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Author Topic: would it make sense to think of space-time as 3.5 dimensional instead of 4?  (Read 8299 times)

Offline McKay

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That is, if one dimension adds two possible directions to move in, would a half a dimension add only one direction to move in? That is - one way in time.


 

Offline PmbPhy

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That is, if one dimension adds two possible directions to move in, would a half a dimension add only one direction to move in? That is - one way in time.
Your confusing possible directions with dimensions. The number of dimensions is the number of integers required to uniquely located a particle. Thus it takes one integer to determine where a particle is on the x-axis, one integer to determine where a particle is on the y-axis, one integer to determine where a particle is on the z-axis and one integer to determine what time it is - that gives 4 integers.
 

Offline jeffreyH

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Well in a 2 dimensional universe gravity would operate as a 1/r force and not an inverse square force as in 1/r^2. The power the radius is raised to is always 1 less than the number of spatial dimensions. So in 4 spatial dimensions the law would be 1/r^3. These are all integer increments.
 

Offline CliffordK

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The number of dimensions is the number of integers required to uniquely located a particle. Thus it takes one integer to determine where a particle is on the x-axis, one integer to determine where a particle is on the y-axis, one integer to determine where a particle is on the z-axis and one integer to determine what time it is - that gives 4 integers.
Of course, dimensions are not limited to "integers", but rather "numbers" which would include rational, real, and perhaps imaginary numbers.

To define the time component as being uniquely positive may require a person to define an absolute zero, such as the theoretical big bang which may be inconvenient.  Yet, what you're trying to say is not necessarily that one can't consider prior events, but rather that the rate of change of time is always positive.

One doesn't necessarily consider a positive only axis as being any different from one that has positive and negative values.  So a box may have 3 dimensions even though it can not have a negative volume.

It is also often handy to define an arbitrary "zero", such as 0 AD, for which there are values before and after that point
 

Offline PmbPhy

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Quote from: CliffordK
Of course, dimensions are not limited to "integers", but rather "numbers" which would include rational, real, and perhaps imaginary numbers.
Thanks for correcting me Cliff. Let me restate the above and speak more precisely. What we're talking about are manifolds. The term manifold is simply a fancy name for the term set or space. A manifold is a special kind of space in that it has certain properties which, simply put, such as smoothness. That's where the term differential manifold comes from. See:
http://home.comcast.net/~peter.m.brown/math_phy/introduction_to_manifolds.htm

A point X in a manifold is a set of numbers X = ( x1, x2, .... , xn) where the xk is either a real number or a complex number. The number n is called the dimension of the space. It's the smallest number which is required to uniquely locate a point in the manifold.
 

Offline JohnDuffield

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would it make sense to think of space-time as 3.5 dimensional instead of 4?

That is, if one dimension adds two possible directions to move in, would a half a dimension add only one direction to move in? That is - one way in time.
No. You don't really move in any direction in time. Yes, people talk about moving forwards through time, but it's just a figure of speech. So is "a clock measures the passage of time". That's not what a clock does. It features some kind of local cyclical motion; it counts how many times some pendulum has swung or how many times some crystal has vibrated, and it shows you a cumulative result called the time. Time is a "dimension" in the old sense, where a dimension is a measure. It isn't a dimension in the contemporary sense, it doesn't offer freedom of motion. I can hop forward a metre but you can't hop forward a second. Or backward. 
 

Offline evan_au

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Quote from: McKay
would a half a dimension add only one direction to move in?
A linear ratchet moves in only one direction, but it takes a whole dimension of space to describe the position of the ratchet.

This dimension allows you to describe the position of the ratchet in the present, the future or the past. It does not imply that in the future you can move back to a position it had in the past.

Quote from: CliffordK
dimensions are not limited to "integers", but rather "numbers" which would include rational, real, and perhaps imaginary numbers.
Imaginary numbers have two dimensions, since they have two components (the "real" and "imaginary" components) which can vary independently.

Quote from: PmbPhy
A point X in a manifold is a set of numbers X = ( x1, x2, .... , xn) where the xk is either a real number or a complex number.
A point X composed of complex numbers could also be described as a point Y composed of real numbers Y = ( y1, y2, .... , y2n) where the yk are real numbers.

Quote from: McKay
a half a dimension
The concept of partial dimensions do occur with fractals.
We are used to the area of a circle increasing with the square of the radius, and the volume of a sphere increasing with the cube of the radius.

With certain fractal shapes, it is possible to find functions in nature that increase at an intermediate rate.
 

Offline yor_on

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Interesting Evan.

"With certain fractal shapes, it is possible to find functions in nature that increase at an intermediate rate." Although, I'm not sure how you mean there. Can you give a example of it?
 

