A simple question on a very very complicated matter.

[easelic125]

I have a question on the directions of the fourth dimension.
And I already know that those directions are named as Ana and Kata, what I'm asking about is what direction it goes in.
I hear a lot about the fourth dimension being a dimension of time, and I have a bit of trouble wrapping my head around the idea of the geometry of time, therefore I ask, could the fourth dimension be a combination of Geometry and Time?

[jinydu]

Some people have thought about time as the fourth dimension, but in this forum, we assume that the "fourth dimension" refers to a fourth spatial dimension (hence the name "tetraspace").

If you ask what direction Ana and Kata are in, all I can say is that they are perpendicular to all directions that we have experienced. As for what directions they actually are, there are two points:

1) Nobody has seen them before, so we don't know. Ana and Kata are simply names we give to them.

2) Even if we did have 4D perception, it is doubtful whether a truly meaningful answer could be given. For example, imagine placing a 2D object on a square table. You are seated along one of the sides of the table, and decide which direction to call length and which to call width. However, the person who is sitting right next to you (NOT across from you), will have exactly the opposite idea. The direction you call length, he will call width. Similarly, the direction you call width, he will call length. Hence, which direction you (if you were a 4D being) choose to call Ana and Kata simply depends on your orientation (which way you are facing), and is hence quite arbitrary. Possibly the only exception is height. All observers living on a small part of a planet will agree on what direction to call height, because it is the direction in which gravity acts.

[RQ]

What most of you people don't realize when talking about perception in a higher dimension is that it's impossible. If you take Fred for example, and he has perception of a 3D universe instead of just a 2D one. Where would he look? This is assuming higher dimensions can't interact with lower ones. It's a bit more than an assumption, but nothing's for sure yet.

[jinydu]

RQ, I think you misunderstood what I was trying to say. I mentioned in parenthesis: if you were a 4D being. That is, I was thinking about how Emily would see things.

[RQ]

Yeah I admit I didnt read all of it... :?
I think higher dimensional perception is possible, but only if the person imagines a diagonal perpendicular at a given point to his dimension of forward perception. Of course all this has to be in the mind, so withouth visual aid, this is a tough job.

[easelic125]

jinydu! I do agree with you, in that this is very shakey ground.
It's sorta like giving two primative humans names, and then expecting them to have a complex conversation.
I think the secret to visualization is within the curvature of our 3 dimensional space.
If you think of how Riemann thought about it: flatlanders living on a piece of paper; if you crumple that paper, what will happen? The flatlanders will feel a "force". They will feel like they are moving in a straight line, but because that paper is in 3d space, they are actually moving in that 3rd dimension along that plane...
I believe the same thing is happening here, in the 3rd dimension. According to Einstein, matter-energy causes a cuvature in space-time. One could say that mass crumples space, and this is why we feel the force of gravity. We are actually moving in a direction that we cannot yet conceptualize.
ugg.

[jinydu]

According to General Relativity (GR), yes, gravity is actually a curvature in spacetime. Hence, the shortest distance between two points is a curved line (a geodesic). However, there is one thing to keep in mind.

A curved space does not necessarily need to have something to "curve into". For example, any point on a sphere can be determined using only two coordinates. In fact, the whole of spherical geometry can be formulated without assuming the existence of a surrounding 3 dimensional space.

The feature that distinguishes spherical geometry from regular (Euclidean) geometry is the "fifth postulate". In Euclidean geometry, for any given line and any given point not on the line, there is only one possible line that passes through the point and is parallel to the line.

However, in spherical geometry, it is postulated that for any given line and any given point not on the line, there are no lines passing through the point that are parallel to the line.

Notice the way that this postulate is stated. There is no reference to a sphere, let alone a 3D space surrounding the sphere. There is no need to "look outside the plane". In fact, this goes for all the theorems in spherical geometry. Fred could fully understand spherical geometry. He may find some of the ideas counterintuitie, but he would never have to visualize any 3D shape. We may also think that spherical geometry is counterintuitive; but as 3D beings, we have a privelege that Fred does not. We notice that the postulates are satisfied if we imagine everything taking place on a sphere, and reinterpret "line" to mean "great circle". Thus, we name the theory, "spherical geometry". We use this picture because it is more intuitive to us than the image of a line having no parallels. However, it is only a tool to help with visualization. It has no logical necessity. There is no postulate or theorem in spherical geometry that says "This is the geometry on the surface of a sphere."

Thus, saying that "spacetime is curved" is a more intuitive (or to some people, less counterintuitive) way of saying that the equations of Euclidean geometry don't apply, and different geometric equations must be used instead. Analogous to the situation before, there is no need to "look outside spacetime." Unfortunately, some of the gains in visualization are lost because this time, we cannot (or at least, find it very difficult) visualize objects with more than 3 dimensions. Still, people find this easier to swallow than simply throwing away the equations of Euclidean geometry without giving a reason.

