The parable of the two travelers

[stevmg]

In a 2D world, two travelers at the equator of a sphere at different points on the equator travel North starting parallel to each other. As they move up the latitude, they gradually approach each other as they maintain their longitude geodesic. They interpret the closing linear distance as acceleration towards each other and a "force of attraction." In reality, it is the curved 2D surface that draws them together.

The question is: what makes them move north in the first place. If they sat still on the equator there would be no relative motion towards each other and no "gravity."

My take is that with referfence to this 2D world, the northward movement would be due to the time coordinate in the third dimension not seen in the 2D world and worldline expansion, which would occr even if both bodies were totaly at rest on the surface with respect to each other and the movement would not be averted and the "gravity" would still appear.

I assume, analogously in our 3D world, the unseen 4th dimension, time, marches on and there is always a worldline created in spacetime which is ever getting longer and this line, if subject to an uneven or curved spacetime would likewise come out as gravity as the world lines would not be "straight."

 

[JesseM]

Are you familiar with the concept of a "geodesic"? On a curved 2D surface this would be the path that minimized the distance between any two points on the path (like a section of a great circle on a sphere), while in spacetime this would be the worldline that maximized the proper time between any two events on it. So yeah, when people say that according to GR objects follow geodesics and this explains gravity, they're talking about geodesics in curved spacetime rather than geodesics in curved space, with a geodesic being a set of events that constitutes the object's worldline. In this sense you're right that it wouldn't make sense to consider an object fixed at single position on a curved globe rather than tracing out a continuous path, since that would correspond to an object that only existed at a single instant in spacetime rather than tracing out a continuous worldline.

 

[DrGreg]

The question is: what makes them move north in the first place. If they sat still on the equator there would be no relative motion towards each other and no "gravity."

Remember, in this analogy "distance north" is equivalent to "time". So you are asking why do things move forward in time, or rather, why do they persist over time. Why don't objects spontaneously appear from nowhere and then immediately disappear again?

You shouldn't think of things moving through spacetime. Things (can) move through space, but a point-object in space becomes a line in spacetime (not a point moving along a line, just a line). Time is one of the dimensions within spacetime; there is no external-time over which you can watch spacetime change.

 

[stevmg]

Are you familiar with the concept of a "geodesic"? On a curved 2D surface this would be the path that minimized the distance between any two points on the path (like a section of a great circle on a sphere), while in spacetime this would be the worldline that maximized the proper time between any two events on it. So yeah, when people say that according to GR objects follow geodesics and this explains gravity, they're talking about geodesics in curved spacetime rather than geodesics in curved space, with a geodesic being a set of events that constitutes the object's worldline. In this sense you're right that it wouldn't make sense to consider an object fixed at single position on a curved globe rather than tracing out a continuous path, since that would correspond to an object that only existed at a single instant in spacetime rather than tracing out a continuous worldline.

Yes, I am. That is EXACTKY what I am referring to here. The point I was trying to make is that in spactime, there are no "fixed points" (unless they existed instantaneously) and that the two travelers, at the equator, would move along a geodesic even if they were in the same time frame of reference and didn't "move" at all. Their timelines or worldlines would necessarily "grow." This would generate the motion which would b necessity be curvilinear with respect to each other nad "gravity" (or at least acceleration) would appear.

That is my supposition. I hope it is correct in a general sense. Your reply above indicates that you might agree with me.

 

[stevmg]

Remember, in this analogy "distance north" is equivalent to "time". So you are asking why do things move forward in time, or rather, why do they persist over time. Why don't objects spontaneously appear from nowhere and then immediately disappear again?

You shouldn't think of things moving through spacetime. Things (can) move through space, but a point-object in space becomes a line in spacetime (not a point moving along a line, just a line). Time is one of the dimensions within spacetime; there is no external-time over which you can watch spacetime change.

Yea, I agree with that. In my case "North" is not time but one of the two dimensions. Time is along the vertical radius from the center to the N Pole and everything slips "up" as time marches on in its own dimension forcing the curvilinear path on the globe.

We are on the same sheet of music, believe it or not. I am not that glib at explaining it. It would be easy to draw in person but, that is not possible.

I am just discovering what has been obvious to you folks all along but never really emphasized in books, that all points in 3D are lines (curved or straight) in spacetime. As I have no one locally to talk this over with I have to rely on this forum to set me straight.

To put it in your terms, I need to get to another frame of reference of knowledge and that will require a lot of hop, skip and jumping.

 

[DrGreg]

Yea, I agree with that. In my case "North" is not time but one of the two dimensions. Time is along the vertical radius from the center to the N Pole and everything slips "up" as time marches on in its own dimension forcing the curvilinear path on the globe.

Well, it depends what you mean by "time"! If you are talking about the time coordinate of a coordinate system, that could well be measured along the axis as you say. But the "proper time" as an object measures itself (i.e. as recorded by the object's own clock) is just the length measured along the curve of its worldline. The angle between the Earth's axis and the curved line at any point is indicative of time dilation, i.e. the distances aren't the same. (In this analogy.)

 

[stevmg]

Thank you folks for all your ellucidation.

Of course proper time is measured "by itself" with all the spatial distances "removed."

tau² = SQRT[(ct)² - x² - y² - z²] (I can't get the Latex to work correctly)

But I got it! I got it! This stuff is not easy for a novice like me with no friends who know anything about it to get...

This almost puts me where Einstein was when he was age three...

I am 67 years old and I feel as excited about this as I was when I was in high school learning something new and esoteric.