If gravity is in reality spacetime geometry why when I drop an object on the surface of the Earth does it fall to the ground? Does spacetime push it?
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What is exactly your thinking behind "gravity is space-time geometry"?
In layman's terms, general relativity tells us that a particle with some defined mass will curve space-time, causing a gravitational attraction to be felt by the objects around it. When you drop an object on the surface of the Earth, the curvature of space-time caused by the large mass of the Earth, relative to the small mass of the object, causes it to be attracted by gravity and fall to the ground.
(source: space.com)
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$\begingroup$ In your graphic there is a spacecraft presumably in orbit - presumably following a geodesic path - it has velocity relative to the Earth - If it had no velocity why would it travel to the earth? Does spacetime push it? $\endgroup$ Commented Dec 6, 2015 at 22:17
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$\begingroup$ @PaulStanley As you pass through time, the distortion of space-time accelerates the satellite towards the centre of the Earth. If the satellite had no velocity, it would accelerate towards the Earth. It is not sensible to say that space-time pushes the satellite since it is just an abstract mathematical model. For further notice, I think you should read through the similar questions above, since this question is a duplicate. $\endgroup$ Commented Dec 6, 2015 at 22:25