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### Something thats always confused me about "The 4th Dimension"

Posted: Mon Feb 16, 2009 7:23 pm UTC
If you ask any astrophysicist or semi-intellectual person what the fourth dimension is, they'll say that it's spacetime.

But what has always confused me was the "geometrical" fourth dimension. What does this have to do with spacetime.

In other words, which of these examples of "fourth dimensions" is the correct usage of the fourth dimension, and why are the others wrong?
1. Einsteins space geometry. Curved Space in accordance with gravity.
2. A physical fourth dimension, Height Width Length and some fourth property that we can barely wrap our minds around.
3. Time. Like how the Tralfamadorians (from slaughterhouse five) can see in four dimensions, meaning they can see every instant of someones life when they look at someone.
4. Time compression. Like how time will move slower when you travel faster

My explainations may barely make sense, but hopefully someone has an answer that will clear up this mess for me.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Mon Feb 16, 2009 7:49 pm UTC
Time is, in my opinion, the pseudo-intelligent cop-out with a look-how-smart-i-am-to-know-this-and-you-haven't-even-considered-it effect. It's like misunderstanding the question and then acting smug about it.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Mon Feb 16, 2009 8:31 pm UTC
Your 1-4 list are all the same. Let me see if I can be at all helpful. Spacetime is four dimensional, three of them are the familiar space dimensions and time is different.
Your #2 would suggest that, in addition to length, width and height there is also a fourth. There is! Let's call the first three [imath]\Delta x[/imath], [imath]\Delta y[/imath] and [imath]\Delta z[/imath]. If you do some familiar Pythagoras, you get
$\Delta s ^2 = \Delta x^2 + \Delta y^2 + \Delta z^2$
for s the total separation between two points. If you add in time, there is now a measurement in the spacetime distance between two events, where an event is characterized by a position in space in time. Now the spacetime separation becomes
$\Delta s ^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 - \Delta t^2$
See the minus sign? That is because time is different. The fact that it is different is probably clear from your experience of time being not quite the same as space, but the fact that this difference is completely described by a sign change is quite neat.

The fact that this sign difference is there is what gives you your #4, the time (and length) dilation and compression effects. You can derive the Lorentz factors from the above consideration. It comes from the fact that every two observers agree on how much [imath]\Delta s[/imath] separates two events, but they may disagree on how it splits up into x, y, z and t pieces. So I may disagree with your measurement about the spacial separation, and about the elapsed time, but our [imath]\Delta s[/imath] will always agree.

Now the above is all in flat space. Regular Pythagoras works just fine. In curved space you distort these distance measurements a bit. It takes a bit more math to describe this, some calc and linear algebra and you could be on your way to differential geometry if you wanted to be. But your #1 is essentially just the curvy version of the above.

Your #3 is a little tricky, because the way we think about time, even if you took our 4 dimensional spacetime and plunked it down, for example, inside an 11 dimensional spacetime, someone who was more aware of all the others would still live inside time. They couldn't step out of it and survey the whole thing at once, I don't think this is remotely possible. But you could easily posit that the whole of spacetime with a complete past and future "exists."

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Mon Feb 16, 2009 8:35 pm UTC
Neither is the correct one. Like most things, "the fourth dimension" means different things in different contexts. In my experience, when people use the phrase, either they don't know what they're talking about, or it's tongue-in-cheek.

Physicists don't usually refer to time as "the fourth dimension".

When you're talking about a four-dimensional object, you wouldn't say it "lives in the fourth dimension", unless you want to sound like a douchebag.

There are lots of things that can be nicely described an a four-dimensional manifold. The universe is one of them (that's your 1 and 3). Theoretical "physically four-dimensional objects", like a tesseract, require four spacial dimensions (that's your 2). Your 4 has almost nothing to do with the number of dimensions of space-time.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Mon Feb 16, 2009 10:36 pm UTC
pernero wrote:If you ask any astrophysicist or semi-intellectual person what the fourth dimension is, they'll say that it's spacetime.

Start from the start, and it's not so hard.

Start with a point on a number line. You can describe to someone where it is easily by telling them which number the point is on.

What about a point on a plane. Well, draw arrows at 90 degree angles to each other. Label one "x" and the other "y". Then draw out a grid to the point in question. Then give the (x, y) coordinate of the grid. That's two dimensions.

What about a point in three dimensions? You can't do it on a chalkboard or on paper any more. But you can use three rulers in the corner of a room at 90 degree angles. Label the rulers x, y, and z. Give the (x, y, z) coordinate.

What about four dimensions? Well, sadly, we've run out of physical dimensions, so we can't do it in our room. Space only has three dimensions, so we're out of luck in creating an actual point in four dimensions. But mathematically, we just continue what we were doing and ignore the limitations of real life:

1 dimension - number line - one real number
2 dimesions - a chalk board - two (ordered) real numbers
3 dimensions - a room - three (ordered) real numbers

Take a guess. How do we describe a point in 4 dimensions? You guessed it, four real numbers. And similarly, we can talk about 5 or 6 or 7 or 29. In fact, as mathematicians, when a theorem applies to any finite-dimensional real space, we just call it "Rn". Lots of things that are true in R2 and R3 are true in Rn for any n. Lengths are measured using the "root-means-squared" function we derive from the Pythagorean theorem. So if we have a point (x, y, z, w), the distance from the origin is the value sqrt(x^2 + y^2 + z^2 + w^2). Of course, we could use any letter for w, but w and t are the two most common ones.

