Why are orbital paths elliptical if gravity is spacetime curvature?

[Osvaldo]

Though it is hard not to believe in the spacetime curvature that cause planets to follow curved path arround massive objects, I wander how come these paths are eliptical, the object change velocity when moving arround the massive object and what is more obeys the Keppler laws. If there is not such a gravitational force (as said byexperienced physics) which would really caused these conditions, how come these orbital movements do not belong to a central force? Can somebody give a clear explanation?

 

[sweet springs]

Kepler laws or eliptic orbit of planets is derivied from Newton's inverse square law of universal gravitation.
I understand this Newton's inverse square law of central force is a fairy good approximation of GR theory or curved spacetime around the Sun, a massive body.

 

[Dale]

I wander how come these paths are eliptical

They are actually not quite elliptical. GR explains both why they are almost elliptical and also correctly predicts how much they deviate from being completely elliptical.

 

[A.T.]

Though it is hard not to believe in the spacetime curvature that cause planets to follow curved path arround massive objects, I wander how come these paths are eliptical, the object change velocity when moving arround the massive object and what is more obeys the Keppler laws. If there is not such a gravitational force (as said byexperienced physics) which would really caused these conditions, how come these orbital movements do not belong to a central force? Can somebody give a clear explanation?

Locally and for weak fields the space-time geometry just reproduces the same effect Newton's force would have, and would also result in an elliptical orbit:



But globally you get an additional effect from space curvature, so the orbit is not closed:
http://demoweb.physics.ucla.edu/content/10-curved-spacetime (Figure 2)

 

[Osvaldo]

Locally and for weak fields the space-time geometry just reproduces the same effect Newton's force would have, and would also result in an elliptical orbit:



But globally you get an additional effect from space curvature, so the orbit is not closed:
http://demoweb.physics.ucla.edu/content/10-curved-spacetime (Figure 2)

 

[Osvaldo]

Wonderful demonstration. Never saw it like the one in the video. Thanks a lot