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I know according to classical theory, its centrifugal force.

Also, I need to assure that mass curves spacetime, not space.

Now, if mass curves spacetime, how can an apple which follows geodesic in curved spacetime (due to earth) fall straight onto the ground while the moon which follows geodesic in curved spacetime (due to earth) go around the earth in an elliptic path.

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    $\begingroup$ The apple has no angular momentum, the moon does. This is the same explanation as in classical gravity. $\endgroup$ – Ryan Unger Sep 22 '15 at 13:49
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    $\begingroup$ It's centripetal force! $\endgroup$ – Physics Moron Sep 22 '15 at 13:52
  • $\begingroup$ There is no such thing as a centrifugal force. In classical physics gravity causes a distant dependent acceleration towards the center of mass and things are falling around each other, or, in case of the apple right towards that center. In general relativity they are moving on geodesics, the straightest of all possible lines. $\endgroup$ – CuriousOne Sep 22 '15 at 13:53
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    $\begingroup$ In classical physics, gravity acts as a centripetal force! $\endgroup$ – Sean Sep 22 '15 at 13:58
  • $\begingroup$ Spacetime is curved by Earth in such a way that an apple falls to the ground and the Moon revolves around the Earth. Yes, this is true. $\endgroup$ – Prof. Legolasov Sep 22 '15 at 16:10
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In good old Newtonian mechanics the equation of motion of an object is:

$$ \frac{d^2x}{dt^2} = \frac{F}{m} $$

where the left hand side of the equation is just the acceleration. In general relativity the analogous equation is called the geodesic equation and it's:

$$ {d^2 x^\mu \over d\tau^2} = - \Gamma^\mu_{\alpha\beta} {dx^\alpha \over d\tau} {dx^\beta \over d\tau} $$

The left hand side of this equation is the four-acceleration, just as in the Newtonian case the left hand side is the (coordinate) acceleration. The right hand side has two bits. The symbols $\Gamma^\mu_{\alpha\beta}$ are the Christoffel symbols and they describe the spacetime curvature. The $dx^\alpha/d\tau$ is the four velocity of the object whose path we are calculating.

The key point is that the four acceleration isn't just dependant on the spacetime curvature, it's also dependant on the four velocity. That means if I throw a ball fast the four-acceleration will be different to if I throw it slowly, and the two balls will travel different paths even though both paths are geodesics.

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