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(I apologize for my naivete in advance.)

The Lagrange points around the earth and the moon provide a semi-stable space to put objects because the gravitational pull (combined with centripetal force) can be equalized there. Does that also equalize out the effects of time dilation say compared to an object at an equal altitude on the other side of the Earth?

Another example is that the closer something gets to a black hole the more time dilation one would expect. While it would tear an object apart if something was equal distance between two equal sized black holes, would that position for an instant negate the time dilation of both black holes? Or would it be doubled or neither?

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Gravitational time dilation is proportional to the gravitational potential, not the gravitational force. So two objects with equal and opposite force do not cancel time dilation. Roughly speaking (in the weak field limit) they would double it.

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