Gupta, Law and Levin (Phys. Rev. D 104, 084059, 2021) showed at test-particle level that a Kerr black hole immersed in an asymptotically uniform magnetic field supports an electromagnetic Penrose process: a charged parent particle can decay into a negative-energy daughter — possibly trapped in novel toroidal zero-energy regions outside the ergosphere — and a boosted outgoing ('superradiant') daughter, with the boost set by an effective charge coupling χQ=qˉ(Q/2M−aB)\chi_Q=\bar q(Q/2M-aB) that vanishes as the hole charges up to the Wald value QW=2aMBQ_W=2aMB, at which point extraction shuts off. The paper explicitly defers the astrophysical question: whether the process ever operates at meaningful rates. The open problem is to determine that effectiveness quantitatively: compute the decay and collision rates and extractable power in a specified magnetized environment; the synchrotron radiation reaction (including the non-local DeWitt–Brehme tail term) of the charged daughters, especially those trapped in the toroidal negative-energy regions; the detailed numerical modeling of a realistic environment (plasma effects and collective electromagnetic processes) beyond the clean vacuum, test-particle dynamics; and the resulting observable signatures of the enhanced emission power in flux and spectrum, with implications for the polarization of light near the event horizon of M87* noted by the authors as a further consideration. The paper's demonstrated charging of the hole toward the Wald value — a saturation level since argued to differ from QWQ_W — sets the coupling history χQ(t)\chi_Q(t) that enters any such prediction. No work quantifies these components for a magnetized black-hole environment.

Background

The Penrose process is the classic mechanism for mining energy from a rotating black hole. In the Kerr geometry — the stationary vacuum solution describing a spinning black hole of mass MM and spin parameter aa — there is an ergosphere: a region outside the event horizon where spacetime dragging forces every observer to co-rotate with the hole, and where a particle can have negative conserved energy as measured from infinity. A particle that decays inside the ergosphere can therefore send one fragment into the hole with negative energy while the other escapes with more energy than the parent, the difference coming from the hole's rotational energy. The purely mechanical process is limited, with a maximal efficiency of about 20.7% for an extremal Kerr hole. Electromagnetic analogues change this picture. Wald (1974) showed that a Kerr black hole immersed in an asymptotically uniform magnetic field BB — treated as a test field on the fixed Kerr geometry — develops an induced electric field, and that the energetically favored charge of the hole is the Wald charge QW=2aMBQ_W=2aMB (geometrized units, G=c=1G=c=1): a spinning hole in an ambient magnetic field is inclined to charge up rather than swiftly neutralize. Building on 1980s work on charged-particle energy extraction in magnetized ergospheres, the magnetic Penrose process (MPP) — ionization or decay of a parent into oppositely charged fragments in a threading magnetic field — was shown to allow efficiencies formally exceeding 100%, since the external field does part of the work; Dadhich, Tursunov, Ahmedov and Stuchlík (2018) and Tursunov and Dadhich (2019) argue it could power ultra-high-energy cosmic rays, relativistic jets, and fast radio bursts, though only at test-particle level. The established field-theoretic extraction channel in the strong-field regime is the Blandford–Znajek mechanism, in which a force-free magnetosphere extracts rotational energy; it is studied with relativistic magnetohydrodynamics (GRMHD) simulations rather than discrete particles. Gupta, Law and Levin (2021) analyzed the complementary discrete-particle setup: a charged parent particle decays near a Kerr hole carrying charge QQ in a Wald test field. They showed, first, that the regions allowing energy extraction — the zero-energy surfaces, where the conserved particle energy (per unit mass, e∗e_*) can vanish or turn negative — extend beyond the ergosphere and can include disconnected toroidal regions that trap negative-energy daughters on orbits from which they can neither escape to infinity nor fall through the horizon. Second, the energy boost delivered to the outgoing daughter — termed superradiant, in analogy with the amplification of waves scattered off a rotating black hole — is set by the effective charge coupling χQ=qˉ(Q/(2M)−aB)\chi_Q=\bar q(Q/(2M)-aB), where qˉ\bar q is the daughter's charge-to-mass ratio; natural astrophysical values are enormous, χQ∼1010\chi_Q\sim10^{10}–102110^{21}. Third, the process itself drives the hole's charge toward the Wald value, so χQ→0\chi_Q\to0 and extraction is impeded — an electromagnetic analogue of mechanical spin-down. All of this is test-particle dynamics in a vacuum test field. The paper's closing section explicitly defers the astrophysical questions: it makes 'no attempt to quantify the probability of energy extraction'; a predictive model of the enhanced power of observable emission 'would entail detailed numerical modeling'; light emission from the trapped negative-energy daughters 'would require the full synchrotron radiation-reaction problem' — in curved spacetime the equation of motion of a radiating charge is the DeWitt–Brehme equation, which contains a non-local tail term from the backscatter of the emitted radiation off spacetime curvature — and implications for the polarization of light emitted near the event horizon of M87*, the supermassive black hole imaged by the Event Horizon Telescope, are left unexplored. One proposed environment is a black hole battery: a black hole threaded by a companion neutron star's dipole field, powering a luminous electromagnetic circuit (Mingarelli, Levin and Lazio, 2015).

