Infinite Dimensions

[Nick]

I have no kind of formal education in astronomy and I don't have any solid grasp of any physics beyond what I learned in high school, but I do a lot of thinking about all this universe stuff and this looks like a good place to put some of my ideas down and have them discussed. I don't have a lot of time right at the moment to write much (I have a lot of calculus homework to do), so I'll start with an essay I recently wrote for one of my college applications:


A couple years ago I became interested in the geometric principles of dimensions. After studying the subject for a little while, I realized that, as far as I could determine with my limited knowledge, there could theoretically be an infinite number of dimensions. Human beings knowingly observe only four - breadth, height, depth, and time - but that may only be four out of infinity. To me, this seems like something physicists and mathematicians should be seriously looking into to solve the mysteries of the universe, yet I, a somewhat scientifically plugged-in individual, hear nothing about it.
Before I continue, let me clarify that the extra dimensions conjectured in String Theory, Brane Theory, and assorted other theories are not the same sorts of dimensions as the ones I?€™m interested in. I only understand these dimensions enough to know that they have properties which are not even possible in simple Euclidean dimensions. I view these theories, therefore, as irrelevant to any of my questions concerning dimensions.
What concerns me is that a relatively uncomplicated condition appears to be perfectly possible in the universe which would open the door to infinite possibilities in physics. It seems possible that many of the most puzzling behaviors of the universe might be so puzzling because their driving mechanisms exist elsewhere along some unknown dimension, undetectable from where we are.
If there are more dimensions out there, there must be something interesting going on in them. This is one of the many intriguing ideas I would like to explore as I move into higher education.


To summarize: The GEOMETRIC properties of the four observable dimensions can be compounded infinitely. To ME, that seems like just the sort of thing the universe would be likely to take advantage of, and it would mean that the universe is a lot bigger than most people seem to comprehend.

Discuss.

[Evil Dr Ganymede]

Um, physicists ARE seriously looking into other dimensions to "solve the mysteries of the universe" - that's what string theory and membrane theory is all about. For example, it seems that the physical laws of this universe are determined largely by how the other 7 (at least) dimensions that are all curled up at a subatomic level interact with eachother - so in a way the behaviours of the universe ARE driven by what's going on in unknown dimensions.

You may want to read The Elegant Universe by Dr Brian Greene for a primer on the subject, it explains things quite well and is a fascinating read.

[Sky Pilot]

How about a brand NEW theory (at least, I've never heard this one before -- I just made it up).

The theory would be that at the time just prior to the big bang, there was NO matter in our universe. All matter existed only in other universes. Our universe didn't exist at all. It came into existance only because of a massive hemorage from another universe (let's call it "the Big Fat Universe"). The membrane that seperates our universe from BFU got a tiny hole in it and the matter in the BFU "spewed" into our universe in a very infinitessimally small instant.

This created instant relief for the BFU, and created a brand new universe which we live in. So, the Big Bang was really a GBF of the BFU.

NOTE: GBF = Great Big F*rt.

[Nick]

Um, physicists ARE seriously looking into other dimensions to "solve the mysteries of the universe" - that's what string theory and membrane theory is all about. For example, it seems that the physical laws of this universe are determined largely by how the other 7 (at least) dimensions that are all curled up at a subatomic level interact with eachother - so in a way the behaviours of the universe ARE driven by what's going on in unknown dimensions.

You may want to read The Elegant Universe by Dr Brian Greene for a primer on the subject, it explains things quite well and is a fascinating read.

As I said in the essay, the types of dimensions proposed in string theory and membrane theory don't concern me, because they have properties that aren't even possible in simple Euclidean dimensions. For example, in basic geometry, an object must exist in at least two dimensions to have curvature. Yet the dimensions in string theory are THEMSELVES curved. Also, basic dimensions are infinite, conisisting only of two opposite directions without size or scale. String theory's dimensions are, as you said, all confined to a finite subatomic size. There is some sort of complicated mathematics or advanced concepts involved in the dimensions of string theory, which I do not understand, and which I do not need to understand or be concerned with to ask my questions about dimensions.

And I already own a copy of The Elegant Universe, although I've only read a few pages.

[Evil Dr Ganymede]

I don't even understand what your issue is with dimensions and why you're "not concerned with them". What do you want here? You claim that scientists should be looking into SOMETHING to figure out the "mysteries of the universe" but when it's pointed out to you that this is what they're doing, you say it doesn't concern you. For example:

What concerns me is that a relatively uncomplicated condition appears to be perfectly possible in the universe which would open the door to infinite possibilities in physics. It seems possible that many of the most puzzling behaviors of the universe might be so puzzling because their driving mechanisms exist elsewhere along some unknown dimension, undetectable from where we are. If there are more dimensions out there, there must be something interesting going on in them.