The key aspect of something being a dimesion is that we need an additional coordinate to describe it. It is "independent" of the rest. If you have a cube, certainly it's height has no bearing on its width or its length. This "independence" is why we have to use rulers at 90 degree angles. Right angles are very, very strongly connected to this notion of independence.

So where else can we find this independence in real life? Time comes to mind. Say we put a box in the room and take it out shortly thereafter. HOW LONG we keep the box in the room has no bearing on its height or its width or its length. It satisfies this independence, so we can just go ahead and call it a dimension. We can describe the box by four parameters: its (x, y, z, t) where x y and z are its physical dimensions and t represents (in seconds) how long the box stayed in the room. It's a "length of time." So there, it's a dimension.

Now time, even in classical physics, isn't like the other dimensions. The spacial dimensions are directions we can move stuff back and forth. You go to school, you go back home, and the next day, you're at school yet again. But time doesn't allow you to really "move back". It's a much less obvious dimension. Not only that, but it seems that your spacial position is a function of time. That is, at any given point in time, you are at exactly one location. This means, geometrically, you are a CURVE in 4-D space. This isn't so weird, though, because this is the essence of a graph. In calculus or physics, you might graph a ball's height as a function of time. What you're really drawing is a graph of spacetime, using ink to represent the ball.

Just to rid your idea of the specialty of time as a unique forth dimension, let's take another example of a 4D space. As you know, to describe a point on a sheet of paper, you need two numbers, the (x, y) coordinate. Well, how do you describe the location of TWO points, point A and point B? You need four coordinates, (x_A, y_A) and (x_B, y_B). But the positions of either point doesn't depend on another, so you could just as easily write (x_A, y_A, x_B, y_B). And what's in a name? We can just rename that as (x, y, z, w) and voila. A "point" in four dimensions. Now it's *really* two points in two dimensions. But even more *really* than that is it doesn't matter. Words like "real" and "exists" are pretty stupid words. What matters is what you can describe, and you can describe a pair of points just as easily as two planar points as you could as if it were a point in 4 dimensional space.

Let's talk about relativity for a bit. There are three steps in relativity. Galileian Relativity, discovered in the 1600s was that there is no distinction between "motion" and "rest". A kid in an airplane can't tell if he's on the ground of soaring through the air (at a constant speed). The air may be more turbulent or something, but it could just be an Earthquake. Galileo gave us this theory that one man's rest is another man's motino.

Einstein's Special Relativity invented 300 years later made an important correction based on experiments in light and electromagnetism. Light waves can move through space at a constant speed. But very strangely, this constant speed did NOT depend on the observer. Take this example. A man watches a plane fly overhead. He sees the plane moving at 600mph. Another man is on a train and sees the same plane. He measures the speed at 500mph. Do the same experiment with light instead of a plane. The man on the ground sees a light wave moving at 670 million miles per hour. The man on the train will also see the light wave moving at the EXACT SAME SPEED. That is creepy if you think about it. The man on the ground will swear to god that the man on the train is wrong. He sees the light moving at 670 million mph. If the man on the train, moving 100mph sees the light moving at 670 million mph, then the light must actually be moving 670 million PLUS 100 mph. There's a paradox to be resolved.

Einstein's resolution was Everything You Know is Wrong. It is taken as an axiom that light moves at a constant speed relative to the person measuring it. From that axiom, we can prove that time slows down for moving observers, length contracts, and no two events separated in space can happen "at the same time". But conversely, if I see you moving, you see me moving. To me, YOU are the one who is slowing down in time. To you, *I* am slowing down in time. It's called relativity because you have to pick a biased observer and your measurements are "relative" to that observer in accordance to the theory. Who is *really* slowing down? You can't answer that question without picking a frame of reference -- a bias observer.

So, that was our physics interlude. What does this mean for time's "dimension status"? Well, it turns out it affects how you translate from one frame to another. To any fixed observer, time works just like it does in classical physics. But when you want to change your POV (point of view), you have to use a different set of operations. In Galileo's theory, you used Galileian transformations, which basically subtracted your new POV's velocity from everything. In Special Relativity, it's not a simple subtraction. There's an extra factor based on the new POV's velocity in terms of the percentage of the speed of light. This new way to change POV is called a Lorentz Transformation, if you want the full details.

Now special relativity has limitations. It only works when velocities are fixed. When all objects are just cruising about space with no forces acting on anything. The problem is that that doesn't really happen in life. It took Einstein over a decade to work out the rest of his theory, General Relativity.

