On the (Im)possibility of Electrically Charged Planck Relics

TL;DR

This work shows that black-hole relics carrying SM electric charge cannot survive cosmological history due to two robust exterior-channel neutralizations: ambient capture enhanced by Coulomb and gravitational focusing, and QED Schwinger pair production near the horizon. It provides closed-form neutralization times via a trigamma-summed capture rate and derives an explicit integrated Schwinger discharge law, yielding a minuscule residual charge $Q_{\rm stop}^{(e)}$ for Planck-scale horizons. The analysis spans sub-Planckian objects, Planck relics, and astrophysical black holes, including extensions to dark-sector charges with heavy carriers and kinetic mixing. It also discusses a conservative no-Schwinger limit and shows that ambient exposure during reionization and later epochs excludes any integer SM charge, with gravity reinforcing neutralization for heavier relics. Three loopholes—extreme near-extremality, hidden-sector charges, or discrete/topological charge—remain as potential exceptions, but otherwise SM electromagnetism disfavors charged relics as dark matter candidates. The results reinforce Gauss-law exterior-field behavior and support the view that charged remnants must be electrically neutral with respect to the SM across cosmic history.

Abstract

I revisit whether black-hole remnants, from sub-Planckian compact objects to Planck relics and up to (super)massive black holes, can preserve Standard-Model (SM) electric charge. Two exterior-field mechanisms -- Coulomb-focused capture from ambient media and QED Schwinger pair production -- robustly neutralize such objects across cosmic history. I first derive the general capture rate including both Coulomb and gravitational focusing, and sum the stepwise discharge time in closed form via the trigamma function, exhibiting transparent Coulomb- and gravity-dominated limits. I then integrate the Schwinger rate over the near-horizon region to obtain an explicit $\dot Q(Q)$ law: discharge proceeds until the horizon field falls below $E_{\rm crit}$, leaving a residual charge $Q_{\rm stop}^{(e)}\!\propto\! r_h^2$ that is $\ll e$ for Planck radii. Mapping the mass dependence from sub-Planckian to astrophysical scales, I also analyze dark-sector charges with heavy carriers (including kinetic mixing and massive mediators). In a conservative ``no-Schwinger'' limit where vacuum pair creation is absent, cumulative ambient exposures alone force discharge of any integer SM charge. Three possible loopholes remain. (i) A fine-tuned SM corner in which the relic sits arbitrarily close to Reissner-Nordström extremality so greybody factors suppress charged absorption, while Schwinger pair creation is absent due to Planck-scale physics. (ii) Charge relocated to a hidden $U(1)_D$ with no light opposite carriers, e.g. if the lightest state is very heavy and/or kinetic mixing with $U(1)_{\rm EM}$ is vanishingly small. (iii) Discrete or topological charges rather than ordinary SM electric charge. Outside these cases, the conclusion is robust: within SM electromagnetism, charged black-hole relics neutralize efficiently and cannot retain charge over cosmological times.

Figures (3)

• Figure 1: No-capture boundary with extremality suppression $\epsilon=0.1$. Shown is the maximum integer relic charge $|Z|$ that avoids Coulomb-focused capture of ambient electrons (solid) or protons (dashed) as a function of relic mass. Three effective exposures are considered: today only ($\mathcal{E}=0.45$), today plus reionization ($\mathcal{E}=8$), and an illustrative larger exposure ($\mathcal{E}=50$), each uniformly suppressed by $\beta_{\rm abs}=\epsilon=0.1$. The black horizontal line marks $|Z|=1$. Shaded regions below the lowest electron curve are excluded, as any relic in that range would have been neutralized across cosmic history.
• Figure 2: No-capture boundaries with extremality suppression, split by charge sign. Left: electron capture ($Z>0$). Right: proton capture ($Z<0$). In both panels Schwinger discharge is switched off, and I assume $\mathcal{E}=50$. The parameter $\epsilon \equiv 1 - Q/Q_{\max}$ quantifies proximity to extremality, and different mappings $\beta_{\rm abs}(\epsilon)$ are shown: $\beta=\epsilon$ (solid), $\beta=\epsilon^2$ (dotted), and $\beta\propto\sqrt{\epsilon(2-\epsilon)}$ (dashed). Each color corresponds to a different value of $\epsilon$: $10^{-1}$ (blue), $10^{-3}$ (orange), and $10^{-5}$ (green). The black line marks $|Z|=1$.
• Figure 3: Left: Hawking temperature $T$ of Reissner--Nordström black holes as a function of the distance from extremality, $\delta \equiv M/|Q|-1$, for charges $Q=0.1,\,0.5,\,0.9$. The curves illustrate the generic behavior: $T\to 0$ at extremality, a $\sqrt{\delta}$ rise, a maximum at intermediate $\delta$, and Schwarzschild-like cooling at large mass. Right: Time evolution of the Hawking temperature $T$ for near-extremal RN black holes with charges $Q=0.9,\,0.99,\,0.999,\,0.9999$. After an early plateau, all trajectories converge to the universal $T \propto t^{-1/2}$ decay, reflecting the asymptotic approach to extremality. Together, the two panels summarize the mass-domain and time-domain perspectives on how evaporation drives RN black holes toward extremality.