Physicist: In a flat space local ideas about “parallel” and “perpendicular” are global. That is, if two lines are parallel, and you follow them for a while, then they’ll still be parallel. (By “flat” here I mean exactly this property, parallel is parallel forever. Not just “flat like paper”. So you can have 2-d flat space, 3-d, 4-d, whatevs)
An example of curved 2-d space is the surface of a ball (just the surface, don’t worry about the inside and outside). If you draw two parallel lines on the ball, then eventually they will cross. The curvature forces the two lines to come together.
A straight line on a sphere always traces out a "great circle", like the equator. These lines are parallel twice, but also intersect twice.
If an object is not experiencing any force, then it will travel in a straight line through space. This is true for space-time as well. So if you’re sitting still (traveling forward in time), and no one applies a force to you, you’ll continue to sit still (travel forward in time).
Now imagine two people hovering above opposite sides of the Earth. I say hovering in place because this means the lines they trace out in space-time are (initially) parallel. As you run time forward you’ll notice that, even though no force is acting on them (don’t say gravity) and they are traveling in straight lines through space-time, they still move together (fall toward the Earth).
This is due entirely to the Earth curving space-time around it. Literally, it takes the original “flatness” of empty space, and curves it. It’s a little more complicated because the time dimension and the spacial dimensions are fundamentally different, but not as much as you’d think.
Another, slicker-sounding way to describe gravity is: “Things fall because time points a little bit down”. That’s not creative prose, I mean that literally.
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I’m not sure i follow this explanation. It’s simple to see that you can have two parallel lines on a globe too. The difference between a ‘curved space’ and flat is that they will look different when ‘folded out’, but I can’t see why they should cross each other?
Drive you car around the globe and you will get two ‘parallel lines’ that never meet, from your wheels. Then imagine that you flatten out that globe, would those lines now cross each other? I don’t expect them to?
Do you?
It doesn’t seem right?
You can’t state that you have parallel lines if they intersect each other. Either you are right, and then you have defined a axiom that states that there are no parallel lines as they all can be made to cross each other, depending on how you fold ‘space’, which in a way makes sense, but not when comparing a globe to a two dimensional surface.
There’s a trick to it I’m sure, a certain kind of definition you set, but it’s not my understanding of parallel lines.
Yoron.
And what do you mean by a curved 2D space?
A bent paper might be defined as ‘curved’, but only when looked at from a 3D perspective. If I was to draw parallel lines on that paper, before bending it I do not expect them to intersect. And from the perspective of someone living on that surface I do not expect it to be seen either?
I think I can see what you’re getting at, but it’s easier for me to imagine ‘space’ as something plastic, with gravity as the metric ‘shear’, acting in it. I can understand that it’s tricky to describe it in words, from a mathematical standpoint, but there should be some way to make the idea more understandable. And with that I just want to state that I do not hold you responsible, you have a lot of companionship:)
That as I’ve seen similar definitions at several places, using a globe, drawing lines, comparing them to lines of longitude, and then stating that they will ‘cross’. But lines of longitude is a definition made to fit two ‘points’ (south and north pole) where they meet as I understands it, and are no explanation of why gravity would make parallel lines cross to me. Also it seems to me that if I was to draw two lines in space following gravity they would not ‘cross’ each other in any point, they would cross in the same manner as a bridge cross a river, not meet in any point, if you see what I mean. That a triangle won’t hold in a 3D reality is quite clear to me, but parallel lines crossing each other like as I seen described is not.
(ranting a little:)
Cheers
Yoron
@Yoron (last two)
Subtle point!
If you start walking in a straight line on the Earth (a perfectly spherical Earth), then eventually you’ll come back to where you started. More than that, your straight path will have divided the Earth exactly in half. Mathematicians like these paths so much they call them “great circles”.
You’ll find that the only straight paths on a sphere follow great circles. As a extreme example, imagine walking around the north pole at 89°N latitude (1° from the north pole). You’d be hard pressed to find anyone who doesn’t agree that you’re walking on a curved, not-at-all-straight, line. A more subtle curve could be found by walking around the Earth at 1°N (just above the equator). This looks like a straight line, and it’s very close, but it is slightly curved. If you were to walk along it you’d have to continually turn just a little north (like with the 89°N path). A quick way to see that is to remember that a straight line on a sphere divides the sphere in half, and 1°N doesn’t quite do that.
So the car tires in your example don’t quite run parallel. Given the chance (on a perfect sphere) they would eventually run into each other (90° around the planet). You don’t notice the effect because, while science has determined that the Earth is round, it has also determined that it’s not that round when you’re up close.
One of the more mathy way to talk about curvature is to look at the ratio of the circumference to the diameter of a circle. In flat space you always get pi.
In “positively curved” space you get something less than pi. For example, if you take the equator as your circle, and a line that goes through the north pole as your diameter, you’ll find that the circumference divided by the diameter is 2 (less than pi).
In negatively curved (hyperbolic) space you get something larger than pi.
So, in your example with the paper (beautiful example, by the way) the geometry on the paper stays the same even if you bend or fold the paper. Specifically, if you draw a circle, you’ll find that pi is 3.1415…
Stretching is a whole other thing though. Stretching is how you go from a flat piece of paper to a piece that’s curved like a sphere, and that’s the kind of stretching that gravity does to spacetime.
So….
Space can be stretched and the more it stretches, the more “time” (length) there seems to be….
Yet mass is never changed…. but then that means it’s slowed down….
but if something is slowed down, but in only one part of space…. then space really isn’t as flat as we are used to understanding the word flat…
mass, time, space… it’s like without those three, there wouldn’t be anything….
but then…. that almost sounds like what a black hole is…. nothing…. but then it isn’t nothing…. time and mass and space has just “changed” into something we have yet to grasp….
I have just went in complete circles here ^^;
One more thing ^^;
I also see that a Tesseract could be a nice example of never ending time, space, movement?
The car example is interesting but can be explained.
Because of the relatively small size of a car relative to the size of the earth the two tyretracks can be considered as one, or alternatively the cars path is effectively a cylinder not a sphere on these scales.
If you think now about a car with a wheelbase significantly wide in respect to the earth then each wheel would need to be set to be perpindicular to the earth surface in order to be cotact the road and be ‘parallel’ in terms of a sphere. We now set them down pointed north on the equator. They are now parallel and we do not change their direction in any way we can effectively remove the connecting axle. Run the car forward and you will find that the tyretracks converge and intersect at the north pole.
My thoughts on this issue are only guesses. Problem is, I simply can’t wrap my head around gravity. While its effects have been documented for centuries, gravity is still an enigma as to what it really is. Magnetism is also well documented and has many known functions. While an enigma in its own right, scientists state that each atom in the universe is imbued with this phenomenal trait of magnetism, making them in a sense; little magnets. The asteroid belt has chunks of matter with no particular shape, yet with smaller pieces of matter orbiting around them. Why? Our moon, while being lifeless, coreless and with little or no atmosphere is still magnetic because of an atoms phenomenal power. I know there is an old saying: “if it ain’t broke, don’t fix it”. But if only to give praise to where praise is due, both magnetism and gravity should be given another close look.