General Relativity goes and makes things complicated. A lot of Einstein's time working on the theory was learning the math behind it. It's not simple Euclidean geometry and calculus. It requires a branch of mathematics called Riemann Geometry. Riemann Geometry started off as a way to work out how to do calculus on the surfaces of 3D objects. The Earth is a good example. We can talk about "distance" between New York and Los Angelos, but the truth is this distance isn't just the distance of a straight line between the two cities. Such a line would travel through the Earth's crust. Not a path we can take on a plane. Instead, we restrict our motion to the surface of the sphere. The surface, on a very small scale, looks almost like a plane, even though it's part of a three dimensional object. This concept is generalized into what's called a manifold. Riemann Geometry gives us the tools to do natural geometric things in this "richer" space. What is the shortest distance between two points? What is the angle of a triangle embedded on the surface? How "curved" is this point on the surface? It's not a subject for the casually curious, sadly. It requires a pretty good understanding of calculus, analysis, and topology.

So bottom line, even if that didn't all make sense. "The forth dimension", along with "division by zero", "infinity", and "time travel", is just one of those phrases which sounds a LOT more exotic than it really is. There's interesting stuff to space and time, but it's MUCH more interesting than trying to count higher than three. If you ever hear anyone talk about "four dimensional space", they guaranteedly don't know as much about the subject as you think they know.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Mon Feb 16, 2009 11:04 pm UTC
Tac-Tics wrote:If you ever hear anyone talk about "four dimensional space", they guaranteedly don't know as much about the subject as you think they know.

I'd be careful with making that guarantee about people on these fora. Not everyone here is a poseur, after all.

One thing with four dimensional space - people especially in physics will often refer to spacetime this way, and specifically mean the four dimensional space equipped with this spacetime structure. You can add a fourth spatial dimension if you want. This doesn't describe our universe (because we very definitely have time) but it's certainly interesting mathematically. If a math person starts talking about 'four dimensional space' without saying anything more specific, they probably aren't fixing a particular geometry.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Mon Feb 16, 2009 11:20 pm UTC
Saying "time is the 4th dimension" is kind of like saying "width is the second dimension". It doesn't make sense unless you include the context in which you are saying these things. I'd like to think of each dimension as a coordinate you need to use when specifying where something is. You need 4 - 3 in space and 1 in time - to describe really where something is. So we appear to live in 4-dimensional space-time. But to really separate one "dimension" out of the deal is kinda weird.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Tue Feb 17, 2009 12:39 am UTC
If you'd like a nice introduction to the 4th geometrical/mathematical dimension, i think that Carl Sagan is the one that has summed it up best for now.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Tue Feb 17, 2009 1:21 am UTC
Carl Sagan? Who's that? I think you mean Agent Smith...

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 7:38 am UTC
It's at a right angle to reality.

Which doesn't make too much sense to me.

The interesting thing, I find, is that dimensions can be used to explain forces. Einstein used three-plus-one-makes four dimensions to explain gravity, and that worked out well. I forget the name of another guy who got excited, and then assigned the electrostatic force the 5th dimension, and that worked out well.

Taking the extra-dimensions-carry-fundamental-forces thing a bit farther, the current hope is that we can, using the LHC, experimentally find energy moving from our dimensions into these postulated extra dimensions. We'll know this happened if the debris of a collision have less energy than the things that collided.

This will be exciting. We will then use this to try and cleverly guess the size of the extra dimensions, and compare that with what string theory predicts.

If it all syncs up, we'll have a remarkably wonderful theory of everything that's been in the works for over 50 years.

Or it might all be wrong and we'll have to start over. That will be exciting.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 1:10 pm UTC
Now see here I thought the Fourth Dimension was a hair salon "where you keep on looking like you look when you left"...

no but srlsy I think the real argument is which order we number dimensions in... time can be called a dimension, one thru which we travel in a single direction at a velocity based upon our relative velocity.. (as far as I know, I'm kinda an ihjut sometimes) But there could also be a fourth spatial dimension ala the whole apple in flatland thing... its just a matter of which one you label as the fourth? maybe time is the fourth and hyperspace (or you know whatever) is the 5th? I don't think the two concepts are mutually exclusive.

on a four dimensionally unrelated note.. I now have that stupid jungle stuck in my head and cant find a single link in google to prove to you guys that I didnt imagine the place so someodd years ago.. I fail at google.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 2:42 pm UTC
Rentsy wrote:It's at a right angle to reality.

It's not literally at a right angle. It's simply independent of the other degrees of freedom an object has.

The interesting thing, I find, is that dimensions can be used to explain forces. Einstein used three-plus-one-makes four dimensions to explain gravity, and that worked out well. I forget the name of another guy who got excited, and then assigned the electrostatic force the 5th dimension, and that worked out well.

Gravity is special among forces because given a gravity field, an object in the field feels acceleration independent of all of that object's physical properties. Take an object on the surface of the Earth. Disregarding air resistance, it doesn't matter if it's a human, an elephant, a drop of water, or an airplane, all objects feel the same acceration. This indifferent attitude that gravity has, that it doesn't care about WHAT you are, it still pulls on you the same, is what allows it to be explained geometrically.

