Is my intuitive way of thinking about non-Euclidean geometry valid?

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Summary:

I try to separate in my head the idea of a curve from the actions taken to create that curve. This clears up the unintuitive concept of parallel lines intersecting.

Main Question or Discussion Point

I always tend to get confused when thinking about non-Euclidean geometry and what straight lines and parallel lines are. If I think of a sphere, I get how two people driving north would almost mysteriously intersect at the North Pole and how the angles of a triangle would not add up to 180 degrees. But in my imagination, and I presume other people’s as well, the two-dimensional surface of the sphere is embedded in a three-dimensional universe and from the three-dimensional point of view, there is no mystery. Those lines which met at the North Pole were never really parallel or straight to begin with. In fact, I find it impossible to visualize two parallel lines intersecting, but I’m told that they are really parallel and the three-dimensional space around them is not supposed to exist.

However, I do have a mental trick that I use to help me. I imagine people in cars who are each able to perform only two actions – step on the gas and/or turn the steering wheel. Then I think about what would happen if they were placed on the surface of a sphere, pointed north, and told to step on the gas without turning their steering wheels. They would meet at the North Pole. The reason this helps me is because I have mentally separated the path with the actions required to move along that path. The paths these two motorists moved along were still curved, but their actions (beginning in the same direction and stepping on the gas without turning the steering wheel) may be defined as “parallel actions”. So rather than thinking about parallel lines that intersect, I think about parallel actions resulting in curved lines that intersect.

Is this a valid way of thinking about the subject?
 

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PeroK
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Define "line" and define "parallel". If someone drives round the equator and someone else drives round a line of latitude, then their paths never cross. Are they "lines"; are they "parallel"lines?
 
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SamRoss
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A straight drive means stepping on the gas while keeping the steering wheel centered. Two drivers are parallel when they are pointing in the same direction from the point of view of an observer in three dimensions. A line is a collection of points in three dimensions such that a vector between any two distinct points is either parallel or anti-parallel to the vector between any other two distinct points. In the situation you described, the two drivers begin parallel and remain parallel (assuming they are at the same longitude the whole time) as they drive straight, though they are both driving in circles. In the situation I originally described, the two drivers were also driving straight although they would only have been parallel momentarily while they were both at the equator.
 
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Orodruin
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You generally should not think at all about any embedding space. For the sphere I know that it is easy to do, but you really shouldn’t. It is also not a good idea to create your own terminology as that will hinder you when you try to communicate with others.
 
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PeroK
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A straight drive means stepping on the gas while keeping the steering wheel centered. Two drivers are parallel when they are pointing in the same direction from the point of view of an observer in three dimensions. A line is a collection of points in three dimensions such that a vector between any two distinct points is either parallel or anti-parallel to the vector between any other two distinct points. In the situation you described, the two drivers begin parallel and remain parallel (assuming they are at the same longitude the whole time) as they drive straight, though they are both driving in circles. In the situation I originally described, the two drivers were also driving straight although they would only have been parallel momentarily while they were both at the equator.
There is a mathematical definition of lines and parallel, which is not dependant on the internal combustion engine!

For spherical geometry, for example, a "line" is taken only to be one of the "great circles". Lines of latitude don't count. The reason is that lines are defined (in this case at least) to be the shortest path between two points - which is always a segment of a great circle .

The set of great circles obeys Euclid's first four postulates, but not the fifth (parallel postulate). Which is the defining characteristic of non-Euclidean geometry.

The funny thing is that there is nothing strange or magical about this. It's rather obvious that any two great circles intersect at a pair of opposite points and that they meet the requirements of Euclid's first four postulates. You might ask why it took so long (2000 years!) for mathematicians to notice this. The only answer I can give is that they had strong preconceptions about what a line must be, and confused the physicality of a line on a plane with the raw Euclidean definitions they were dealing with.

You have the advantage of being able to learn from the last 300 years of mathematical maturity and do not have to fall into the same trap.
 
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SamRoss
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#PeroK #Orodruin

Can you recommend a beginner-level book on differential geometry? My main motivation is to understand GR.
 
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PeroK
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#PeroK #Orodruin

Can you recommend a beginner-level book on differential geometry? My main motivation is to understand GR.
If you want to learn GR at undergraduate level, there are three main prerequisites:

Calculus (single variable, multivariable, vector, diferential equations)
Special Relativity (at undergraduate level)
Lagrangian mechanics

All things calculus is here, for example:

http://tutorial.math.lamar.edu/

GR can only be at the end of a road of several years of study.

You could look at Susskind's theoretical minimum:

https://theoreticalminimum.com/courses

I have the book by Hartle:

https://www.goodreads.com/book/show/17301.Gravity

This is aimed at an undergraduate audience. He goes very easy on the differential geometry, but it's still an advanced undergraduate subject.
 
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SamRoss
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If you want to learn GR at undergraduate level, there are three main prerequisites:

Calculus (single variable, multivariable, vector, diferential equations)
Special Relativity (at undergraduate level)
Lagrangian mechanics

All things calculus is here, for example:

http://tutorial.math.lamar.edu/

GR can only be at the end of a road of several years of study.

You could look at Susskind's theoretical minimum:

https://theoreticalminimum.com/courses

I have the book by Hartle:

https://www.goodreads.com/book/show/17301.Gravity

This is aimed at an undergraduate audience. He goes very easy on the differential geometry, but it's still an advanced undergraduate subject.
I am familiar with undergraduate math and special relativity. I have read all of Susskind's Theoretical Minumum books. In fact, I have even read through Einstein's orginial papers on special and general relativity. I know the basic ideas. "Distance" is measured in proper time (being invariant for all observers) x^2-t^2. This holds when no mass is involved. Otherwise, the field equations set the stress-energy tensor equal to functions of the metric tensor, whereby the metric tensor can be solved for and geodesic lines determined which tell us the path of an object.

