Electric Penrose Process

• Electric Penrose process is an energy extraction mechanism that uses electromagnetic coupling and negative Killing-energy states in black-hole spacetimes.
• It distinguishes between rotating spacetimes, where frame dragging enables extraction, and static charged setups, where Coulomb interactions create generalized ergoregions.
• Astrophysical applications include using weakly charged particle ionization and magnetized environments to accelerate particles and power relativistic jets.

The electric Penrose process is the class of black-hole energy-extraction mechanisms in which electromagnetic coupling, rather than purely mechanical fragmentation in a Kerr ergoregion, supplies the negative-energy channel. In the broadest covariant sense, it includes electromagnetic fields and charged matter in rotating spacetimes, where frame dragging and the horizon generator permit negative Killing-energy and negative angular-momentum flux into the horizon. In a narrower but now widespread usage, it denotes charged-particle energy extraction in static charged spacetimes, where the electrostatic term $qA_t$ creates a generalized negative-energy region even in the absence of an ergosphere. These two usages are mathematically distinct but physically continuous: both rely on an ingoing degree of freedom carrying negative energy-at-infinity, thereby allowing an outgoing fragment, wave, or Poynting flux to emerge with excess energy (Lasota et al., 2013, Zaslavskii, 2024, Alloqulov et al., 2024).

1. Conceptual scope and nomenclature

The historical prototype is the mechanical Penrose process in Kerr spacetime, where a particle splits inside the ergoregion and one fragment crosses the horizon with negative Killing energy. The electric Penrose process generalizes this structure in two directions. First, in rotating spacetimes, electromagnetic fields and charged particles can realize the same horizon bookkeeping without literal particle disintegration. Second, in static charged spacetimes such as Reissner–Nordström-type geometries, the Coulomb interaction itself can render the conserved Killing energy negative, so that energy extraction can occur with no rotational ergosphere at all (Lasota et al., 2013, Zaslavskii, 2024).

This immediately distinguishes the electric Penrose process from superradiance and from purely mechanical Penrose splitting. Superradiance is a wave-amplification phenomenon with mode constraints and boundary conditions, whereas the electric Penrose process is usually formulated in terms of charged particles, local conservation laws, and electromagnetic contributions to the canonical energy. It also differs from the neutral collisional Penrose process, because the term $-qA_t$ or, in rotating electrovacua, $E-\Omega_H L-\Phi_H q$, opens negative-energy channels without requiring the same degree of fine tuning in kinematics or extremality (Hejda, 26 Nov 2025).

A frequent source of confusion is the role of the “ergosphere.” In the rotating case, the process may indeed be tied to the Kerr ergoregion or to its electromagnetic analogue on the horizon. In static charged spacetimes, however, the relevant structure is a generalized ergoregion for charged particles: a region outside the horizon where the minimum admissible conserved energy can be negative because the electrostatic term dominates. Several papers therefore use “electric Penrose process” for nonrotating charged black holes, explicitly emphasizing energy extraction “with no ergosphere” in the geometric Kerr sense (Alloqulov et al., 2024, Chen et al., 18 Aug 2025).

2. Covariant formulation in rotating spacetimes

For stationary, axisymmetric spacetimes with stationary Killing vector $\eta^\mu$ and axial Killing vector $\xi^\mu$, the horizon generator is

$\ell^\mu=\eta^\mu+\omega_H\,\xi^\mu,$

with $\omega_H$ the horizon angular velocity. For arbitrary matter or fields with stress tensor $T_{\mu\nu}$, the Noether energy and angular-momentum currents are

$$P^\mu:=-T^\mu{}_\nu\,\eta^\nu,\qquad M^\mu:=T^\mu{}_\nu\,\xi^\nu.$$

The integrated horizon fluxes are

$$\Delta E_H=\int_{\Delta H}T_{\mu\nu}\,\eta^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2,\qquad \Delta J_H=-\int_{\Delta H}T_{\mu\nu}\,\xi^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2.$$

The generalized Penrose criterion is then exact: the exterior gains net energy if and only if

$-qA_t$0

Assuming the null energy condition on the horizon, one also has

$-qA_t$1

so any extraction event implies $-qA_t$2. In this covariant sense, rotational energy extraction is equivalent to absorption by the horizon of negative Killing energy and negative Killing angular momentum; no horizon torque is required or even well-defined, because the event horizon is a stationary null hypersurface that only absorbs fluxes (Lasota et al., 2013).

