What Do I Observe Moving Around in Curved Space

[solarblast]

I understand some of the basic concepts of curved space, flat, positive, negative, spheres, saddles, etc. In flat space, if one measures the angles of a triangle, the sum adds to 180 degrees. In spherical space, the sum is greater than 180.

OK, how do things work in spherical space? If I shine a light in front of me, does it curve around as on a great circle and hit me in the back? How do I determine the length of a curve (I suppose assuming I'm not attached or standing on anything, but am free to float with a jetpack) or the radius of the circle it makes? Put down a marker and walk (jet) through "space" trying somehow to keep a "straight" path? What's straight here? Does it need to be a great circle or can I pick any old circle?

If you and I are in flat space, and I see you, then someone throws a switch to turn it into spherical space, do you look the same to me?

 

[tiny-tim]

Welcome to PF!

Hi solarblast! Welcome to PF!

… If I shine a light in front of me, does it curve around as on a great circle and hit me in the back?

If nothing gets in the way, and if you live long enough, yes.

How do I determine the length of a curve (I suppose assuming I'm not attached or standing on anything, but am free to float with a jetpack) or the radius of the circle it makes? Put down a marker and walk (jet) through "space" trying somehow to keep a "straight" path? What's straight here? Does it need to be a great circle or can I pick any old circle?

What's straight? Straight is straight!

There isn't a problem.

Do you have any difficulty walking a straight line on the Earth (or sailing a straight line on the sea)?

If you and I are in flat space, and I see you, then someone throws a switch to turn it into spherical space, do you look the same to me?

If I'm large enough, my circumference will be smaller than expected, compared with my radius.

 

[solarblast]

As I have begun to realize, straight is a geodesic, the shortest distance between two points. To achieve it in space one ties a cord to something and stretches it out. I can certainly walk along a line of latitude that might be considered straight, but it is not the same as a geodesic, or straight in the sense used by physicists and mathematicians in describing these ideas. However, putting that aside, I see that I confused myself somewhat. When they speak of spherical space, they mean 2-D, the surface itself. Hang onto that for a moment.

A 2-D flat surface is like a piece of paper. The idea of it being defined by triangles having the sum of its angles is 180 degrees is clear there, but some books add to this that our 3-D universe behaves as though these ideas are true in it. In other words, if we are in a 3-D space making measurements of a triangle, the results are those one observes in a 2-D world. This is where the confusion is. What I was thinking is that authors were saying is that the geometric laws of the sphere might apply to our world in the same sense. Something like one space embedded (immersed) in another. Assuming this is correct, then my question has to do with the effects with what we would see in our 3-D universe.

 

[tiny-tim]

What I was thinking is that authors were saying is that the geometric laws of the sphere might apply to our world in the same sense. Something like one space embedded (immersed) in another. Assuming this is correct, then my question has to do with the effects with what we would see in our 3-D universe.

The change in the perimeter/diameter ratio of a triangle or circle is the same in 3D as it is in 2D … the authors are saying that, in 3D, any 2D plane will show the same geometrical laws.

(And we don't see in 3D … wee see a 2D projection of whatever's in front of us.)