How do you 'travel' in a single dimension? [closed]

Question

I do not know if I should ask this on the 'theoretical physics' stack, but I'm afraid I know nothing about Physics so I'll look like a clown there.

There are many stories in which people travel through a 4th dimension. What happens to your other two dimensions when you do that? To my limited, high-school level knowledge, when a three dimensional object is moving, it makes changes to its coordinates in the x,y,z axes.

Did the Harry Potter series get this right? When Harry apparates, he feels that everything is pressing down toward him, like he is being forced inside a very tight tube. Is J K Rowling hinting that Apparition is one-dimensional travel through wormholes?

Answer 1

The basic concept of travelling in a single dimension is that your coordinates change only in that dimension, but stay the same in every other dimension. For example, if you're going up in an elevator, you're changing your height, but staying in the same latitude and longitude.

The problem with asking about travelling in a 4th dimension is the question of what exactly that fourth dimension is. Some physicists claim that time is the 4th dimension -- so the time turner would take you back in time but leave you in the exact same place.

I'm not sure that apparating would count as travelling in one dimension only, but it's one possibility. It's also possible that the mechanism of travel matters more than the direction (which is what dimensions refer to). Travelling by Portkey, you feel like you're being dragged from around the navel. Travelling by Floo is a different set of sensations. As far as I know, Rowling never specified exactly how any of these mechanisms work.

Answer 2

You're question seems reasonable to me (I'm a physicist :-) so I'm not sure why you've been downvoted.

The usual analogy is to consider a 2D creature living on a sheet of paper. If you draw a circle round him, the creature is trapped because he can't get out of the circle. But suppose he can travel in the third dimension, that is up above the paper. Now can can fly over the circle then drop back onto the paper. By travelling in the 3rd dimension he has apparently travelled through an impenetrable barrier. From a mathematical perspective exactly the same argument can be applied to 3D creatures like me (and you? :-). If I can move in a fourth dimension I can get out of a 3D box by moving some distance in a 4th spatial dimension, bypassing the box then dropping back. The trouble is that this is impossible for us 3D creatures to visualise, which is why the 2D analogy is useful.

So far so good, but note the the 4th dimension isn't a shortcut. Even if it existed, which it probably doesn't or at least not in the form described above, travelling in the 4th dimension would take just as long as travelling in the other three. When SciFi authors babble about the fourth dimension they're probably thinking about something like a wormhole. See http://en.wikipedia.org/wiki/Wormhole or Google for endless stories involving wormholes.

Wormholes are a possible, though so far purely hypothetical, outcome of General Relativity. GR is formulated using a mathematical structure called a four dimensional manifold, so it's very common to think of time as the fourth dimension, but this is a bit misleading. For example String Theory assumes that there are 9 spatial dimensions plus time, so if it turns out to be true should we describe time as the 10th dimension instead of te 4th dimension? Also, while the 4D anifold used in GR works very well, the nature of the time dimension is very different to the 3 spatial dimensions. In Physics speak it has a "different signature". Thinking of time as a dimension just like 3 spatial dimensions can lead you down blind alleys.

Answer 3

There is a novel dealing with this subject, entitled "Flatland; A Romance of Many Dimensions." It deals with a two-dimensional figure, A. Square, who exists in a two-dimensional society. A sphere drops in one day to show him other dimensions, notably Pointland and Lineland (one-dimensional travel). All of the beings in Lineland know only their neighbors, as they can only travel in one line. http://en.wikipedia.org/wiki/Flatland Interestingly, there is speculation about four-dimensional beings, but the Sphere dismisses this as crazy.

Answer 4

John Rennie actually said this in his answer already, but only as a sidenote.

The crucial distinction to think about, IMO, is that between a vector space (or an affine space) and a manifold. This is actually pure mathematics, but I dare to give an overview here anyway.

A vector space is the kind of mathematical space that we're familiar with; in such a space "3-dimensional" means that you can pick any point and describe its location very simply with just 3 numbers: you need some conventional basis of three orthogonal¹ vectors, and then you just say "go 4' in direction e_x , then 7' in direction e_y and finally 2' in direction e_z ." Important: this description is unique, i.e. once you've fixed your basis there's no different alternative way to describe that point. That means that, to get from one point to another one, you always have to travel that distance. There are no short-cuts, in the sense that you can always find the shortest route by just walking as staighly forwards as possible, namely in a straight line.

A manifold is more general. A manifold is locally a vector space. The usual example is the earth's surface: if you're just interested in a small area, you can easily make a map of it, which is a (bounded) 2-dimensional vector space. Still, the earth surface as a whole is not a vector space, but a 2-sphere. That's a very simple manifold: its fundamental group is trivial, which means that there is still a unique shortest way that can be found by going as straightly as possible, i.e. to be found by tightening a string between the two points on a globe. However there exist more complicated manifolds, for example the 2-torus; think of a doughnut's surface. On such a manifold, there are multiple topologically different ways to go, and it's not possible to, and it's not possible to determine the shortest route by pulling a single string. Now it might be possible that one knew only about one of the possible routes, and be utterly surprised that there was a shorter way all the time: a shortcut.

What does this have to do with dimensions? The thing is, we know pretty certain that "our normal" 3-dimensional space has no such notrivial topological features, so if there were to be shortcuts they'd need to be embedded in a higher-dimensional space that we can't observe right now. That's the "extra-dimension".

¹Actually, linearly independent is sufficient.