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This question is sparked with an answer to another one. We know that fields have mass-energy and gravitational field is no exception, some share of BH mass should be contributed by its gravitational field.

Anixx
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The energy in a gravitational field cannot be localized in GR. The Newtonian expression for the energy density of the gravitational field is proportional to $g^2$, where $g$ is the gravitational field, but the equivalence principle tell us that $g$ is not observable in GR. For example, we can make $g=0$ at any point we like, simply by adopting an inertial frame of reference. There is a detailed discussion of this sort of thing in Wald, section 11.2. When we talk about the mass of a black hole, we're talking about some quantity such as its ADM mass, which is essentially the mass measured by a distant observer.

  • A more recent thinking is that energy (or rather, energy–momentum) in GR is properly quasi-local, i.e. associated with a closed 2-surface. For an overview see lrr-2009-4. With that in mind, OP's question sort of makes sense. – A.V.S. Dec 10 '18 at 18:35
  • This does not answer the question :( – Anixx Dec 11 '18 at 13:34
  • @Anixx: The answer to the question is that there is no answer to the question. –  Dec 11 '18 at 16:32
  • @A.V.S.: That sounds interesting. How about writing up an answer along those lines? –  Dec 11 '18 at 16:32