Problem Statement

Determine the astrophysical effectiveness of the electromagnetic Penrose process for a Kerr black hole of mass MM, spin aa, and charge QQ immersed in an asymptotically uniform magnetic field BB (the Wald test-field regime): replace the test-particle kinematics of Gupta, Law and Levin (2021) with a quantitative prediction of whether, how often, and with what observable output the process operates in a realistic magnetized black-hole environment. For a specified environment (ambient field strength and orientation, charged-particle population, plasma state), a complete answer jointly delivers the following facets of that single objective: (1) Rates and power — the decay and collision rates at which parent particles produce a trapped or captured negative-energy daughter together with a boosted superradiant daughter, and hence the extractable power as a function of MM, aa, QQ, BB and the environment. (2) Radiative losses — the energy loss, orbital evolution, and emitted spectrum of the charged daughters under synchrotron radiation reaction in curved spacetime (the DeWitt–Brehme equation including its non-local tail term, or a controlled approximation to it), in particular for daughters trapped in the toroidal negative-energy regions, whose radiation is the only observable channel for particles that never escape. (3) Realistic environment — the detailed numerical modeling of the specified environment that the source defers in place of its clean vacuum solutions: plasma effects and collective electromagnetic processes beyond the test-particle treatment, while the setup itself remains the Wald test-field regime. (4) Signatures — the observable imprints of the enhanced emission power for a candidate source class such as a black hole battery or a supermassive hole like M87*: predicted fluxes and spectra; the source additionally notes that implications for the polarization of light emitted near the event horizon of M87* would be intriguing to consider. Because the process itself drives the hole's charge toward the Wald value — argued in later work to be a saturation level that sufficiently strong fields overshoot — the charging history shapes χQ(t)\chi_Q(t) and the duration of the predicted power output; this is context for facets (1) and (4) rather than a separate required deliverable. All four facets are mandatory for an affirmative resolution, and it suffices to fix a single concretely specified environment. A defensible negative answer — for example, a quantitative demonstration that the rates are negligible, or that radiative losses or backreaction kill the process, across an explicitly bounded class of physically motivated environments — equally resolves the problem, since the source question is precisely whether a system will avail itself of these boosts.

