Lines in higher than 3D

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  • #1
ftr
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I read somewhere that a unique line connecting two points does not exist in higher than 3D, is this correct?
 

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  • #2
jambaugh
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No. It isn't higher dimensions but curvature that makes non-unique (locally) shortest paths. What is the shortest path from the north pole to the south? (on the surface please!)
 
  • #3
jambaugh
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Oops, I slightly misread but my statement and example still applies. What is not unique in higher dim is a line unmoved by rotations. Elementary rotations are really not "around an axis" but rather within a plane. In N dimensions the "axis" will be N-2 dimensional while the plane is the plane.
 
  • #4
ftr
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No. It isn't higher dimensions but curvature that makes non-unique (locally) shortest paths. What is the shortest path from the north pole to the south? (on the surface please!)
Thanks for the reply. I am new to differential Geometry. I have quoted this post because it is a good example to further explain my question. Now, if you run a straight line from NP to SP that is unique. these other Longitude lines that run on the surface are not unique. So my question is (I guess) can we have a unique STRAIGHT line in higher than 3D.

also, I have read that the concept of line(I guess they mean straight line) in higher than 3D does not make sense or something to that effect.

If my question is not making sense please tell me.
 
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In "regular flat space", i.e. not a sphere or something like that (the name for this is Euclidean space), two distinct points always uniquely determine a single straight line. This holds no matter how high the dimension, even in infinite dimensions.

In curved spaces it may not be true for all points (the case already given of antipodal points on a sphere is an example). And in some spaces, the notion of "straight line" is not meaningful, for instance a manifold without a defined metric. But these considerations don't have anything to do with dimension.
 
  • #6
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Thanks for the reply. I am new to differential Geometry. I have quoted this post because it is a good example to further explain my question. Now, if you run a straight line from NP to SP that is unique. these other Longitude lines that run on the surface are not unique. So my question is (I guess) can we have a unique STRAIGHT line in higher than 3D.

also, I have read that the concept of line(I guess they mean straight line) in higher than 3D does not make sense or something to that effect.

If my question is not making sense please tell me.
Somehow, I don't think you're meaning to say "straight line", but rather "geodesic". A geodesic is a shortest line connecting two points (this isn't the definition of a geodesic, but I'm being informal). Geodesics make sense in a very general context. The concept of "straight line" only really makes sense in a vector space (this is basically just ##\mathbb{R}^n##, so ##n##-dimensional flat space).
 
  • #7
ftr
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Thanks micro and tiny ( what coincidence of names). Please bear with my ignorance. Now, a line element length can be described as

s^2=dx1^2 +dx2^2 +dx3^2 +dx4^2 + ....

for 3D dx=1 line element length =sqrt(3)

for 4D dx=1 line element length =sqrt(4)

for 3D that is clear but for 4D do I actually walk that much if I go into that space, sounds very strange.
 
  • #8
WannabeNewton
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And in some spaces, the notion of "straight line" is not meaningful, for instance a manifold without a defined metric.
Hi Tiny! If by "straight line" you mean geodesics then you can in fact define geodesics without any riemannian metric as long as you have an affine connection ##\nabla## endowed on your smooth manifold. The curves satisfying ##\nabla_{\dot{\gamma}}\dot{\gamma} = 0## are called affine geodesics and intuitively these are the curves that parallel transport their own tangent vectors. This definition is much more intuitive geometrically in my opinion than the notion of a geodesic as something that has vanishing first variations of arc length or energy. If you don't have a smooth manifold (i.e. just a topological manifold) then yeah you're perfectly right. However if we are talking about smooth manifolds then you can always give the manifold a Riemannian structure so it really isn't much of an issue. I hope that bit of trivia was worthwhile, cheers!
 
  • #9
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Thanks micro and tiny ( what coincidence of names). Please bear with my ignorance. Now, a line element length can be described as

s^2=dx1^2 +dx2^2 +dx3^2 +dx4^2 + ....

for 3D dx=1 line element length =sqrt(3)

for 4D dx=1 line element length =sqrt(4)

for 3D that is clear but for 4D do I actually walk that much if I go into that space, sounds very strange.
Do you mean straight line or geodesic [as the previous posters pointed out?] Assuming you really do mean straight line, then I am still not exactly sure what you mean: let's back down a bit, how about line in 1D [just line by itself], line in 2D [line in a plane], and line in 3D? You see that any line in 3D space can really be contained in a plane, so there is really no difference between a line in 2D or 3D or 4D -- they are one-dimensional entity, what exactly is confusing you?
 
  • #10
ftr
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Do you mean straight line or geodesic [as the previous posters pointed out?] Assuming you really do mean straight line, then I am still not exactly sure what you mean: let's back down a bit, how about line in 1D [just line by itself], line in 2D [line in a plane], and line in 3D? You see that any line in 3D space can really be contained in a plane, so there is really no difference between a line in 2D or 3D or 4D -- they are one-dimensional entity, what exactly is confusing you?
yes, I meant a straight line and not a geodesic. Of course I can understand it in the sense of the mathematics, well sort of. But when in 4D and higher you cannot see it as a line (or anything else for that matter) in the usual "physical" representation, so I don't know really what to make of it.

So, in that sense whether it is GR or string it seems what they imply is that our reality is indeed a mathematical entity, at least this my naive interpretation.
 

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