Offline evan_au

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Quote from: yor_on
Can you give a example of [partial dimensions]?
The partial dimensions come about from the way we measure the object. Some examples from nature:
  • Dimension of the Coastline of UK: about 1.25
  • 3D Polymer: 1.66
  • Cauliflower: 2.33
  • Lightning strike pattern: 2.5
  • Surface area of the human brain: 2.79
  • Surface area of the lung: 2.97
The lung is a 2-dimensional surface, but its recursive structure has a dimension close to 3, which means that it almost fills a 3D volume.
 

Offline evan_au

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Another example of fractal dimensions that may be of more interest in a Cosmology/Astronomy thread is that the fractal dimension of galactic super-clusters in the universe seems to be around 2, as indicated by the Sloan Digital Sky Survey.

We normally think of physical space occupying 3 dimensions, but the amount of visible matter in this volume only seems to grow as the square as you consider larger and larger radii.

So the number of galaxies grows like the surface of a sphere, not the volume of a sphere, as you might expect. This has implications for the density of matter and the gravitational future of the universe.

It is thought that visible matter is clustered around denser aggregations of dark matter, so perhaps dark matter also has a similar fractal dimension?

The recent announcement of X-Rays that are a possible indicator of Dark Matter may eventually allow us to determine the fractal dimension of Dark Matter.
 

Offline yor_on

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What is your definition of a dimension there Evan? I'm not following it as you present it? presently some single malt might limit my understanding, but I still hope you will find a description that works, even so :)
 

Offline yor_on

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The real point there is that one certain definition is the one Pete gave.  "The number of dimensions is the number of integers required to uniquely located a particle. " I do not see why anyone would have to excuse that one, because that one is the one we can prove, at any time. The others may follow, mathematically, choosing a logic, but they are not what we can prove.

Can anyone see the difference between a mind game, and what we have? What we have decide your life.
I know, welcome to locality :)
« Last Edit: 02/01/2015 18:58:15 by yor_on »
 

Offline yor_on

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All depending on what 'dimensions' should be defined as naturally, and that's also why I guess Pete, and me too I'm afraid, are willing to listen. Experimentally though, Pete must be correct, as locally defined.
 

Offline yor_on

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The question that you need to consider is simple :)

what is real?

Your local definition, or a mathematical space?

tell me where you live.
 

Offline lightarrow

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What is your definition of a dimension there Evan? I'm not following it as you present it? presently some single malt might limit my understanding, but I still hope you will find a description that works, even so :)
http://en.wikipedia.org/wiki/Hausdorff_dimension
http://en.wikipedia.org/wiki/Fractal_dimension

examples:
http://en.wikipedia.org/wiki/Cantor_set
http://en.wikipedia.org/wiki/Koch_snowflake
http://en.wikipedia.org/wiki/Sierpinski_carpet
http://en.wikipedia.org/wiki/Menger_sponge

a list where you can unleash your fantasy:
http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

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« Last Edit: 03/01/2015 13:20:25 by lightarrow »
 

Offline Bill S

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presently some single malt might limit my understanding

Single Malt is the key to all wisdom.

For “is” and “is-not” though with Rule and Line,
 “Up-and-down” without, I could define,
I yet in all I ever cared to know,
Was never deep in anything but—Wine.  Single Malt.

With apologies to Omar.

 
« Last Edit: 03/01/2015 21:19:41 by Bill S »
 

Offline yor_on

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:)

thanks friends, and yep, single malt is indeed the water of life. And Evan, consider me as arguing :) so make me see what you think yourself seeing. Ah well. might have been a little too hasty, let's call it 'inspired' :). It depends on how you define it, mathematically or, let's call it, astronomically. Would our Omar be that famous Persian poet Bill? Anyway, you can use manifolds, but when I try to translate that into the way I observe dimensions, those we measure normally, a dimension of 2.5 becomes hard to imagine. But it might also be a question of what makes dimensions? I want to define a universe from a local 'point', and it is with that as with a Big Bang, also defined from 'local points' as I read it. You won't find a origin in some singular center, but you do have a accelerating expansion everywhere, which if you turn it around, in my thoughts can be defined as happening in each point, counteracted by gravity (as well as the other four, well three then, forces.)

Doing that the next step then becomes to ask what makes the 'dimensions' we measure. There Pete's definition suits me perfectly. We define those by the amount of coordinates we need to agree on a position, in time and 'space'.

(there is a subtlety to that one, depending on if you want the exact same answer, relatively speaking. Generally speaking though it doesn't matter if our clocks differ for this, as long as we agree on the same amount of coordinates needed.)
==

You can think of it this way, as a set of points, each point a center of your locally described universe, meaning all other 'points' you observe. Translated into relativistic terms it's always 'your local clock and ruler' defining it. To that one then need to add some principle allowing points to casually coexist. Maybe that can allow 2.5 dimensions to exist? It depends on what you mean by it, and how you will prove it.
« Last Edit: 04/01/2015 20:07:07 by yor_on »
 

Offline yor_on

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Let's use a simple one. You can translate a plane (2D) into a line (1D). There exist a correspondence in 'cardinality' meaning that instead of two parameter you can use one. That doesn't state that you from this line can translate it back to that exact position in the 2D plane you first measured, does it? If it does, it describes the dimensions I consider, if not, it's firstly a mathematical concept. Let's take it a step further, we live in three 'room dimensions', how would you translate a position defined in those, to two? And then be able to translate it back into the exact same three? Because that seems to be necessary for me, to accept it.
=

Evan, could it be here we differ? " A fractal has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space." some things as the lung and possibly also brain matter use a fractal behavior, but to me they still do it inside a three dimensional topology, four, if you include time.