References

  1. Kshitij Gupta, Y. T. Albert Law, Janna Levin, 'Penrose process for a charged black hole in a uniform magnetic field', Phys. Rev. D 104, 084059 (2021). DOI: 10.1103/PhysRevD.104.084059; arXiv:2106.15010. https://doi.org/10.1103/PhysRevD.104.084059
  2. Robert M. Wald, 'Black hole in a uniform magnetic field', Phys. Rev. D 10, 1680–1685 (1974). DOI: 10.1103/PhysRevD.10.1680. https://doi.org/10.1103/PhysRevD.10.1680
  3. Chiara M. F. Mingarelli, Janna Levin, T. Joseph W. Lazio, 'Fast Radio Bursts and Radio Transients from Black Hole Batteries', Astrophys. J. 814, L20 (2015). DOI: 10.1088/2041-8205/814/2/L20. https://doi.org/10.1088/2041-8205/814/2/L20
  4. Naresh Dadhich, Arman Tursunov, Bobomurat Ahmedov, Zdeněk Stuchlík, 'The distinguishing signature of Magnetic Penrose Process', Mon. Not. R. Astron. Soc. 478, L89 (2018). DOI: 10.1093/mnrasl/sly073; arXiv:1804.09679. https://doi.org/10.1093/mnrasl/sly073
  5. Arman Tursunov, Naresh Dadhich, 'Fifty years of energy extraction from rotating black hole: revisiting magnetic Penrose process', Universe 5(5), 125 (2019). DOI: 10.3390/universe5050125; arXiv:1905.05321. https://doi.org/10.3390/universe5050125
  6. Prakruth Adari, Roman Berens, Janna Levin, 'Charging up boosted black holes', Phys. Rev. D 107, 044055 (2023). DOI: 10.1103/PhysRevD.107.044055; arXiv:2111.15027. https://doi.org/10.1103/PhysRevD.107.044055
  7. João S. Santos, Vitor Cardoso, José Natário, 'Electromagnetic radiation reaction and energy extraction from black holes: The tail term cannot be ignored', Phys. Rev. D 107, 064046 (2023). DOI: 10.1103/PhysRevD.107.064046; arXiv:2303.03411. https://doi.org/10.1103/PhysRevD.107.064046
  8. João S. Santos, Vitor Cardoso, José Natário, 'Floating orbits and energy extraction from magnetized Kerr black holes', Phys. Rev. D 110, 124054 (2024). DOI: 10.1103/PhysRevD.110.124054; arXiv:2409.17225. https://doi.org/10.1103/PhysRevD.110.124054
  9. O. B. Zaslavskii, 'General properties of the electric Penrose process', Phys. Rev. D 109, 124053 (2024). DOI: 10.1103/PhysRevD.109.124053; arXiv:2403.12879. https://doi.org/10.1103/PhysRevD.109.124053
  10. Conor Dyson, David Pereñiguez, 'Magnetic black holes: From Thomson dipoles to the Penrose process and cosmic censorship', Phys. Rev. D 108, 084064 (2023). DOI: 10.1103/PhysRevD.108.084064; arXiv:2306.15751. https://doi.org/10.1103/PhysRevD.108.084064
  11. J. A. Rueda, R. Ruffini, 'Kerr black hole energy extraction, irreducible mass feedback, and the effect of captured particles charge', Eur. Phys. J. C 84, 1166 (2024). DOI: 10.1140/epjc/s10052-024-13459-1; arXiv:2410.04776. https://doi.org/10.1140/epjc/s10052-024-13459-1
  12. Duarte Feiteira, José P. S. Lemos, Oleg B. Zaslavskii, 'Recursive Penrose processes in electrically charged black hole spacetimes: Backreaction and energy extraction', Phys. Rev. D 114, 024062 (2026). DOI: 10.1103/3m3p-1kwq; arXiv:2603.19354. https://doi.org/10.1103/3m3p-1kwq
  13. Ethan Berreby, Avner Okun, Shahar Hadar, Amos Ori, 'Net charge accretion in magnetized Kerr black holes', Phys. Rev. D 113, 124080 (2026). DOI: 10.1103/3xqq-f93g; arXiv:2511.22356. https://doi.org/10.1103/3xqq-f93g