What "relatively uncomplicated condition"? As far as we understand it, our universe has three spatial dimensions, one temporal dimension, and seven curled up dimensions, and the physical laws of the universe can be explained by the properties of those curled-up dimensions. That's your answer, right there.

The GEOMETRIC properties of the four observable dimensions can be compounded infinitely.

What do you even MEAN by this statement?!

The Elegant Universe (and other books) pretty much explains this, in relatively uncomplicated terms. Yes, it can be tricky to get one's head around, but that's just how the universe is. Why don't you try reading that book, since you have it, and then ask questions based on that? It'll probably give you a better basic grounding of the subject than anyone here could.

[t00fri]

Um, physicists ARE seriously looking into other dimensions to "solve the mysteries of the universe" - that's what string theory and membrane theory is all about. For example, it seems that the physical laws of this universe are determined largely by how the other 7 (at least) dimensions that are all curled up at a subatomic level interact with eachother - so in a way the behaviours of the universe ARE driven by what's going on in unknown dimensions.

You may want to read The Elegant Universe by Dr Brian Greene for a primer on the subject, it explains things quite well and is a fascinating read.

As I said in the essay, the types of dimensions proposed in string theory and membrane theory don't concern me, because they have properties that aren't even possible in simple Euclidean dimensions. For example, in basic geometry, an object must exist in at least two dimensions to have curvature. Yet the dimensions in string theory are THEMSELVES curved. Also, basic dimensions are infinite, conisisting only of two opposite directions without size or scale. String theory's dimensions are, as you said, all confined to a finite subatomic size. There is some sort of complicated mathematics or advanced concepts involved in the dimensions of string theory, which I do not understand, and which I do not need to understand or be concerned with to ask my questions about dimensions.

And I already own a copy of The Elegant Universe, although I've only read a few pages.

Are you sure you understand what you are writing here?

Well, it is quite obvious that you do not understand string theory. Many people don't. But people like yourself, contemplating about related matters, definitely should try to. Please consider that in string theory the brightest minds of this globe are producing exciting results since > 20 years...

It is very hard to compete. Believe me...


Bye Fridger

[Nick]

First of all, don't think I presume to CHALLENGE string theory. I'm just saying that, from what I know, there seems to be OTHER possibilities as well. String theory is still only a theory, so proposing other things is still allowed. Besides, as far as I know, my issue does not interfere with string theory anyway.

Now, let me try to clarify my what my issue is. The three dimensions of space all share the same geometric principles. Each dimension is comprised of two opposite directions that are perpendicular to the directions of all other dimensions (up and down, left and right, forward and back). They are infinite, and they have no shape (except, perhaps, that of a straight line, since that is the only shape that can exist within a single dimension). You cannot start at one point, go one direction, and end up at the same point you started. Although the dimension of time has additional properties that makes it unique, it is nonetheless a dimension because it fullfills all of these criteria.

Thus, the three spacial dimensions and the dimension of time are the only known examples of the "kind of dimension" I am concerned with.

Now, correct me if I'm wrong, but it's my understanding that the extra dimensions in string theory do infact allow one (oh, lets just say it's an ant) to move in ONE direction through these dimensions and eventually return to the point where it began. According to the criteria I laid out above, that is not "my kind of dimension".

All I am saying with respect to string theory is that it does not examine "my kind of dimensions". My kinds of dimensions are the kinds exemplified by space and time. What I figured out a couple years ago was that the properties that define one of these dimensions can be repeated infinitely, making four-, five-, and infinite-dimensional space a geometrical possibility. The process of this discovery was a simple one, and I am by no means the first to understand it. You should be able to comprehend it fairly quickly if you tried, Ganymede, since you obviously have a very large mind (though perhaps not a very open one).

So, once again, what I discovered is that there is a mathematical possibility that our observable four-dimensional world (space-time) may only be a plane in an infinite-dimension universe, consisting ENTIRELY of "my kind of dimension". I did not attempt to actually come up with any physical theories, I only recognized the possibility that some of the unexplained behaviors of our universe might be caused by mechanisms outside our observable plane.

[t00fri]

Nick,

you mix up a number of things. Dimensions of spaces have neither a direction, nor are they "othorgonal" on each other etc. You have to consider different space(-time)s with their associated geometries, instead. Mathematicians call these 'manifolds'.

What you express in lengthy words is just the difference between compact and non-compact spaces.

The role-model of compact spaces is the surface of a sphere, or even simpler, all points on a circle! Homework: figure out their respective dimensions . How about an infinitely long 3d-cylinder? Compact or non-compact? . How about a 3d-doughnut? What's special with a 'doughnut' shaped space? How to "make" a 'doughnut'-space from a cylinder?

