N Dimensions

[Ashish]

First of all i dont have much knowledge of string theory.

 

 

String Theory talks about 11-Dimensions.

 

On the other hand all the things or object to which human can observe are 3D and for more details 1 for time, even this too make sense.

 

But according to string theory all the dimensions more than (3+1) are infinitesimal.

 

OK now consider a any mathematical function, f(x,y,z) which when plotted, you can see it in 3D and if you wanna see it more from that than you can zoom up to the limit of your computer performance, memory than also you'll not be available to see any further dimensions. This is not only the case with computer but in real paper and pencil also.

 

This is the beautiful pure world of theoretical mathematics where u can go upto any zoom or infinite.

 

But then also you cant see or imagine of any further dimensions.

[Bignose]

I can imagine many, many more than just 3 dimensions. You just have to know what you are looking for:

 

Let me give you an example:

 

Consider a single particle in space. It will use 3 dimensions just to describe its location, x, y, z. Now, consider further that that particle is moving. So it will will have velocities in the x,y,z directions as well, call them v_x, v_y, v_z. That velocity can really be considered a second set of dimensions.

 

A new dimension is needed when the current dimensions cannot adequately describe different states that can exist. In this case, using only position, one cannot adequately discriminate between a particle at rest at 0,0,0 or a particle zipping along at half the speed of light that just happens to be at 0,0,0 at a single instance in time. So, to adequately describe the differences in these two particles, you have to introduce the velocity dimensions as well.

 

This can be naturally extended again where the particle's accelerations could be another 3 dimensions. So, a single particle could need up to 9 dimensions to describe it. This is really only just about limited by your imagination. What if the size of the particle mattered? The particle volume could then be another dimension. What if it were made of different substances? A dimension capturing the concentration of the different materials in the particle could also be important.

 

Now, let's further expand this idea of the single particle to a multi-particle system where position and velocity are the properties of the particles we want to study. Say we were studying a system with N particles. To describe that system, we would actually need 6N dimensions to describe the state of that system. A system of 500 particle would need 3000 dimensions!

 

Let [math]P^{[N]}(\mathbf{x}_1,\mathbf{v}_1,\mathbf{x}_2,\mathbf{v}_2,...,\mathbf{x}_N,\mathbf{v}_N,t)[/math] denote the probability of finding the system in a state where particle 1 is within [math]d\mathbf{x}_1[/math] of position [math]\mathbf{x}_1[/math] with a velocity within [math]d\mathbf{v}_1[/math] of velocity [math]\mathbf{v}_1[/math], and so on for each of the N particles at time t.

 

The equation that describes the system is known as the Liouville Equation:

 

[math]\frac{\partial P}{\partial t} + \sum^{N}_{i=1}\mathbf{v}_i \cdot \frac{\partial P}{\partial \mathbf{x}_i} =0 [/math]

 

And, again, P is a function with 6N+1 (+1 for time) dimensions.

 

What "extra" dimensions come down to is the need to describe things that cannot be described by the "old" dimensions. They aren't necessarily some magical or unimaginable things. You just have to know what the math is saying they are, and know how to interpret them.

[Ashish]

but could you make me to visualize it so as to make it comprehensive

[toastywombel]

but could you make me to visualize it so as to make it comprehensive

 

Watch this video http://www.youtube.com/watch?v=JkxieS-6WuA. It is obviously hard to explain how one can visualise beyond the third dimension, my visualisation kinda follows this video. It is a really good watch if your interested in string theory/ super-string theory/ m-theory/ f-theory.

[Ashish]

I can imagine many, many more than just 3 dimensions. You just have to know what you are looking for:

 

Let me give you an example:

 

Consider a single particle in space. It will use 3 dimensions just to describe its location, x, y, z. Now, consider further that that particle is moving. So it will will have velocities in the x,y,z directions as well, call them v_x, v_y, v_z. That velocity can really be considered a second set of dimensions.

 

A new dimension is needed when the current dimensions cannot adequately describe different states that can exist. In this case, using only position, one cannot adequately discriminate between a particle at rest at 0,0,0 or a particle zipping along at half the speed of light that just happens to be at 0,0,0 at a single instance in time. So, to adequately describe the differences in these two particles, you have to introduce the velocity dimensions as well.