« Last Edit: 04/01/2015 19:59:24 by yor_on »
 

Offline evan_au

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What is your definition of a dimension there Evan?
I've been trying to get my head around fractals since I heard about Benoit Mandelbrot's 1982 book The Fractal Geometry of Nature.

In my simple understanding:
  • You can take a 1-dimensional measure (like length) and apply this to a path in 2 or 3 dimensional space. As the complexity or roughness of the fractal line increases, the length also increases. The fractal line may fill a 1-dimensional space, a 2-dimensional space, or a 3-dimensional space, or (more likely), somewhere in-between.
  • You could extend this to a path in 4-dimensional space. Einstein spoke of Geodesics in spacetime, the shortest distance between two points; a fractal must be about the least direct path between two points!
  • You can also take a 2-dimensional measure like area, and apply this to a 2, 3 or 4 dimensional space.
To link back to the original question: to specify the position of a point on a path in a 4-dimensional spacetime, you need to specify 4 numbers: x, y, z and t.
 

Offline jeffreyH

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You can think of a zero dimensional space containing scalars. Then a one dimensional space containing only the line. In this case you can only have two points, one that defines the start of the line and another defining the end of the line. This description includes the zero dimensional scalar as the line has a length. In 2 dimensions there is the plane. No more than 3 points can definitely fall on the same plane if the plain is a slice of a 3 dimensional space. Therefore we can associate 3 points with the plain via this relationship. If we look at 3 dimensional space we have the cube, the sphere and countless other shapes that can be described. To find out if we have limits in 3 dimensions this would have to relate to a space with 4 dimension.

EDIT: It can be shown that for any 3 randomly distributed points a sphere can be constructed so that all 3 points become coincident with the surface of the sphere.
« Last Edit: 04/01/2015 23:05:22 by jeffreyH »
 

Offline PmbPhy

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From what I gather from what Evan said in this thread it appears to me that he has the wrong idea of how  the term "dimension" is defined. First of all when we speak of the term "dimension" it's important to keep in mind just what it is that we're talking about.

From Wiki - Dimension
Quote
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.
So the dimension could be describing an vector space, a manifold, a physical object etc.

Regarding what Evan posted: He made a few mistakes.

The dimension describes the space/manifold that the point is in and not something like a subspace that you have in mind like a geodesic in spacetime.

Regarding the terminology on world-lines and geodesics:

1) Not all world-lines are geodesics. However, all geodesics are worldliness.

2) If (a) the 4-force on the particle is not zero then the worldline not a geodesic and (b) if the 4-force is zero then the worldline is a geodesic.

3) Assume the spacetime is not flat and the 4-force is zero. Define S as

967c4703309ad2d8ea43a094fe20ee85.gif

where 268684c8e7cd5f12895426621ced3326.gif

(This is merely called the integral of the interval. Some people call it the "length" of the world line. That is a widely used misconception.)

While S is not necessarily the smallest possible value, it is a stationary value.
 

Offline lightarrow

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3) Assume the spacetime is not flat and the 4-force is zero. Define S as

967c4703309ad2d8ea43a094fe20ee85.gif

where 268684c8e7cd5f12895426621ced3326.gif

(This is merely called the integral of the interval. Some people call it the "length" of the world line. That is a widely used misconception.)

While S is not necessarily the smallest possible value, it is a stationary value.
What about calling S the proper (interval of) time? 😊

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lightarrow
 

Offline PmbPhy

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3) Assume the spacetime is not flat and the 4-force is zero. Define S as

967c4703309ad2d8ea43a094fe20ee85.gif

where 268684c8e7cd5f12895426621ced3326.gif

(This is merely called the integral of the interval. Some people call it the "length" of the world line. That is a widely used misconception.)

While S is not necessarily the smallest possible value, it is a stationary value.
What about calling S the proper (interval of) time? 😊

--
lightarrow
The problem with "the proper interval of time" is that it's not unique. It really is "a" proper time interval along "a" particular worldline. But you are right in the fact that it's a proper time interval.
 

Offline yor_on

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Don't know why Evan, but 'a fractal must be about the least direct path between two points!' catch my imagination. It's a very interesting idea, so in what way can we make this come true? Is there a way to prove such an idea?
=

and naturally, as it to me seems to be about symmetries :) we then should need to assume a opposite. Which to me makes the idea more than interesting.
« Last Edit: 23/01/2015 19:11:46 by yor_on »
 

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