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View Full Version : Flow through a 5 dimensional ( 4d 1t ) space-time

tommac
2010-Nov-18, 09:42 PM
How would flow be represented in 4d space / time

Shaula
2010-Nov-19, 12:43 AM
Represented in what way? Flow can be represented as a vector field in which case it would be a 4D vector field not a 3D one. I don't understand what you want to know from this question?

antoniseb
2010-Nov-19, 12:49 AM
How would flow be represented in 4d space / time

What do you want to use this representation for? Just with anything else, the purpose provides the framework to define utility, and hence actual usage.
When describing a 3d flow graphically, on a flat page, it's done with a 2d diagram. Same would be true for a 4d 1t flow. If all you have is a flat page, you'll make a 2d diagram. Need something better? Define it, and an answer will probably present itself.

tommac
2010-Nov-19, 05:17 AM
What do you want to use this representation for? Just with anything else, the purpose provides the framework to define utility, and hence actual usage.
When describing a 3d flow graphically, on a flat page, it's done with a 2d diagram. Same would be true for a 4d 1t flow. If all you have is a flat page, you'll make a 2d diagram. Need something better? Define it, and an answer will probably present itself.

Well lets say a liquid flow through a 4d pipe and wanted to represent it in 3d ? would that be possible?

tusenfem
2010-Nov-19, 05:51 AM
sure just project it on the three unit vectors of your 3d space. basic math, just the same as you project 3d flow on a 2d piece of paper.
and then you even get to chose which three vectors make up your 3d space, just like projecting 3d flow on a 2d piece of paper.
to save typing I will used the 3d-2d example, then you just plot the flow in the XY - XZ - YZ plane.
but we are probably not told something significant and you want to turn it into a hologram most likely.

tommac
2010-Nov-19, 12:40 PM
sure just project it on the three unit vectors of your 3d space. basic math, just the same as you project 3d flow on a 2d piece of paper.
and then you even get to chose which three vectors make up your 3d space, just like projecting 3d flow on a 2d piece of paper.
to save typing I will used the 3d-2d example, then you just plot the flow in the XY - XZ - YZ plane.
but we are probably not told something significant and you want to turn it into a hologram most likely.

Not really a hologram but the hologram idea got me thinking and i wanted to do a thought exercise that required a 5d flow. here is the logic if you are curious.

I have always seen some sort of "special" relationship between a black hole and the edge of the visible universe. So I have been reading a little about holograms and it got me thinking a little.
For example ... lets take an expanding universe where there is just a super massive black hole and some distant stars marking the edge of the visible universe ( to be used as reference ).

From the perspective of the black hole, things around it are being pulled into it and distant stuff is being pushed away from it. So we can create a field using positive numbers ( + ) for all points where space is warped so that stuff is moving towards the black hole and ( - ) for all points where stuff is moving away from the black hole. each point would have a scalar associate with it also.

So we have a field and can see that it goes from -c to +c in all directions ... to me this seems like a flow of sorts. and maybe this is where I am wrong ... but anyway ... if we wanted to show this flow as a continuous flow as through a pipe we would need another dimension since the 3d flow is coming in from all directions.

To simplify ( or maybe to confuse ) lets take 40 days and 40 nights of rain ... we could say that there is a flow of water coming from the sky to the surface of the earth from all directions. The problem here is where does the water flow to? I think that water can have a flow if we were able to manipulate a 5th dimension here.

I am probably just confusing things more but trying to work through the mind exercise and was just curious what flow through a 5 dimensional pipe would "look" like.

Shaula
2010-Nov-19, 03:47 PM
I am probably just confusing things more but trying to work through the mind exercise and was just curious what flow through a 5 dimensional pipe would "look" like.
It wouldn't look like anything familiar. We work in 3+1D, trying to imagine a fourth spatial degree of freedom is pretty much impossible except by analogy. So to represent higher dimensional entities we slice them and project them into more familiar two and three dimensional representations. But to take a simple example - take any number of two dimensional slices through a cylinder. Which one represents the cylinder? None of them. The cylinder is represented by a mathematical relationship we cannot express in a 2D world. All we can do is project and try to form a mental image of the 3D reality. So it is for 4+1D objects.

tommac
2010-Nov-19, 04:11 PM
Well that was going to be my approach. Lets take a plane of space ( as mentioned above ) that cuts through the center of the black hole.

So we are reducing the 3d space to 2d. Now to add back the third dimension to create flow we can pull the singularity into a 3d and the edges of the universe on the other end. To create a 3d pipe with a flow of ??? (gravitational potential??? ) from one end of the pipe to the other.

Forget about the what is flowing ... but could it make sense to project the 3d+1t space-time to a 4d+1t space-time in this fashion? Basically by taking a slice then adding a dimension?

It wouldn't look like anything familiar. We work in 3+1D, trying to imagine a fourth spatial degree of freedom is pretty much impossible except by analogy. So to represent higher dimensional entities we slice them and project them into more familiar two and three dimensional representations. But to take a simple example - take any number of two dimensional slices through a cylinder. Which one represents the cylinder? None of them. The cylinder is represented by a mathematical relationship we cannot express in a 2D world. All we can do is project and try to form a mental image of the 3D reality. So it is for 4+1D objects.

Shaula
2010-Nov-20, 03:15 AM
Why would you do this? You are starting off by throwing data away. It would make more sense to reproject the 4+1D data into a different representation/set of basis vectors. Otherwise you are effectively only generating a new view of the three dimensional slice, not saying anything about the 4+1D original. You seem to be mixing spatial and more abstract degrees of freedom as dimensions pretty freely as well - which is dangerous ground, mathematically. NB: I am not a mathematician.

tommac
2010-Nov-20, 02:21 PM
what data am i throwing away? I do agree after further analysis that the 5d flow could probably be represented in a better way the the 5d pipe that I thought would be an interesting way to view it. Especially since it is a flow of vectors ( vector fields ? ) .

Why would you do this? You are starting off by throwing data away. It would make more sense to reproject the 4+1D data into a different representation/set of basis vectors. Otherwise you are effectively only generating a new view of the three dimensional slice, not saying anything about the 4+1D original. You seem to be mixing spatial and more abstract degrees of freedom as dimensions pretty freely as well - which is dangerous ground, mathematically. NB: I am not a mathematician.

The reason for the slicing is because it is hard to envision the 4d space. To simplify I removed a dimension in 3d space and then replaced it with the 4th spacial dimension. This helps me understand the 4d flow that I was trying to envision. In 5d ( 4s +1t ) I believe you would be able to have the entire 3d universe flowing through another dimension ( I think and thus the question ).

For a quick sample of what I was trying to do was take a bouncing ball as a function of time. One could take a plane of space lets take x,y. and ignore the z dimension and replace it with t ( with some space to time ratio of our choosing ) you could then represent 4d space-time in a 3d graph or visualization.