But it's NOT as simple as "adding a dimension". Adding time doesn't cause gravity to pull things inward. There's a lot more to it. Roughly, it happens because all objects in spacetime have a constant velocity or 4-velocity. This 4-velocity can be broken up into "spacial" velocity, which is what a traffic cop measures when you speed past him, and a "time" velocity, which is the rate at which you see an object moving through time. An object at rest is feeling time at 100% its normal rate and has no spacial velocity. As an object's spacial velocity approaches the speed of light, it feels more and more time dilation, and its "time" velocity approaches 0% the normal rate. The trick used in general relativity is that curved spacetime allows one person's time to look like another person's space. In flat spacetime, where an object can casually drift in time without moving, in a curvy spacetime, an object near a massive object drifts in time, but that time from another frame looks like space, so there is a spacial drift and a time dilation. This spacial drift is gravity.

If you were to talk about electrostatic forces, one additional dimension wouldn't be sufficient. Electricity, unlike gravity, affects an object based on its charge, and so the interaction is much more complicated.

You could call charge on a particle an extra dimension. There's no harm in that, as long as you don't make it sound deep or intriguing, because it's really not. Charge is independent of a particle's position and its timeline, so it's a degree of freedom and a candidate for "dimension-hood." But the charge on an object, in our understanding of physics, is conserved, so a single particle moving through spacetime would always have a constant charge coordinate. Now if you had two objects which could exchange electrons and vary in charge, it might make sense. Either way, stick to the standard interpretation with four dimensions and you'll have plenty to learn.

This will be exciting. We will then use this to try and cleverly guess the size of the extra dimensions, and compare that with what string theory predicts.

String theory is bs. Coming up with theories is not science. I have a theory that my dryer contains hidden extra dimensions where my socks go when I lose them, but that's not science. Until someone tests a hypothesis or there is strong evidence for it over the other alternatives, you're just doing math.

I also think people get too caught up in the idea of a "theory of everything." Isn't a "theory of most things" good enough? Or a "theory of practical things"? It's quite plausible that a theory of everything is simply not possible to discover. I think more important for our society is to take the good science we have an put it into a form that is easily digested by the public. There are too many Discovery channel specials about string theory and not enough about electromagnetism or mechanics. If you don't understand EM (and most people don't), you're effed when it comes to understanding any modern scientific theories. And the end result is people getting all excited about "extra dimensions" and "parallel universes" when in reality, they have no fricking clue what those things might even be or how they are at all useful in explaining the world. They are just popular terms because they sound like something from a science fiction novel, but the way they are presented in those kinds of shows is less than useless.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 3:39 pm UTC
Tac-Tics wrote:If you were to talk about electrostatic forces, one additional dimension wouldn't be sufficient. Electricity, unlike gravity, affects an object based on its charge, and so the interaction is much more complicated.

The one Rentsy is talking about is Kaluza Klein. You get radiation too, not just statics! There are some problems with sources though, if I recall.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 4:21 pm UTC
Tac-Tics wrote:String theory is bs. Coming up with theories is not science. I have a theory that my dryer contains hidden extra dimensions where my socks go when I lose them, but that's not science. Until someone tests a hypothesis or there is strong evidence for it over the other alternatives, you're just doing math.

Isn't that what every theory is until it gets tested? Try saying that about the status of the theory of relativity in 1910. String theory actually has a lot of precedent in already proven physics.
Spoiler for long:
Spoiler:
The Black Hole War, by Leonard Susskind wrote:A half century of experimenting on nucleons has made it certain that they are elastic strings that can stretch, rotate, and vibrate when excited by adding energy. In fact, all hadrons can be spun out into long, stringlike objects. Evidently, they are all made of the same sticky, stringy, stretchable stuff - something like maddeningly stubborn bubble gum that just won't let go.
..
Mesons are the simplest hadrons. Many different kinds of mesons have been discovered, but they all share the same structure: one quark and one antiquark, joined by a sticky string.
..
Nucleons contain three quarks, each attached to a string, and the three strings are joined at the center like a gaucho's bolo. They can also twirl and vibrate.
..
--This is the important part
One more kind of hadron exists: a family of "quarkless" particles made only of string, closing on themselves and forming a loop. Hadron physicists call them glueballs, but to a string theorist, they are just closed strings.
Quarks don't appear to be made of smaller particles. Like electrons, they are so small that their size is undetectable. But the strings that bind quarks together are definitely made of other objects, and those objects aren't quarks. The sticky particles that combine to form strings are called gluons.

Coming from the guy who embarrassed Stephen Hawking, I would call this a credible precedent for string theory. Another quote (from the same page, no less):
The String Theory of nucleons, mesons, and glueballs is not an idle speculation. It has been extremely well confirmed over the years and is now considered to be part of the standard theory of hadrons. What is confusing is whether we should think of String Theory as being a consequence of Quantum Chromodynamics - in other words, should strings be thought of as long chains of the more fundamental gluons - or whether it is the other way around - that is, gluons are nothing but short segments of string.

