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We do know that black holes can and sometimes do have angular momentum, as described by the Kerr metric. Though I have not found anything about the description of the angular momentum of the contained singularity.

Since a point cannot support rotation or angular momentum in classical physics (general relativity being a classical theory), the minimal shape of the singularity that can support these properties is instead a ring with zero thickness but non-zero radius, and this is referred to as a ringularity or Kerr singularity.

https://en.wikipedia.org/wiki/Ring_singularity

The question I believe is not easily answered, because singularities are said to have no spatial extent, and so should not possess classical angular momentum, but rather quantum spin (that elementary particles have because they do not have any spatial extent either).

In our mathematical model of particles, the intrinsic spin of a fundamental particle behaves exactly like that centre point.

Could a non-pointlike structure of elementary particles explain their spin?

The only thing I found was the ringularity, but that still has zero thickness. In many cases on this site, singularities are compared to elementary particles (both being pointlike, zero dimensional), so the question comes up whether black holes' singularities can have classical or quantum spin.

Electrons - and all other elementary particles - may be viewed as microstates of very tiny black holes.

Are electrons just incompletely evaporated black holes?

As far as I understand, there are many similarities between singularities and elementary particles, so their spin might be related too.

Question:

  1. Can singularities' spin be related to the quantum spin of elementary particles?
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  • $\begingroup$ @safesphere thank you so much! $\endgroup$ Aug 18, 2022 at 21:28

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Can singularities' spin be related to the quantum spin of elementary particles?

“Related” in this instance is somewhat nebulous concept. But one thing that hints at the existence of such relation is gyromagnetic ratio of Kerr–Newman metric, which coincides with that of the Dirac electron.

To quote the Scholarpedia article on KN metric (also available at arXiv):

Interpretive issues

It was first noted by Carter (Carter, 1968) that the gyromagnetic ratio associated with the Kerr-Newman metric is that of the Dirac electron. The classical gyromagnetic ration $γ_\text{class}$, which is defined by the ratio of the magnetic moment to the (spin) angular momentum, turns out to be $γ_\text{class}=\frac{Q}{2M}$ for simple systems. The Dirac theory of the electron yields a gyromagnetic ratio that is twice as large: $γ_\text{Dirac}=g γ_\text{class}$, with $g=2$. From the asymptotic magnetic field of the Kerr-Newman solution, one sees that its magnetic dipole moment is given by $μ=Q a$, while the angular momentum is $J=Ma$. Hence, we immediately recover $γ_\text{KN}=Q/M$, so $g=2$ and the Dirac value is obtained.

This has led several authors to speculate that the Kerr-Newman black hole could provide some sort of classical “model” for the Dirac electron (c.f.,Israel, 1970; Burinskii, 2005). It remains unclear the extent to which this proposal can be fully realized. One potential issue is the fact that the Kerr-Newman metric also has an associated electric quadrupole moment, which cannot be incorporated into the Dirac theory and has no basis in experimental observation. However, in more general asymptotically flat space-times, $g=2$ can still be recovered when the centers of mass and charge coincide (Kozameh, Newman and Silva-Ortigoza, 2008; Adamo, Newman, and Kozameh, 2012; Adamo and Newman, 2011).

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    $\begingroup$ Wouldn’t another “relation” be the fact that nothing rotates in the Kerr solution, so the angular momentum can be only intrinsic, which normally is not a classical concept? +1 $\endgroup$
    – safesphere
    Aug 18, 2022 at 17:13
  • $\begingroup$ Thank you so much! $\endgroup$ Aug 18, 2022 at 21:29
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Can singularities' spin be related to the quantum spin of elementary particles?

The word "spin" comes from the everyday observation of a "spinning top" and really is the everyday word for what in mathematical physics is angular momentum.

It may be that the mathematics of the ring_singularity's angular momentum of general relativity has the same theoretical form as spin in the theories of particle physics, but one must not forget that the observed particles are assigned a spin so as that the law of conservation of angular momentum in the interactions of particles holds also at the quantum level. In order for the law to be valid at the quantum level, elementary particles are assigned a fixed spin./

As the particles have fixed spin and there is no such constraint on the "ringularity" as far as I see, the answer is "the values cannot be related". Of course if/when gravity is quantized a different interpretation may emerge.

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  • $\begingroup$ Thank you so much! $\endgroup$ Aug 17, 2022 at 15:29
  • $\begingroup$ @safesphere clarified I am talking of the ring singularity , the Kerr singularity that is referred in the question, $\endgroup$
    – anna v
    Aug 18, 2022 at 16:56

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