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gjm comments on Stupid Questions December 2014 - Less Wrong
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Yes, I saw an article a few years ago a back of the envelope estimate that suggested this would be doable if one could turn mass on the moon more or less directly to energy and use the moon as a gravitational tug to slowly move Earth out of the way. You can change mass almost directly into energy by feeding the mass into a few smallish blackholes.
How do they propose to move the blackholes? Nothing can touch a blackhole, right?
Black holes feel gravity just like any other massive body. And they can be electrically charged. So you can move them around with strong enough gravitational and/or electric fields.