Curvature of Space-Time: Why is Covariant Derivative Nonzero?

[TimeRip496]

I recently watched Susskind video on general relativity. I am unsure why the commutator of the covariant derivative of the vectors is nonzero when there is curvature. E.g. DrDsVm-DsDrVm
In flat space, that difference is zero. But why is it non zero in curved space? Someone please enlightened me!

Sorry for being so vague as I don't really know how to express this in words and how to use this forum.

 

[Nugatory]

On the curved surface of the earth, moving one kilometer east and then one kilometer south does not take you to the same place as moving one kilometer south and one kilometer east - that is, movements in different directions do not commute.

 

[TimeRip496]

On the curved surface of the earth, moving one kilometer east and then one kilometer south does not take you to the same place as moving one kilometer south and one kilometer east - that is, movements in different directions do not commute.

How? I find that difficult to visualise and I even try drawing it on my tennis ball but they do end up at the same place.

 

[stevendaryl]

How? I find that difficult to visualise and I even try drawing it on my tennis ball but they do end up at the same place.

Suppose you are standing on the North Pole, facing south along the line of 0 degrees longitude. Your right hand is straight out from your side, pointing south along the line of 90 degrees west longitude. Now walk south along 0 degrees longitude to the equator, keeping your arm pointing in the same direction. At this point, you are at the equator, and your right hand is pointing west. Now walk west along the equator, keeping your right hand pointing in the same direction, until you reach the line of 90 degrees west longitude. Now walk north along this line of longitude until you get back to the North Pole. At this point, your right hand will be pointing south along the line of 180 degrees west longitude.

 

[Nugatory]

How? I find that difficult to visualise and I even try drawing it on my tennis ball but they do end up at the same place.

If you are standing at the equator and walk one kilometer east, your longitude changes by a certain amount. If you start at a point one kilometer south of the equator, that same one kilometer east will change your longitude by a larger amount, so the south-then-east walk will leave you farther to the east than the east-then-south walk.

If you still don't see it, try it with a larger distance - the effect will be very clear at, say, one-eighth the circumference.

 

[A.T.]

On the curved surface of the earth, moving one kilometer east and then one kilometer south does not take you to the same place as moving one kilometer south and one kilometer east

Unless you start 1/2km north of the equator. Maybe that's what the OP draws on his tennis ball.

 

[TimeRip496]

If you are standing at the equator and walk one kilometer east, your longitude changes by a certain amount. If you start at a point one kilometer south of the equator, that same one kilometer east will change your longitude by a larger amount, so the south-then-east walk will leave you farther to the east than the east-then-south walk.

If you still don't see it, try it with a larger distance - the effect will be very clear at, say, one-eighth the circumference.

Thanks a lot!

 

[Nugatory]

Unless you start 1/2km north of the equator.

Good point. :)