Of course in this example, there is 'breathable air' and no 'lava core' for the sake of limiting the question only to the question of gravity. If you jumped in this hole, what would happen? Would you fall to the center slowing down as you reached? Fall past it and then fall back up eventually remaining in the center? Fall past it from momentum and go to the other side? If you stayed in the center would you IMPLODE or EXPLODE? You could implode because gravity is pushing in both directions. But you could also explode because gravity is pulling in both directions. Would it cancel and you would just sit there and float until you died of starvation?
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