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I know a Black hole can have angular momentum, described by the Kerr metric. It can also have an electric charge, described by the Kerr-Newmann metric. I read it cannot have a non-abelian charge. I would like to know why.

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  • $\begingroup$ Interesting question and I look forward to an answer. You may wish to make your title more specific - there are lots of interesting things about the physics of black holes, and the nature of its charge is by no means the most obvious. $\endgroup$ – IanF1 May 25 '17 at 20:24
  • $\begingroup$ Related: physics.stackexchange.com/q/148374 $\endgroup$ – Dvij Mankad May 25 '17 at 21:14
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    $\begingroup$ I don't have time to write an answer, but look up the works on BMS symmetry by Steinhardt and Mathur's work on quantum hair. Entanglement symmetry of quantum states on black holes can involve nonabelian symmetries. $\endgroup$ – Lawrence B. Crowell May 26 '17 at 0:32
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If the matter fields include nonabelian gauge fields, then black holes actually can have nonabelian charges, in violation of the no-hair theorem: https://arxiv.org/abs/gr-qc/9606008v1. But as far as I know, all known such black holes are unstable.

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    $\begingroup$ Strictly speaking it's not a violation, it simply not fulfill the no hair theorem hypothesis (gravity + Maxwell EM). $\endgroup$ – Rexcirus May 26 '17 at 17:33
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Thanks to Lawrence B. Crowell, I found a paper by Hawking, Perry and Strominger which explains everything. I haven't had time to write a clean answer but I will shortly, using this Soft Hair on Black holes paper. More soon.

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    $\begingroup$ Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. $\endgroup$ – Qmechanic May 26 '17 at 22:46
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The proof of the no-hair theorem, which says that Black Holes cannot carry any vector field, nor charge other than the electric charge relies on the asymptotic properties of spacetime, far away from the source. It was originally believed spacetime would be flat and the symmetry group would be the Poincare group. The Bondi Mezner Sacks group includes transformations known as super-translations and super-rotations. These are part of the Black Hole's hair.

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  • $\begingroup$ Write a new answer only if you have another, essentially different answer to the same question (which is very rare, but possible). If you cite something, detail its content at least in some sentences. The goal is here to make your answer understable without making it dependent from any external source (links can disappear, books can be unavailable, etc). $\endgroup$ – peterh May 29 '17 at 6:34
  • $\begingroup$ This is unrelated. @tparker's answer is the right one. $\endgroup$ – Mitchell Porter May 29 '17 at 13:07

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