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I understand that gravity is viewed as flowing as a river pushing objects down on the body of a planet. If that is the case and earth is a sphere, where does the gravity go when it hits the center of the planet? Does it get compressed? Is it destroyed?

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closed as not constructive by Waffle's Crazy Peanut, akhmeteli, Brandon Enright, Emilio Pisanty, user1504 Jun 7 '13 at 20:18

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    $\begingroup$ Gravity is curvature of spacetime, not a river. The river descrition is very strange to say the least, where do you have that from? $\endgroup$ – Dilaton May 27 '13 at 12:05
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    $\begingroup$ This may be (yet another) bad analogy (it is, and a real stinker at that), but it appears in popular explanations of GR. $\endgroup$ – dmckee May 27 '13 at 19:13
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    $\begingroup$ Although the OP's description isn't very good, the river model is sometimes used to describe the "flow of three space" e.g. here $\endgroup$ – twistor59 Jun 7 '13 at 16:19
  • $\begingroup$ Yes, rivers flow from the mountains due to gravity. They end up in the sea and never hit the centre of the planet. $\endgroup$ – Abhimanyu Pallavi Sudhir Jun 18 '13 at 8:52
  • $\begingroup$ I guess he has been mislead by Michio Kaku's misleading popula r - science explanation of general relativity. $\endgroup$ – Abhimanyu Pallavi Sudhir Jun 19 '13 at 7:03
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I take it you're referring to the river analogy that's sometimes used in connection with Painleve Gullstrand coordinates.

That is often applied to a black hole, where the "river" is a fanciful description of the "flow" of space into the black hole. This flow increases as you approach the horizon, at which point its speed is the speed of light.

You're worried that if we apply this to an object like a planet, where the surface isn't inside its Schwarzschild radius, that the thing that's flowing in the river, i.e space, would somehow "pile up" at the planet surface or center? There is no problem with this as spacetime points are not physical objects, but mathematical abstractions.

The apparent "flow" of space is just a way of describing the way you associate points on a slice of space with points on the slice of space an instant of time later. This flow is the shift vector you get when express your spacetime metric in the form defined in this question.

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