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How do primordial black holes have an event horizon?

Sorry for the stupid question, but I know primordial black holes have just a handful of mass in a cosmic scale compressed in a singularity. But if the mass is so small, compared sometimes with the Moon, how do they can have a gravity that event the light can’t scape at a point?

At the beginning, I thought it was because of the density, but if it was the case, if the Sun was replaced by a black hole with the same mass, the planets orbits would be destabilized, wouldn’t be? But this is not the case: if we replace the sun with a black hole, everything would continue to be stable. That means the problem isn’t the density as I thought, and the primordial black holes would have to have the same gravity as moon. So please, what is happening?

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It is neither the mass nor the density that determines black hole formation. Instead, the most relevant quantity is the depth of the gravitational potential. For a sphere of mass $M$ and radius $R$, the magnitude of the potential depth is $\Phi=GM/R$. This can be made arbitrarily large by making the radius $R$ of the object arbitrarily small.

That the potential is the relevant quantity is easy to understand from a Newtonian perspective. To climb a potential difference $\Phi$, an object needs kinetic energy (per mass) $\frac{1}{2} v^2=\Phi$. Hence a "critical" potential depth can be defined, such that something needs to travel at the speed of light, $c$, to escape: $\Phi_\rm{c}=c^2/2$. Of course, Newtonian physics aren't valid under such conditions, but we should expect things to at least scale in the same way under general relativity.

(Actually, these considerations predict exactly the correct Schwarzschild radius, but we couldn't have anticipated this. Other predictions are off; for example the "critical potential" being $c^2/2$ suggests that we can extract half of an object's rest mass as mechanical energy by lowering it into a black hole, but according to general relativity we can actually extract all of it.)

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  • $\begingroup$ Wow, thanks! In that case, the gravity is still the same, but for the light, that travels at the space-time, it matters more the gravitational energy. Did I get it correctly? $\endgroup$
    – Artur Carneiro Barroso
    Commented Jul 15, 2023 at 17:59
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    $\begingroup$ @ArturCarneiroBarroso "the gravity" would usually refer to the (Newtonian) gravitational acceleration (or field), which is $GM/R^2$ -- different from all of the quantities discussed above! The reason black hole formation isn't linked to the gravitational acceleration is that the area over which the acceleration is large matters. Something could have a large gravitational pull in a small area, and then it is still easy to escape. The potential accounts for this. $\endgroup$
    – Sten
    Commented Jul 15, 2023 at 19:27

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