[RQ]

First of all gravity does bend space-time, but not in the higher dimension as you refer to crumbling the flat piece of paper. It merely curves space.
Though no interaction can be achieved with different dimensions, whether a flatland citizen's flat sheet of paper is rectangular or a big paper ball, as long as all the information inside is the same, bending it will not cause any observable features. Bending a paper lengthwise or widthwise does not make it have 3 dimensions, because it still has no thickness (2D paper, not the ones we use).

This is easily shown by the fact that a triangle (2D figure as you all know) in a flat 2D world has from both sides a sum of 180 degrees of its internal angles. If we bend this triangle on a saddle, in our universe one side will have a sum of more than 180 degrees and the other less than. Now these effects are observable only to the 3D universe, and if they had an effect on its 2D citizens, then the triangle's sum of internal angles would have to add up to both less than and more than 180 degrees. Now this can't be, otherwise it would create paradoxes. This can only show, that even if a lowerspace universe were to be bent in the higher dimension to its higher dimensional observers, it would have no effect on its objects inside, and therefore not enter the higher dimension, but infact be of no observable capacity to its insiders, and using Occam's razor, they would not be a higher dimension!

Now, if for example as in Abott's Flatland, an orange were to pass through a 2D planiverse, and the people would see a dot instantaneously become a widening circle and then back to a dot and nothing. Now let's say that happened in reality to a lower dimensional universe. We have to agree that an orange has 2D incorporations, and in turn 1D ones, and dots. Now If initially the orange had those circles incorporated in itself when it entered the planiverse, where would the circles in the planiverse go to that the orange's circles are occupying? The answer is that Abott's Flatland is not to be taken literally but to help you visualize the higher extended spatial dimension. An orange isn't going to magically pop up in the air start enlarging and start getting smaller and vanish quite simply because the space would vanish magically when the orange enters our universe.

[jinydu]

Though no interaction can be achieved with different dimensions, whether a flatland citizen's flat sheet of paper is rectangular or a big paper ball, as long as all the information inside is the same, bending it will not cause any observable features. Bending a paper lengthwise or widthwise does not make it have 3 dimensions, because it still has no thickness (2D paper, not the ones we use).

Not exactly. It IS possible for flatlanders to detect the shape of their universe. One way to do is continue walking in (what they consider to be) a straight line. If they eventually return to their starting position, they can rule out the possibility of living on an Euclidean plane. Another way would be to go out into space, far away from any significant gravitational fields, and shine a pair of (initially) parallel beams of light. If the beams converge, remain parallel, or diverge, this would also tell them something about the shape of their universe. For more information about that, please see http://map.gsfc.nasa.gov/m_mm/mr_content.html And of course, they could draw a triangle out in space and measure the sum of the angles.

This is easily shown by the fact that a triangle (2D figure as you all know) in a flat 2D world has from both sides a sum of 180 degrees of its internal angles. If we bend this triangle on a saddle, in our universe one side will have a sum of more than 180 degrees and the other less than.

No, the sum of the angles on any triangle on a saddle is always less than 180 degrees. More information can be found here: http://mathforum.org/library/drmath/view/65121.html However, you are right in saying that the flatlanders would be unable to detect what's going on outside their "plane". In the example of flatlanders living on a sphere, they would be justified in saying "We live on a sphere, but there's nothing inside or outside the sphere."

Still, there are two conditions, which if both satisfied, would make it impossible (or at least very difficult) for flatlanders to detect the curvature (assuming it exists) of their universe:

1) Their universe is far larger than anything their technology deal with.

2) The shape of their universe is smooth.

If these conditions are met, the region of space that the flatlanders are able to study approximates Euclidean space very accurately. For example, flatlanders might try drawing what is to them a very large triangle and measuring the sum of the angles. They may end up with 180 degrees when the real sum is actually 180.000000000000000000001 degrees, well beyond their capabilities of measurement.

[RQ]

Yes flatlanders would only be able to detect that their universe was connected if they came back to the same spot by traveling in a straight (to them) line, but not any other way.
No, parallel light rays would remain parallel unless something deviated either one of them.
I know a saddle is always less than 180degrees, I was talking about the other side, the surface of a sphere, but didn't bother to write it.

I read some stuff about worm holes and "white" holes, but thought it was all bull so I left it alone.

The reason I consider getting to different universes and higher dimensions impossible is:
a) You would have to travel in the higher dimension to get there which would predict your own downfall.
b) The above. Having no "tridth" means you're litterally nothing.

[RQ]

If the names length width and height remain as defined as:
length: forward and backward
width: left and right
height: up and down

Then... height would be the first dimension, length second, and width 3rd if gravity was present in all three situations, since up and down would be height, then forward and backward (even if left to right is your second dimension to your personal choice, then you would have to move in those as if they were forward and backward. Then width.

Then again this is just naming of names...