Now, string theory has to be initially written down for consistency in a space-time of 10 (or 11) dimensions. In order to be able to describe our 3+1 dimensional world, some kind of 'dimensional collapse' has to take place, that is called 'compactification'. The idea is that 6 dimensions 'curl up' into (tiny) compact spaces (6d-"spheres"), leaving a 3+1 dimensional non-compact space-time for us to live in.

Our 4 dimensional space-time world is considered non-compact since (at least) TIME, is thought to extend infinitely...

Now the process of 'compactification' leaves us with (~ infinitely) many inequivalent possibilities and consitutes the very reason why testable predictions of string theory are lacking at present.

In summary, in ten dimensions, string theory is beautiful and comparatively simple, a real candidate for the "theory of everything", including gravity. 'Compactification' introduces at the present time a large amount of arbitrariness, since 10 dimensional spaces may be 'decomposed' in many ways into our 3+1 dimensional world and 'something else'...

Bye Fridger

[dirkpitt]

Complicating the arena is the fact that in recent years, there have been a lot of talk about membranes ("branes"), not strings.
This relatively new "super" theory, called M-theory, is supposedly had some nice successes recently in the area of black holes
but it's still all theoretical and open to argument. There was even talk of detecting x-dimensional branes via gravity
measurements but so far the results are all negative.

Perhaps the new LHC under construction at CERN will provide new, badly-needed experimental evidence for M-theory.

[t00fri]

Complicating the arena is the fact that in recent years, there have been a lot of talk about membranes ("branes"), not strings.
This relatively new "super" theory, called M-theory, is supposedly had some nice successes recently in the area of black holes
but it's still all theoretical and open to argument. There was even talk of detecting x-dimensional branes via gravity
measurements but so far the results are all negative.

Perhaps the new LHC under construction at CERN will provide new, badly-needed experimental evidence for M-theory.

M-theory is much more than "membrane" theory!

Among string theorists it is actually a matter of taste what the 'M' is supposed to stand for. Many imagine 'M'=Mother, M-theory, as the embedding of all known string realizations. Others do not attribute any significance to the 'M'...

There are two different known ways how higher dimensional surfaces ("branes") may come in.

a) One carries the generalization one step further, by considering the smallest entities of the theory not to be strings, but rather membranes (surfaces). This approach is considered problematic by most string theorists and is currently not much pursued.

b) "Branes" of varying dimension are NOT fundamental objects in string theory, but rather are 'formed dynamically' if the interaction among strings becomes strong (and hence non-perturbative). The formation of such branes has been shown to be of crucial significance for the fact that different known realizations of string theory are actually 'dual' to each other, i.e. describe the same physics in terms of different degrees of freedom!

Our Universe is typically considered to be such a brane, embedded in a higher-dimensional (bulk) space.

M(other)-theory therefore encompasses /all known/ string-models and interpretes them as different manifestations of one hopefully unique string theory.


Indeed, if the energy scale of string theory was sufficiently low, tiny black holes could be produced at the forthcoming LHC proton collider.
They would show up as rather spectacular 'fireball' kind of events with very many particles in the final state.


Bye Fridger

[Vincent]

Just for information :

"The elegant Universe", Brian R. Greene, 1999

Grace to this book, I learnt a lot of things about M-theory... Quite simply...

@+
Vincent

[rthorvald]

you mix up a number of things. Dimensions of spaces have neither a direction, nor are they "othorgonal" on each other etc. You have to consider different space(-time)s with their associated geometries, instead

Hm... Just to throw some gasoline on the fire here, i think what Nick means is the type of geometrical dimensions exemplified in RAH??s "And he built a crooked house" - very funny story about an architect that built a house as a four-dimensional cube. It proved very difficult to leave the house after getting in...

For those not familiar with it, it??s the classic Hypercube, and it looks like this:

The point is that all angles in this picture are 90 degrees, and the inner cube has the same dimensions as the outer, giving us six cubes in total. Quite outlandish, but a fun thought experiment... In fact, it seems to me such a cube in reality just describes a regular sphere...

-rthorvald

[Evil Dr Ganymede]

You should be able to comprehend it fairly quickly if you tried, Ganymede, since you obviously have a very large mind (though perhaps not a very open one).

My mind is quite open thank you very much - you just haven't been explaining yourself clearly. Like I said, it's better to read something like the Elegant Universe book to get some basic knowledge of the subject, and then ask your questions.