This is getting into religious war territory though.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 4:31 pm UTC
The Boz wrote:Time is, in my opinion, the pseudo-intelligent cop-out with a look-how-smart-i-am-to-know-this-and-you-haven't-even-considered-it effect. It's like misunderstanding the question and then acting smug about it.

Time is the fourth dimension. It's not a pseudo-illectual bullshit answer at all. A basic course in general relativity would explain that.

OP: "Space-time" isn't a fourth dimension at all. It's just 'time' is the 4th dimension. The story goes: space-time is a 4 dimensional manifold equipped with a non-positive definite metric. Pairs of points with time-like and space-like separations are defined by the sign of their distance (conventions depend on who you ask, but relativists generally take distance<0 implies time-like and distance>0 implies space-like). In some tangent space on space-time, whichever basis vector points in a time-like direction is generally taken to point in the direction of increasing 'time.' That's 1 dimension of the tangent space. There are 3 other basis vectors one can choose that are linearly independent and are space-like. These account for the other 3 dimensions of space-time.

tl;dr or didn't understand: Time is on the same footing as space. It's just another coordinate, but one that has special properties. Space-time is a 3+1 dimensional manifold: 3 space-like dimensions and 1 time-like dimension.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 6:13 pm UTC
Tac-Tics wrote:String theory is bs. Coming up with theories is not science. I have a theory that my dryer contains hidden extra dimensions where my socks go when I lose them, but that's not science. Until someone tests a hypothesis or there is strong evidence for it over the other alternatives, you're just doing math.

"CDF sees dark unified SUSY in lepton jets?"
http://motls.blogspot.com/2008_10_01_archive.html

This news is almost 4 months old now.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 6:19 pm UTC
Behold the string theory shit storm. I swear this thread is turning into /b/ but with fancy words and without the gore pics.

But seriously, cut it some slack. A lot of physics IS coming up with theories. Otherwise all you have is a set of data with no fundamental understanding of it and nothing with which you can make predictions. They have to be verified, otherwise they shouldn't ultimately be believed, but don't diss them less it's simply unverifiable. And theories are motivated by need and usefulness. They're not nearly as capricious and arbitrary as "there are extra dimensions in my dryer." And I haven't had a chance to read the article, but I recently stumbled upon one titled "String Theory finally make an experimental prediction." If it's true, then dis string theory when the prediction is falsified. Otherwise ust hold your horses. Sheesh.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 6:31 pm UTC
Tac-Tics: Notice mass can essentially be expressed as a divergence of a gravitational field? I.e. it's a topological defect in space-time. Suppose we experienced 2+1 dimensions but existed in 3+1 (sort of like what string theory posits). Due to the nonlinearity of space-time, distortions in the 3rd dimension could affect distortions in the 2nd, but if we insisted on describing the physics in 2 dimensions using only the value of the divergence in 2 dimensions, we'd be missing something and would have to fill the void with another force and an extra property of particles. It's entirely possible things like E&M have a geometric description, but would require higher dimensions (Kaluza-Klein) or other geometric quantities (torsion, extrinsic curvature, etc) that aren't currently accounted for. Kaluza-Klein came close. The only problem was it predicted an extra scalar field that isn't found in nature. So don't be so hasty to toss out geometric descriptions of other forces. (ex: E&M can already be described using line bundles - another geometry-related thing - but this isn't necessarily as physical as it is a mathematical convenience.)

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 6:41 pm UTC
For that matter, what is the third dimension? Height? Or is it length? Maybe width?

I don't think it really matters. The fourth dimension is whatever fourth independent value you're measuring. Sure, it could be time. It could also be temperature, pressure, cost, mass, velocity, or phlogiston content; as long as that 4th thing is independent of the other 3.

IANAP/IANAM (physicist/mathematician)

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 6:47 pm UTC
sgt york wrote:For that matter, what is the third dimension? Height? Or is it length? Maybe width?

I don't think it really matters. The fourth dimension is whatever fourth independent value you're measuring. Sure, it could be time. It could also be temperature, pressure, cost, mass, velocity, or phlogiston content; as long as that 4th thing is independent of the other 3.

IANAP/IANAM (physicist/mathematician)

Think geometry. It's time. But yes, generally it's just a degree of freedom.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 6:58 pm UTC
hitokiriilh wrote:I swear this thread is turning into /b/ but with fancy words and without the gore pics.

Fancy words are just as bad as gore pictures. They are both used to shock their audience and aside from that have no real use whatsoever.

The "number of dimensions" is just a very small aspect of the mathematical framework behind a theory. To draw attention to it is pointless, like counting the number of strokes in a famous painting. To know the number of strokes wouldn't help you understand the artist's motivation. It isn't something important to the interpretation. It's a pointless peice of trivia.

The core ideas in special relativity are time dilation, non-simultaneity, length contraction, non-additivity of relative velocities, and increase of mass of moving objects. It is only a coincidence of real life that we are working in a four-dimensional spacetime. The theory works just as well in two or three dimenions. In fact, most classes teach relativity in only two-dimenional spacetime, where objects can only move left or right along a single axis over time. Once the core concepts are mastered, almost no time at all needs to be spent explaining how things work in four dimensions, because it is so easy to generalize. As long as you choose the right basis, everything reduces to a two dimensional case anyway.