 

This can be naturally extended again where the particle's accelerations could be another 3 dimensions. So, a single particle could need up to 9 dimensions to describe it. This is really only just about limited by your imagination. What if the size of the particle mattered? The particle volume could then be another dimension. What if it were made of different substances? A dimension capturing the concentration of the different materials in the particle could also be important.

 

Now, let's further expand this idea of the single particle to a multi-particle system where position and velocity are the properties of the particles we want to study. Say we were studying a system with N particles. To describe that system, we would actually need 6N dimensions to describe the state of that system. A system of 500 particle would need 3000 dimensions!

 

Let [math]P^{[N]}(\mathbf{x}_1,\mathbf{v}_1,\mathbf{x}_2,\mathbf{v}_2,...,\mathbf{x}_N,\mathbf{v}_N,t)[/math] denote the probability of finding the system in a state where particle 1 is within [math]d\mathbf{x}_1[/math] of position [math]\mathbf{x}_1[/math] with a velocity within [math]d\mathbf{v}_1[/math] of velocity [math]\mathbf{v}_1[/math], and so on for each of the N particles at time t.

 

The equation that describes the system is known as the Liouville Equation:

 

[math]\frac{\partial P}{\partial t} + \sum^{N}_{i=1}\mathbf{v}_i \cdot \frac{\partial P}{\partial \mathbf{x}_i} =0 [/math]

 

And, again, P is a function with 6N+1 (+1 for time) dimensions.

 

What "extra" dimensions come down to is the need to describe things that cannot be described by the "old" dimensions. They aren't necessarily some magical or unimaginable things. You just have to know what the math is saying they are, and know how to interpret them.

And that other set of dimension will also be in 3D how hard you try and if time then 4D. Because we cant imagine no..... we can imagine but we cant see.

 

 

And if you think so I'm wrong with my thinking can you show me any real world example of any object having dimensions more than 3. (ONLY IMAGE)

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Merged post follows:

Consecutive posts merged

Watch this video http://www.youtube.com/watch?v=JkxieS-6WuA. It is obviously hard to explain how one can visualise beyond the third dimension, my visualisation kinda follows this video. It is a really good watch if your interested in string theory/ super-string theory/ m-theory/ f-theory.

I've seen it year before also and I'm able to comprehend upto 5D and after that it went off my mind.

 

But whatever its only explanation but hard to believe coz' human cannot visualize dimension more than 3D.

Edited February 14, 2010 by Ashish
Consecutive posts merged.

[Bignose]

And if you think so I'm wrong with my thinking can you show me any real world example of any object having dimensions more than 3. (ONLY IMAGE)

 

What troubles are you having imagining a particle with a position, a velocity, and an acceleration at a certain time that requires 10 dimensions?

 

If you need more dimensions, again, let's consider a special particle.

 

Let's consider it to be a cell. Where we want to describe the cell by its size (volume as another dimension), by its age since it last split (age as another dimension), by the concentrations of 20 different chemicals in it (those 20 concentrations are another 20 dimensions), the location of the nucleus inside the cell may require another 3 space dimensions, etc. etc. etc.

 

No matter how you try, you cannot describe the concentration of lysine in a cell using x,y,z. You need another dimension, concentration of the amino acid lysine. This is a real world example where you need more than 3 dimensions.

[ajb]

But according to string theory all the dimensions more than (3+1) are infinitesimal.

 

They may be large or small, but not infinitesimal.

[michel123456]

That's all in the definition of what a dimension is.

Bignose seems to present a dimension equal as a property.

In String Theory, all dimensions are spatial dimensions (except one for time). i don't think extra dimensions are meant to be properties in S.T.

[vordhosbn]

Why not? If you plot the volt-ampere characteristic of a diode in a x,y diagram, you are essentially doing the same.

[Sagar Gorijala]

Watch this...

3 or more Ds?

 

 

There are three and 3 dimensions only.

FTOE

 

 

There are 3 length and 1 Time dimensions only.

FTOE

We can ignore other dimensions for our convenience but in reality there are 3 and 3 only length Ds and 1 time D. That's all, there is to it.

As there are only 3Ds, the 4thD can't be plotted. Time is an imaginary D. Imaginary in a sense, it has no physical existence. It is change... Time=Change. It can't be touched or eaten or smelled or felt or what ever... It does exist, it's change. We can plot a line - 1D, a 2D graph and a 3D graph but not more than 3D....

Edited November 14, 2011 by way12go