VoiceOfRa comments on Stupid questions thread, October 2015 - Less Wrong Discussion
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More-or-less like a black hole, gravitational signature but no light emitted.
That doesn't work. The emitted energy must have higher entropy content than the initial energy of the star, so it would be less useful to the destination star. In fact, if the original destination star can extract useful work from it, so could the original star.
Sure, but I don't see a problem here. If you have a particular attachment to a star system, you may want to supply it with energy from other stars instead of moving to them.
The point is that since the energy will get "consumed" elsewhere, the waste heat at the origin will be minimal and so the Dyson sphere will remain relatively cold (though higher than the background temperature, to be sure).
However, the destination star would emit large amounts of heat, i.e., infrared.