Joyce Sharp

2022-09-25

Imagine if you have a magnetic material and you fix it into a position between two electomagnets, or perhaps in the center of a sphere of electomagnets. Is it possible that you could apply enough force to crush the magnetic material such that it becomes a black hole?

Kate Martinez

No, and the curious reason is that the field itself would break because of quantum field effects.
Around magnetars magnetic fields can reach around $1-100$ billion teslas. At this point the mass-energy density of the field itself is ${10}^{4}$ times denser than terrestrial lead. Which sounds impressive, but lead (unless you have a very large region of it) doesn't implode to black holes: the magnetar itself is far denser than the field.
That field is close to one approximate limit set by quantum mechanics found by Landau, $\sim 3×{10}^{9}$ T. That one can be loosely interpreted as when magnetic field lines become as narrow as the electron wavelength.
Using full quantum electrodynamics leads to another, firmer limit: above a certain strength ($B={m}_{e}^{2}{c}^{3}/\hslash e\approx 4.4×{10}^{9}$ T) pair production of electron-positron pairs can happen, converting the field energy into matter (which then starts accelerating and annihilating, disrupting everything).
So your field may well squish the metal into weird states (electron orbitals become cylindrical in extreme fields, producing hypothetical "whiskers" of superstrongly bonded atoms on neutron stars), but before the energy density got truly extreme pair production would make the field break down.
Still, what determines whether you get a black hole is the total mass-energy inside a region. While it is hard to get $M$ units of mass into a radius ${R}_{s}=2GM/{c}^{2}$ region when $M$ is small, the critical density
$\rho =\frac{3M}{4\pi {R}_{s}^{3}}=\frac{3{c}^{6}}{32\pi {G}^{3}{M}^{2}}$
scales as $\propto 1/{M}^{2}$: for very large amounts of mass or energy they don't have to be packed very tightly to cause a collapse. So instead of strong electromagnets you can just use a lot of material, and once you have enough it will become a black hole. For steel you would need about ${10}^{-5}$ solar masses.

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