I have dipped into the subject over the years but I have never felt comfortable with it. Specifically, I find tensor calculus and differential geometry difficult. I read "Tensor Calculus: A Concise Course" by Barry Spain a long time ago (and Einstein actually does a good job of going through the basics of it in his paper) but I never mastered it, nor have I read anything specifically on differential geometry.

Many authors discuss tensor calculus and differential geometry simultaneously which I find difficult because I never developed a firm grasp of either one independently. Tensor calculus is concerned with equations that are true in all reference frames. Do you know of any books that discuss differential geometry without tensors (and some pictures would be nice) so I can get a handle on that before worrying about switching to other frames?
 
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PeroK
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I am familiar with undergraduate math and special relativity. I have read all of Susskind's Theoretical Minumum books. In fact, I have even read through Einstein's orginial papers on special and general relativity. I know the basic ideas. "Distance" is measured in proper time (being invariant for all observers) x^2-t^2. This holds when no mass is involved. Otherwise, the field equations set the stress-energy tensor equal to functions of the metric tensor, whereby the metric tensor can be solved for and geodesic lines determined which tell us the path of an object.

I have dipped into the subject over the years but I have never felt comfortable with it. Specifically, I find tensor calculus and differential geometry difficult. I read "Tensor Calculus: A Concise Course" by Barry Spain a long time ago (and Einstein actually does a good job of going through the basics of it in his paper) but I never mastered it, nor have I read anything specifically on differential geometry.

Many authors discuss tensor calculus and differential geometry simultaneously which I find difficult because I never developed a firm grasp of either one independently. Tensor calculus is concerned with equations that are true in all reference frames. Do you know of any books that discuss differential geometry without tensors (and some pictures would be nice) so I can get a handle on that before worrying about switching to other frames?
Well, according to that, you know more than I do!
 
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SamRoss
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Well, according to that, you know more than I do!
I seriously doubt that. Anyway, I just ordered a copy of "Elementary Differential Geometry" by Barrett O'neill. This will be my first book devoted specifically to differential geometry. Let's see if that helps!
 
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lavinia
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@SamRoss I like your idea of putting on the gas and not turning the wheel. If you drive at constant speed then you would feel no acceleration and that could be taken to mean straight. In Euclidean space a straight line is exactly a path of motion with no acceleration perpendicular to the curve. So at constant speed there is no acceleration at all. On the sphere straight lines are great circles precisely because your driver feels no acceleration tangent to the sphere if he follows one at constant speed. It is true that in 3 space the great circles are curves but their acceleration is perpendicular to the sphere so the driver feels no acceleration tangent to the sphere. If you think of the sphere as not in 3 space but just as an abstract surface then the idea of straight needs to be reformulated but it is still the same idea of no acceleration at least when moving at constant speed.

I like to think of the driver's motion as the analogue of inertial motion. It is acceleration free.

Another way of thinking of curves on the sphere or for that matter any surface in three space is that they are elastic bands stretched over the surface. One can ask how to stretch the band between two points with the least amount of stretching. I believe that Gauss called such curves "curves of least constraint" but don't quote me on that. The only contraint on a great circle is that it has to lie on the sphere. There is no way to release the tension further. But a latitude line feels forces pulling it towards a less stretched shape. If the band has wiggles then letting the wiggles relax releases forces and brings it into a less constrained shape. The wiggles would arise if the driver turned his wheel.

These ideas are different than axiomatic ideas of lines. In those one has an intuition of what straight lines look like and then tries to characterize the intuition with axioms. In a zero acceleration path one only knows what it feels like in the moment; no forces are pulling on you. There is no intuition of what the entire line looks like. On some surfaces, zero acceleration paths - called geodesics - can be quite unintuitive. For instance there are surfaces where a geodesic can bend around and cross over itself.
 
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lavinia
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I seriously doubt that. Anyway, I just ordered a copy of "Elementary Differential Geometry" by Barrett O'neill. This will be my first book devoted specifically to differential geometry. Let's see if that helps!
I find that book well written. His book on Kerr Black Holes is not elementary IMO. Also Struik's book on Classical Differential Geometry presents the geometry of surfaces without tensor calculus or differential forms. It only uses vector calculus. I found it a great start and often go back to it.
 
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"... what straight lines and parallel lines are"

Here is what "straight lines" are: For any close-enough points x and y on a straight line, the part of the line between x and y is the shortest possible path to get between x and y. This principle allows any portion of a line to be extended indefinitely in both directions; its maximal extension in both directions is the entire "straight line".

Think of two straight lines L, L' as being "parallel" exactly when there is a third straight line P that both L and L' are each perpendicular to. (Angles between two lines mean essentially the same thing in all geometries.)
 
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Orodruin
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Here is what "straight lines" are: For any close-enough points x and y on a straight line, the part of the line between x and y is the shortest possible path to get between x and y. This principle allows any portion of a line to be extended indefinitely in both directions; its maximal extension in both directions is the entire "straight line".
This is a particular definition. It works in metric spaces but not anywhere else. A more general definition is that a straight line has tangent vector that is parallel along the line itself, as defined by an affine connection. An example in physics of spaces that include straight lines but that are not metric spaces is Galilean spacetime.

Whether or not such a line can be infinitely extended or not depends on whether or not your space is geodesically complete or not.
 
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Yes, I'm discussing non-Euclidean geometry, i.e., geometry of manifolds that have constant curvature equal to +1 or -1 instead of 0. I wasn't looking to answer the original question in maximal generality.
 

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