This short paper looks at a clash between classical gravity and quantum physics. Recent work showed that a self-gravitating charged scalar field can collapse to form an extremal Reissner–Nordström (eRN) black hole in finite time. An eRN black hole is a charged black hole with zero Hawking temperature. If such formation were possible, it would contradict a version of the third law of thermodynamics for black holes, which says you cannot reach zero temperature in finite time.

The authors review two prior results. Kehle and Unger produced classical solutions that appear to reach extremality in finite time. Reall later proved a complementary classical result: for sufficiently regular fields, collapse cannot produce an extremal eRN black hole if the field’s mass‑to‑charge ratio m/e is at least 1. Hod and Piran point out that quantum effects can also help. In a strong electric field the vacuum can spontaneously produce pairs of charged particles (the Schwinger effect). When the black hole’s electric field is strong enough compared with the particle mass and charge, pair production will discharge the system and move it away from extremality.

Using that quantum discharge condition, the authors conjecture that Reall’s classical bound is not the whole story. They propose a strengthened classical limit on the mass‑to‑charge ratio that depends on the black hole charge Q and Planck’s constant ħ. Roughly, their suggested bound says classical collapse should be unable to make an extremal black hole up to the point where quantum pair production becomes effective. For large charges this strengthened bound approaches Reall’s m/e ≥ 1, but for smaller charges it is slightly stronger.

The paper checks this idea against some recent numerical collapse results. Lee reported numerical examples where the maximum mass‑to‑charge ratios that produced extremal black holes were m/e ≈ 0.9944, 0.9961 and 0.9866 in three cases with different overall scales. Those numbers respect both Reall’s original bound and the new conjectured bound, so they do not contradict the proposal. The authors note that a decisive test will require more collapse simulations that explicitly track the quantum scaling parameter ħ in the models.