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I know that the Israel theorem accounts for only non-rotating, non-electrically charged black holes, but as I understand the theorem was then generalized for rotating and charged black holes. And, as I understand the no-hair theorem would basically implies that every static black hole solution to Einstein Equations cam be completely caracterized by the mass, charge and the absolute value of its intrinsic angular momentum. But the no-hair theorem is usually called a conjecture, so what assumptions are necessary in the generalized Israel theorems that must be overcome so that they can become the full no-hair theorem? (Spherical symmetry? Perturbed spherical symmetry? full analyticity of the space time?)

References: The original theorem is in https://journals.aps.org/pr/abstract/10.1103/PhysRev.164.1776 but it is only valid for non-rotating, non-electrically charged black holes, the generalization is in https://iopscience.iop.org/article/10.1088/0305-4470/15/10/021

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The No-Hair Theorem is a general statement about the simplicity of black holes, asserting that black holes are uniquely defined by 𝑀, 𝑄, and 𝐽 without any additional parameters or complexity.

The (Generalized) Israel Theorem provides specific proof of this simplicity for particular cases: Schwarzschild, Reissner-Nordström, and Kerr black holes. It mathematically establishes the uniqueness of these solutions under the assumption of asymptotic flatness and stationarity, thereby supporting the broader No-Hair conjecture.

The Israel Theorem is not as broad as the No-Hair Theorem; it is a proof for certain cases. In contrast, the No-Hair Theorem is a conjecture or general principle that all black holes will similarly have no hair beyond 𝑀, 𝑄, and 𝐽.

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  • $\begingroup$ (+1) Could you sketch the reasoning/motivation for the no-hair conjecture for mass, please? (Wikipedia is often highly abstract on its GR pages) Does the statement "mass distribution does not matter" imply that $T_{\mu\nu}$ is spherically symmetric within the black hole? Suppose one throws a chair the size of a sun into the black hole, what is the process for this spherical homogenization of $T_{\mu\nu}$ according to the no hair hypothesis? $\endgroup$
    – James
    Commented Aug 10 at 10:04
  • $\begingroup$ @James It's better to post this as a separate question. $\endgroup$
    – Lagrangiann
    Commented Aug 10 at 10:11
  • $\begingroup$ Is there no generalised Israel theorem for Kerr-Newman black holes? And doesn't stationarity enter also in the conditions for a No-Hair Theorem (Conjecture)? In the sense that the dynamics of black-holes outside equilibrium regimes would not necessarily be uniquely defined by $M$, $Q$ and $J$, like when a black hole has its event horizon deformed by other massive object in its close proximity, or when the event horizon has formed but the distribution of mass inside the event horizon isn`t yet spherically symmetric? $\endgroup$
    – Felipe Dilho
    Commented Aug 15 at 19:40
  • $\begingroup$ @FelipeDilho Sorry but there is too many questions in one comment $\endgroup$
    – Lagrangiann
    Commented Aug 15 at 22:13

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