Experience Zero-Gravity at Earth's Core

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Imagine you were safely enchambered right at the center of the earth. Here you would "experience" strong gravitational "forces" but since they all point in opposite directions you would float around as if you were in zero-gravity.

My question is, whether you would experience time dilation or not. You certainly got all the matter to exhibit of gravitational force on you, but still you'r not in a region of spacetime that is curved.
 
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