My point is that you shouldn't be interested in the number of dimensions of a theory. It's a peripherial aspect of any theory, and the number is what it is because it is what it is. You should be more concerned about the explainative power of the theory. What phenomena can the theory explain correctly and accurately? What other theories is it founded in? Which does it modify or replace or disprove? What does an experiment to test the theory look like? I think that last one is especially important. You read all about these mystical physics ideas like atoms, electrons, black holes, quarks, neutrinos, nuclear forces, gravity, electromagnetism, and whatnot, but the honest truth is no one has ever in their life seen an electron with their eyes or felt an electromagnetic wave with their fingers. Truth is not determined by authority or popularity. If it was, a few great scientists (*cough*Penrose*cough) would have sent us all to our graves by now.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 8:31 pm UTC
Tac-Tics wrote:
hitokiriilh wrote:I swear this thread is turning into /b/ but with fancy words and without the gore pics.

Fancy words are just as bad as gore pictures. They are both used to shock their audience and aside from that have no real use whatsoever.

The "number of dimensions" is just a very small aspect of the mathematical framework behind a theory. To draw attention to it is pointless, like counting the number of strokes in a famous painting. To know the number of strokes wouldn't help you understand the artist's motivation. It isn't something important to the interpretation. It's a pointless peice of trivia.

The core ideas in special relativity are time dilation, non-simultaneity, length contraction, non-additivity of relative velocities, and increase of mass of moving objects. It is only a coincidence of real life that we are working in a four-dimensional spacetime. The theory works just as well in two or three dimenions. In fact, most classes teach relativity in only two-dimenional spacetime, where objects can only move left or right along a single axis over time. Once the core concepts are mastered, almost no time at all needs to be spent explaining how things work in four dimensions, because it is so easy to generalize. As long as you choose the right basis, everything reduces to a two dimensional case anyway.

My point is that you shouldn't be interested in the number of dimensions of a theory. It's a peripherial aspect of any theory, and the number is what it is because it is what it is. You should be more concerned about the explainative power of the theory. What phenomena can the theory explain correctly and accurately? What other theories is it founded in? Which does it modify or replace or disprove? What does an experiment to test the theory look like? I think that last one is especially important. You read all about these mystical physics ideas like atoms, electrons, black holes, quarks, neutrinos, nuclear forces, gravity, electromagnetism, and whatnot, but the honest truth is no one has ever in their life seen an electron with their eyes or felt an electromagnetic wave with their fingers. Truth is not determined by authority or popularity. If it was, a few great scientists (*cough*Penrose*cough) would have sent us all to our graves by now.

I just got out of my modern field theory class in which we're presently covering dimensional regularization and how the number of dimensions we work a field in affects how badly divergent the loops in Feynman diagrams are. Anyons can exist in 2 dimensions, but nothing higher. There are many analytical solutions to field theoretic problems in 2-d but not in 3-d. Most duality correspondances currently known relate field theoretical results in d dimensional flat space-time to the result of a different theory in D dimensional curved space-time (for very particular d's and D's). Diagram expansions of scattering amplitudes have been rigorously shown to converge in only 1 and 2 spatial dimensions (Thanks to Arthur Jaffe)- a proof that they converge to all orders in 4 or higher dimensions has yet to be shown. Bosonic string theory is only Lorentz covariant in 26 dimensions; fermionic string theory is only Lorentz covariant in 11; and M-theory is only Lorentz covariant in 12 (the # of dimensions in these theories isn't arbitrary at all). General relativity has couplings between different components of the metric, meaning the curvature in one dimension is affected by the curvature in another dimension. In fact, the only reason special relativity can be taught in just 2 dimensions is because we can always find a set of generators of SO(3,1) that mix only 2 parameters at a time. Usually (but not always), the structure of a theory can be written out for an arbitrary N number of dimensions, but it's a FAR cry from the truth to say the number of dimensions in a theory is only 'peripheral.' Yes, it's far more important that the theory be able to spit out results that agree with experiment, but the process of developing such a theory tends to be extremely sensitive to the number of dimensions in which it dwells.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Wed Feb 18, 2009 10:15 pm UTC
hitokiriilh wrote:Usually (but not always), the structure of a theory can be written out for an arbitrary N number of dimensions, but it's a FAR cry from the truth to say the number of dimensions in a theory is only 'peripheral.' Yes, it's far more important that the theory be able to spit out results that agree with experiment, but the process of developing such a theory tends to be extremely sensitive to the number of dimensions in which it dwells.

How can the number *not* be arbitrary in string theory? A casual survey suggests that for any positive integer n, there exists a flavor of string theory that claims the universe is n-dimensional!

As I said, just because there is a number doesn't mean it isn't peripheral. That the universe has 11 dimensions isn't a central idea of M-theory. The central idea has to do with quantum behavior near black holes. The number 11 just pops up if one does enough number crunching with a fixed set of assumptions.

