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The question goes along the lines of understanding how does the entropy-information additivity works when black holes are inside other black holes, including the special case where the inner black holes have stable orbits around it.

Assuming one knows the separate entropy of two black holes $O$ and $I$, can one write down an expression for the entropy accessible on the outside spacetime when $I$ lives inside $O$? Is it an addition or a substraction of areas?

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black holes are inside other black holes

Within the present model of general relativity this is not a possible situation, as anything within a black hole falls into the singularity at the origin.

The LIGO observation showed that when two black holes merge , (fall on each other) a single new black hole emerges, with a mass smaller than the sum of the two, and a lot of gravitational waves carrying off the left over energy.

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  • $\begingroup$ both rotating and charged black holes have stable orbits inside the second horizon physics.stackexchange.com/a/365986/955 $\endgroup$
    – lurscher
    Commented Oct 11, 2018 at 14:40
  • $\begingroup$ The paper referenced in your link talks about planetary orbits. A black hole is not a planet. arxiv.org/abs/1103.6140 $\endgroup$
    – anna v
    Commented Oct 11, 2018 at 15:26
  • $\begingroup$ a black hole can be smaller than a planet, and from afar is not distinguishable from a planet or star, is just a gravitational well $\endgroup$
    – lurscher
    Commented Oct 11, 2018 at 15:52
  • $\begingroup$ This is not my field, but I cannot think of a stable orbit for a black hole even if there are planetary orbits within a black hole ( seems science fiction to me but as John said the planetary stuff is peer reviewed). As a singularity it will be attracting and consuming anything falling in the "mother" black hole and thus destroy the stability, imo. Maybe for micro black holes it could work for a while, until hawking radiation gets them. The LIGO observation excludes any "orbiting each other" set up, imo. $\endgroup$
    – anna v
    Commented Oct 11, 2018 at 16:24

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