are too close to unity to make an appreciable difference.

15

The 10 and 26 figures come from what’s necessary to remove the “quantum anomaly” in string theory. Basically, each point on the string goes to a point in D-dimensional spacetime. That is, a string is described by a function X(s,t) which takes its value in a D-dimensional manifold (usually taken as Minkowski space for a first approximation). s is the parameter along the string and t is the “time” parameter of the string’s “world-sheet” (analogous to the world-line of a particle).

Now, the rotations, translations, and boosts which generate the Poincare group in Special Relativity have D-dimensional counterparts. The generators (“infinitesimal rotations”) obey a certain collection of commutation relations which are well-known from classical Lie theory. When we pass to the quantized bosonic string (take the “position field” X defined on the world-sheet of the string as bosonic) and calculate the appropriate commutators, there are certain terms which should be zero, but they aren’t.

For instance {pulls out Green, Schwartz, and Witten}, one of the commutators that should vanish has the form of a sum of an infinite number of terms, each constructed from primitives of the theory (which can’t be changed) and a constant C[sub]m[/sub]. A bunch of calculation gives the formula

C[sub]m[/sub] = m((26-D)/12) + m[sup]-1[/sup]((D-26)/12 + 2(1-a))

where D is the dimension of spacetime and a is another (more subtle) constant. To get the whole sum to work out to be zero, each of these constants C[sub]m[/sub] must be zero. The only way for this to work out is for a to be 1 and D to be 26.

A similar calculation goes through when we move to the closed string and allow fermionic excitations. This time the formulas are different, but the upshot is that D must be 10.

So, why bother making sure these commutators are zero? The result on the quantum scale must still respect Lorentz-invariance. To tie this into the earlier discussion, the important thing about the spacetime interval is that it’s invariant under a certain group of transformations. Special Relativity says that all physical laws must be invariant under these transformations. If the bosonic string didn’t obey this commutation relation on the quantum level, it would violate that requirement.

16

I had a feeling this thread would eventually attract your attention, Mathochist.

I do have some questions about your previous post: It seems like you’re saying that D must be 26 for bosons and 10 for fermions – so bosons require a 26 dimensional universe and fermions require a 10 dimensional universe? How are these two results compatible? I mean, do bosonic strings extend into sixteen more dimensions that are for some reason inaccessible for fermionic strings?

Also, I’ve heard it said that M-Theory requires 11 dimensions (specifically, I’ve heard that “11 dimensional super-gravity” is its low-energy limit). But how can that be if the commutators you mention are non-zero in 11 dimensions?

17

No, the bosonic string theory requires 26 and the supersymmetric string theories (which have both bosonic and fermionic excitations) require 10. The thing is, there are four of those.

So there is a theory of open and closed bosonic strings (“Type I”), four different theories with a spacetime supersymmetry and only closed strings (“Type IIA”, “Type IIB”, “Heterotic SO(32)”, “Heterotic E[sub]8[/sub]×E[sub]8[/sub]” (that’s means to be a Cartesian-product sign, in case it’s garbled)), and 11-dimensional supergravity. M-theory contains all these as various limits. If you take such a parameter to go to such a value, it behaves more and more like this limiting theory. I’m not very well versed in M-theory yet, but I believe the supergravity limit is reached by taking the string length to go to zero, recovering something which behaves like a theory of particles in 11 spacetime dimensions.

Why doesn’t M-theory require thsoe commutators? Because those specific ones (and their relations) are specific to the theory of strings. If you take the appropriate limit M-theory behaves like one of these other theories, but the Lorentz anomaly was calculated within string theory. I assume there’s a similar anomaly in M-theory, and if you take its limit you recover the string Lorentz anomaly. Then again, I’m not Ed Witten, so I may well be wrong when it comes to M-theory.

18

If there is a multiple dimension universe we would all transverse every single dimension with every single movement we make, we would just have a hard time thinking about it. Our common sense seems wrong since we only see in two dimensions plus one time dimension, which makes us think in the same way.

As for the obviousness, no it wouldn’t be. That is why it is such a murky subject right now; if there are multiple dimensions they are small and compact, not like the extended dimensions we experience every day. They are too small to see and all that. We can very easily be multiple-dimensional beings without being aware. A small and compact dimension allows for the thing that transverses, in this case a string, to move in 11, 10, or 26 different directions. They curl back on themselves such that where a string starts is where it ends in a cycle, in very simplified terms. Having these at every point of our three extended dimensions of space is much like thinking about atoms. They are everywhere, smaller than the smallest thing you can think of, and has only neglibile impact on our everyday lives.

(Interestingly enough, if there were multiple time dimensions, one would end up where one started, thus being back in time.)

19

This is all true, but in your earlier post you argued the distinction between spatial and temporal dimensions on perceptual grounds. Pick a motivation and stick with it. :slight_smile:

Only if the “higher” temporal dimensions are compactified Even then, it’s a subtle point as to whether it would be “back” in time. Back implies a linear relation and on a submanifold of dimension greater than one (say, the local integrable temporal submanifold through a point if there are other temporal dimensions) there’s no natural linear order.

20

I read somewhere how time may be an imaginary quantity instead of real.

Thus, -(t[sub]2[/sub]-t[sub]1[/sub])[sup]2[/sup]
equals i[sup]2[/sup]=-1
t=i