The motivation behind the theory is much more important than the mathematics. Trying to work the mathematics out from the motivation is easy... it's just a type of engineering. Going backwards -- that is, taking the math and figuring out what problem the theorist was trying to solve to begin with -- is much harder. It's like trying to take apart an electrical device and figuring out what it does without ever turning it on or being allowed to see the thing from the outside.

I apologize if I sound harsh. This kind of thing just bugs me a bit. People throw around the word 'dimension' like it's mysterious and advanced mathematics. It's not. The same goes for infinity and division by zero. On the other hand, physical theories are often much more complicated. Relativity is not what you get just by stapling an extra dimension to the theory of gravity. GR is all about "curved" spacetime. And even though it's only 6 letters long, "curved" is a word that took Einstein over a decade to formulate mathematically. A theory's dimensionality is not its defining characteristic.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 12:11 am UTC
Tac-Tics wrote:
hitokiriilh wrote:Usually (but not always), the structure of a theory can be written out for an arbitrary N number of dimensions, but it's a FAR cry from the truth to say the number of dimensions in a theory is only 'peripheral.' Yes, it's far more important that the theory be able to spit out results that agree with experiment, but the process of developing such a theory tends to be extremely sensitive to the number of dimensions in which it dwells.

How can the number *not* be arbitrary in string theory? A casual survey suggests that for any positive integer n, there exists a flavor of string theory that claims the universe is n-dimensional!

As I said, just because there is a number doesn't mean it isn't peripheral. That the universe has 11 dimensions isn't a central idea of M-theory. The central idea has to do with quantum behavior near black holes. The number 11 just pops up if one does enough number crunching with a fixed set of assumptions.

The motivation behind the theory is much more important than the mathematics. Trying to work the mathematics out from the motivation is easy... it's just a type of engineering. Going backwards -- that is, taking the math and figuring out what problem the theorist was trying to solve to begin with -- is much harder. It's like trying to take apart an electrical device and figuring out what it does without ever turning it on or being allowed to see the thing from the outside.

I apologize if I sound harsh. This kind of thing just bugs me a bit. People throw around the word 'dimension' like it's mysterious and advanced mathematics. It's not. The same goes for infinity and division by zero. On the other hand, physical theories are often much more complicated. Relativity is not what you get just by stapling an extra dimension to the theory of gravity. GR is all about "curved" spacetime. And even though it's only 6 letters long, "curved" is a word that took Einstein over a decade to formulate mathematically. A theory's dimensionality is not its defining characteristic.

Zweibach's introductory book on string theory goes through the whole derivation of how the numebr of dimensions is set. There's a particular commutation relationship that has to vanish for the theory to be Lorentz covariant. It's not at all arbitrary.

The motivation for string theory is overwhelming. The algebraic properties of an open, singley excited string is identical to those of the E&M field; the algebraic properties of the minimally exciting closed string is identical to that of the graviton; etc. The nambu-goto action that lead to bosonic string theory is easy to obtain: just write down the lagrangian for a non-interacting relativistic fluid then constrain it to 1 spatial dimension. So the investigation of the action is easily motivated and the realization that the algebra of various strings is identical to those of various bosonic fields is more than enough to make one say "hey, what if this is something real?" It might not be, but it's too appealing to not look at a second time. Upon further investigation it's discovered the theory only works in 26 dimensions. Now repeat the process for SUSY string theories and you get 11. M-theory dwells in 12. All for the same reasons.

And you don't sound harsh at all. There are many things that bug me. And yes, people do toss around the word 'dimension' too easily, but I assure you I'm not one of them. I've been doing physics too damn long to not know the difference between a dimension and a parallel world and a hole in the ground, etc. I've taken a course in GR as well, have read through several chapters of Carrol's book and Misner, Thorne, and Wheeler's tome. Treating time as another coordinate comes from Minkowski's geometric treatment of special relativity. It was this very suggestion that lead Einstein to consider pseudo-Riemannian geometry in GR. So while "lulz, time is na0 a d1mension" wasn't all there was to GR, it's somewhat central.

GAH! I'd write more, but I gotta meet with my apartment-mate for dinner. Cheers.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 12:18 am UTC
hitokiriilh wrote:Zweibach's introductory book on string theory goes through the whole derivation of how the numebr of dimensions is set. There's a particular commutation relationship that has to vanish for the theory to be Lorentz covariant. It's not at all arbitrary.

$\Sigma_{n=1}^\infty \frac{1}{n^{-1}}=\frac{-1}{12}$

Yeah that's totally legit.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 1:21 am UTC
Oh dear. Yeah, I remember that whole section. That was a heuristic argument that he really should have left out. Did you go on to read the rest and notice the fact that the term he obtains came from a naive calculation and that it's set to -12 by a different, more legit mechanism? If not, go reread it. kthxbye

Edit: actually, in retrospect the fact that that's the only thing you mentioned makes me rage. Do you seriously think the retarded "zeta function-like" heuristic argument would have been enough to get passed in peer review? I hate it when people leave it at a fucking half story. I dislike string theory, just for different reasons. However, I don't care if the end result is the same (people agreeing that it's probably bollox) if the way of getting there is filled with holes.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 1:27 am UTC
It's essentially the same argument Polchinski gives though, no? Do a little analytic continuation, a little renormalization, voila!

