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Let's say I have a fast spinning black hole, and I'm extracting energy from it using a method by which energy can be extracted from a spinning black hole, Penrose Process or something like that. I'm trying to keep the black hole's mass and angular momentum constant by feedind the black hole with photons whose angular momentum points to same direction as the black hole's angular momenum. Is this possible?
If you want to be forever trapped in the ergosphere, yes. In which case you will not be able to transmit the energy you gain to a region outside the ergosphere.
...cause a large increase of spinning energy of the black hole. Energy of the beam does not have to be large...
Depends on definitions. "Inside this ergosphere, the time and one of the angular coordinates swap meaning (time becomes angle and angle becomes time) because timelike coordinates have only a single direction (and remember the particle is necessarily rotating with the black hole in a single direction only). Because of this weird and unusual coordinate swap, the energy of the particle can assume both positive and negative values as measured by an observer at infinity." Now, if that is correct, which I personally hold to be incorrect, then maybe?http://www.gu.se/forskning/publikation?publicationId=195725