But if I understand you correctly, you're wanting to know what it would be like to have universes with different 'uncompacted' dimensions? if so then you may want to take a look at this paper: http://www.hep.upenn.edu/~max/dimensions.html

[Nick]

Now we're getting somewhere. I am not surprised that physicists already had a name for "my kind of dimension", I just didn't know what it was. I looked at that article, Ganymede, and it is definently interesting, except that right now I couldn't possibly understand how Tegmark came to his conclusion. Until I understand, I will continue to have my doubts (which I think you will concede is the proper scientific attitude). It just seems to me that the possibilities of infinite UNCOMPACTED dimensions are more than a single article can put to rest.

Nevertheless, I will try to make a better effort to understand. I have started The Elegant Universe and already I'm beginning to get a better idea of some things.

Also, a correction to rthorvald's comments: A hypercube is composed of eight three-dimensional cubes, not six. And to clarify, none of them, technically, are "inner" or "outer" cubes.

Now, can someone tell me what makes a spatial dimension different from a temporal dimension? The best I could come up with so far is that time has all the properties of space, PLUS some other properties, and that those additional properties are what make it unique.

[JrzyCrim]

[t00fri]

...
It just seems to me that the possibilities of infinite UNCOMPACTED dimensions are more than a single article can put to rest.

Not at all. Non-compact manifolds (extending to infinity, of course) are known both in mathematics and in theoretical physics since a long, long time. There is really nothing special about them! You can easily compactify simple non-compact manifolds yourself:

For example:

Consider the line of real numbers extending from -infinity to +infinity ( imagine this to be the time-axis, for example!). This is a 1d non-compact manifold. It turns into a 'compact' one by simply identifying its -infinity and +infinity ends.

Then you got a 'space' that is equivalent to a circle, a most familiar example of a compact space

...
Now, can someone tell me what makes a spatial dimension different from a temporal dimension?
...
.

The answer depends on the space (or manifold) under consideration (see my explanations further up)! In other words, it depends on the metrics that characterizes distances of points in that space.

In our 4d real world, where we certainly want to incorporate the invariance of our force laws wrto (special) relativity, we are using a so-called Minkowski space-time metrics. It may conveniently be written as a 4x4 - diagonal matrix,

Code: Select all

g:= diag{1,-1,-1,-1}


It is used when we calculate the distance of points ("vectors") in our 4d space-time. Such a point has the general components

Code: Select all

x=(x0,x1,x2,x3)


wiith the first one being proportional to the time coordinate, while x1,x2,x3 stand for the point's x,y,z spacial coordinates.

The length squared of the vector x (i.e. its distance squared from the origin (0,0,0,0) ) is then calculated as

Code: Select all

length^2 = transpose(x).g.x = x0^2 - x1^2 - x2^2 - x3^2


and one can see that the time component x0 enters with a "+" sign, while all spatial components get a "- sign" from our metric matrix g. No matter in which Lorentz frame you measure the length^2 of x, it's always the same due to the peculiar structure of the metric g!.

Apparently, you can have vanishing 'distances' in our Minkowski space, without all components of x having to vanish! A remarkable fact that is directly imposed by special relativity, i.e. the requirement that the speed of light is to be the same in all coordinate frames! Actually, the spatial and time coordinates of light in any frame always satisfy

Code: Select all

x^2=0

Such points ("vectors") are thus also called "lightlike"!

Theoretical physicists often like to consider imaginary (i.e. complex) time t -> I*t. In that case the associated 4d space changes its nature: it becomes a so-called Euclidean space, with the metric matrix g being, instead:

Code: Select all

g_E = diag{1,1,1,1}


Now 'time' and space components are treated completely symmetrically!

There are many other kinds of 'space-times' known or under consideration by scientists: De Sitter spaces or Anti-de Sitter spaces have become again particularly popular recently...


These considerations are to illustrate that the treatment of spatial and time coordinates really depend on the context...

Bye Fridger

[Evil Dr Ganymede]

Until I understand, I will continue to have my doubts (which I think you will concede is the proper scientific attitude).

Having doubts when you know as much about the subject as an author is one thing. But having doubts when you don't know much about the subject is just being stubborn. Lots of people here and on other boards come up with statements and essays about how they think they've found out something great or that contradicts what has been published and discovered so far, when in fact they have completely misunderstood the subject. When those people stick to their guns and claim that their ideas are better or more valid than what's out there, it only makes them look stupid.

The best thing you can do is read up on the subject, try to understand it, and then you'll be in a much better position to decide whether your 'doubts' are valid or not about things. But don't go into it thinking you're right, because in all likelihood you're not.

[Nick]

I never said I thought I was right at all. I fully expect to have my doubts dispelled.

When I try to explain this in writing, it looks silly and overcooked. Suffice it to say I'm not a stupid person, and I understand that I might be out of my league. I only came here to see what people thought of my ideas, not what they thought of me.