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 1:43 am UTC
So then the problem seems to be the questionability of renormalization?

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 2:00 am UTC
Well, I'm generally OK with some renormalization. In fact renormalization pays my rent. But this was like three tricks at once, it was time for me to say "come on" and go see what the loopy folks were doing. Or the noncommutative geometry people. Something with a weird underlying topology, maybe.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 2:03 am UTC
doogly wrote:Well, I'm generally OK with some renormalization. In fact renormalization pays my rent. But this was like three tricks at once, it was time for me to say "come on" and go see what the loopy folks were doing. Or the noncommutative geometry people. Something with a weird underlying topology, maybe.

Hahaha. Well played, sir, well played. =) This might be quite tangential, but do you work with noncommutative geometry? I've recently been getting interested in it.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 2:18 am UTC
Nah, I do quantum fields in curved space. I'm interested in the nc geometry also, and I have Connes' book. It appears to be very difficult stuff. I am pretty good on geometry but it looks like it takes some analysis beyond my ken.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 2:32 am UTC
doogly wrote:Nah, I do quantum fields in curved space. I'm interested in the nc geometry also, and I have Connes' book. It appears to be very difficult stuff. I am pretty good on geometry but it looks like it takes some analysis beyond my ken.

I found a virtual copy of Connes' book. It makes me sad that I can't make it far beyond the definition involving spectral triples. >.< I figure I'll be able to learn some more math in grad school.

So qft in surved space? Liiiike Ads/CFT type stuff or semi-classical gravity or [insert thing involving both here]? You keep peaking my interest here.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 3:24 am UTC
My work might be off topic, but hey, it does take place in 4 dimensions!
Spoiler:
More like semiclassical gravity, but not quite. In the AdS/CFT they actually do their best to get off the curved space, and onto the boundary. (Conjecturally) you are allowed to do this, at least in AdS. But me, I love the curved space! I embrace the curves! In particular I want to prove some kind of energy condition, like
$\int_\gamma T_{\mu \nu} \eta^\mu \eta^\nu ds > 0$
Over a null geodesic, [imath]\gamma[/imath]. Here [imath]\eta[/imath] is the tangent, pointin off in that null direction.

So, the problem is, this isn't true. You can get an arbitrarily negative contribution by switching renormalization scales. But, this only happens when you switch them. So I think it's fine to just say, "Don't do this, pick any scale you want, but you have to stick with it." Then the problem is showing that this scale problem is the only obstacle. If there were no obstacles at all, it might be easy, but showing that there is only one problem is a bit tricky. So this is the plan.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 6:52 am UTC
pernero wrote:If you ask any astrophysicist or semi-intellectual person what the fourth dimension is, they'll say that it's spacetime.

But what has always confused me was the "geometrical" fourth dimension. What does this have to do with spacetime.

In other words, which of these examples of "fourth dimensions" is the correct usage of the fourth dimension, and why are the others wrong?
1. Einsteins space geometry. Curved Space in accordance with gravity.
2. A physical fourth dimension, Height Width Length and some fourth property that we can barely wrap our minds around.
3. Time. Like how the Tralfamadorians (from slaughterhouse five) can see in four dimensions, meaning they can see every instant of someones life when they look at someone.
4. Time compression. Like how time will move slower when you travel faster

My explainations may barely make sense, but hopefully someone has an answer that will clear up this mess for me.

"The Fourth Dimension" usually means time.

The fourth spatial dimension is, obviously, not time, but the space wherein you get hypercubes.

It's pretty much that simple.

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 7:45 am UTC
The motivation for string theory is overwhelming. The algebraic properties of an open, singley excited string is identical to those of the E&M field

To be fair, anything that approaches a field theory in some limit and has a U(1) symmetry will look identical to E&M. Its always hard to say how much of string theory's motivation ACTUALLY depends on the microscopics (i.e. depends on there being strings), and how much depends on symmetries that would survive coarse graining (which essentially removes all details of the microscopics).

### Re: Something thats always confused me about "The 4th Dimension"

Posted: Thu Feb 19, 2009 2:19 pm UTC
Spinoza wrote:
pernero wrote:If you ask any astrophysicist or semi-intellectual person what the fourth dimension is, they'll say that it's spacetime.

But what has always confused me was the "geometrical" fourth dimension. What does this have to do with spacetime.

"The Fourth Dimension" usually means time.

The fourth spatial dimension is, obviously, not time, but the space wherein you get hypercubes.

It's pretty much that simple.

I like hypercubes, but I prefer the term tesseract, just because it sounds cool.

To avoid this confusion, I wish they'd call time the zeroth dimension. But I don't expect this to change in a hurry.