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For a non-spinning (Schwarzschild) black hole:

  • The event horizon (Schwarzschild radius) is at:

    $$r = \frac{2GM}{c^2}$$

    Inside this radius, not even light can escape.

  • Outside that, there is the photon sphere at:

    $$r = \frac{3GM}{c^2}$$

    This is the radius where photons can move in a circular orbit around the black hole.

  • Even farther than that, we have something known as Innermost Stable Circular Orbit (ISCO) at:

    $$r = \frac{6GM}{c^2}$$

    This is the minimum radius required for a "test particle" (which holds mass and charge) to orbit a blackhole in a stable circular orbit.

Therefore as per my understanding:

  1. Massive particles can have circular orbits only for: $$r > \frac{3GM}{c^2}$$

  2. Circular orbits also exist for massive particles in the range:

    $$\frac{3GM}{c^2} < r < \frac{6GM}{c^2}$$

    But those orbits are highly unstable. A tiny inward or outward perturbation will cause the particle to either spiral inwards towards the black hole or move outwards, rather than remain on that same circular orbit.

    Since the electromagnetic force has an infinite range of interaction, it would be safe to say that one would be hard-pressed to find high frequency of these massive charged particles within the aforementioned range.

  3. Massive particles can be extremely easily found orbiting the blackhole at:

    $$r ≥ \frac{6GM}{c^2}$$

Now, here is my question about neutrinos:

A neutrino is electrically neutral, interacts extremely weakly (only via the weak interaction and gravity), but it does have a tiny nonzero rest mass.

I was wondering whether we have a special "Innermost Stable Circular Orbit for Neutrinos" (call it an ISCO$\nu$), that lies at some radius between the photon sphere and the usual ISCO?

In other words, does the fact that neutrinos are extremely light, electrically neutral, and interact only very weakly allow them to stably orbit closer to the black hole than ordinary matter can? Can we find neutrino-only “band” of stable circular orbits which lies between:

$$\frac{3GM}{c^2} < r < \frac{6GM}{c^2}$$

If this is a sound hypothesis, then would it be wrong to say that we should be able to find an extreme concentration of neutrinos orbiting a black hole beneath the ISCO radii but above the photon sphere radii?

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    $\begingroup$ Good question, but you may have answered it yourself. It seems to me a neutrino is the very definition of a test particle, with negligible but nonzero mass. So it should not be expected to accumulate in any region closer than ISCO. The lack of charge is irrelevant for an uncharged black hole. $\endgroup$ Commented Oct 28 at 22:13
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    $\begingroup$ Even if the orbit were possible, there's almost no way for them to concentrate there. Baryonic matter forms rings around massive objects because it's easy for them to interact and lose energy. Neutrinos have no such mechanism. $\endgroup$ Commented Oct 28 at 22:24
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    $\begingroup$ A hand-wavy way of thinking about this: the reason neutrinos are "special" is because while they're massive, they're also relativistic in most situations where we encounter them. But particle motion near the horizon of a black hole more or less has to involve relativistic motion. So I don't think there's really a reason to expect that neutrinos would behave differently from other relativistic massive particles near a black hole. $\endgroup$ Commented Oct 28 at 22:50
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    $\begingroup$ @RC_23 Perhaps change that comment to an answer. We discourage answering in comments as they're not guaranteed persistent. $\endgroup$ Commented Oct 28 at 22:53

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You may have answered your own question during your analysis.

It seems to me a neutrino is the very definition of a gravitational test particle, with negligible but nonzero mass. So it should not be expected to accumulate in any region closer than ISCO. The lack of charge is irrelevant for an uncharged black hole.

The reason unstable orbits decay is not that charged particles feel tiny EM perturbations. This may be one contribution, but absolutely any perturbation – gravity of a planet 10 light-hours away, stray cosmic radiation, a quantum field fluctuation (spontaneous virtual particle) – is enough to disturb the orbit. Like a pyramid balanced on its tip, any deviation no matter how small will tend to grow and move the system away from the unstable point.

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  • $\begingroup$ Thank you for the answer @RC_23. I had assumed that since neutrinos interact so weakly with almost everything, they should be able to (stably) orbit a black hole at radii smaller than the usual ISCO (just like photons are able to). Would it be correct to say that, statistically, neutrinos should be more commonly found than other particles in the region between the photon sphere and the ISCO radius? $\endgroup$ Commented Oct 29 at 2:24

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