ok ,i'll tell what i have understood here
space time is a 4-manifold
the space time around the Earth is locally flat , so the geodesics of freely falling bodies in this region are straight lines , so they fall in straight lines .
but globally the sapce time around the Earth is not flat , so...
http://www.mth.uct.ac.za/omei/gr/chap6/frame6.html" is a derivation of the components of the riemann curvature tensor. the problem is that i can't understand the transition between eq97 and eq89 .
what does "To lowest order " mean ?
forgive my stupidity , but i don't understand
let me put the question in another form ,
why do things fall to the surface of the Earth in straight lines according to GR ?
(and straight lines are not geodesics in curved space time)
http://www.youtube.com/watch?v=8MWNs7Wfk84&feature=PlayList&p=858478F1EC364A2C&index=2" , Edmund Bertschinger is talking about Einstein's field equations .
during the lecture , under the title of : "Gravity as sapcetime curvature (GR viewpoint) " , he wrote :
"Freely falling bodies move along...