This formulation yields a sharp horizon diagnostic. Since the horizon integrand is $-qA_t$3, the condition $-qA_t$4 implies $-qA_t$5 on some part of the horizon. Because $-qA_t$6 is future-directed null, this happens iff the Noether current $-qA_t$7 is spacelike or past-directed timelike/null there. The particle limit reproduces the standard Penrose result: in the Kerr ergoregion, where $-qA_t$8 is spacelike, a geodesic with $-qA_t$9 and $E-\Omega_H L-\Phi_H q$0 can cross the horizon and extract rotational energy (Lasota et al., 2013).

For electromagnetic fields, the same logic gives the electric horizon condition. Defining the corotating pseudoelectric field on the horizon by

$E-\Omega_H L-\Phi_H q$1

one obtains

$E-\Omega_H L-\Phi_H q$2

Because $E-\Omega_H L-\Phi_H q$3 on the null horizon, a necessary condition for electromagnetic Penrose extraction is

$E-\Omega_H L-\Phi_H q$4

on some part of the horizon. In stationary, axisymmetric force-free magnetospheres this reduces to the familiar Blandford–Znajek sign condition

$E-\Omega_H L-\Phi_H q$5

with

$E-\Omega_H L-\Phi_H q$6

The same paper shows that magnetically arrested disk simulations with $E-\Omega_H L-\Phi_H q$7 satisfy the negative-energy and negative-angular-momentum absorption conditions, with time-averaged efficiencies $E-\Omega_H L-\Phi_H q$8 up to $E-\Omega_H L-\Phi_H q$9, thereby identifying the Blandford–Znajek/MAD jet channel as an electromagnetic Penrose process in Noether-current language (Lasota et al., 2013).

3. Charged-particle mechanics in static charged spacetimes

In the static charged case one usually starts from

$\eta^\mu$0

For a charged particle of mass $\eta^\mu$1 and charge $\eta^\mu$2, the canonical conserved quantities are

$\eta^\mu$3

The electrostatic term is the crucial modification: it decouples negativity of the conserved energy from the absence of frame dragging. In Reissner–Nordström-like geometries, one may write $\eta^\mu$4, so for $\eta^\mu$5 an oppositely charged fragment can acquire $\eta^\mu$6 outside the horizon provided the Coulomb contribution outweighs the positive kinematic term (Zaslavskii, 2024, Chen et al., 4 Jan 2026).

The radial dynamics is governed by an effective potential. In one quantum-corrected Reissner–Nordström analysis,

$\eta^\mu$7

and

$\eta^\mu$8

with

$\eta^\mu$9

The generalized ergoregion is then the domain outside the horizon where $\xi^\mu$0, equivalently where a charged timelike trajectory can have negative Killing energy. Its boundary satisfies

$\xi^\mu$1

For $\xi^\mu$2, existence of this boundary requires $\xi^\mu$3, and the outer extent grows with larger $\xi^\mu$4 and smaller $\xi^\mu$5 (Chen et al., 4 Jan 2026).

A more general Reissner–Nordström treatment derives exact daughter-particle formulas for a decay $\xi^\mu$6 in the equatorial plane, with local conservation of $\xi^\mu$7, $\xi^\mu$8, $\xi^\mu$9, and radial momentum. It shows that scenarios in which both daughters are ejected along the trajectory of the parent correspond to extrema of $\ell^\mu=\eta^\mu+\omega_H\,\xi^\mu,$0 or $\ell^\mu=\eta^\mu+\omega_H\,\xi^\mu,$1 with respect to $\ell^\mu=\eta^\mu+\omega_H\,\xi^\mu,$2 or $\ell^\mu=\eta^\mu+\omega_H\,\xi^\mu,$3. It also derives exact bounds on daughter angular momenta and classifies several decay branches, including cases where maximum efficiency is achieved on the horizon and cases where it is not, unless additional constraints on particle parameters are imposed. The same formalism is explicitly model-independent and extends beyond Reissner–Nordström (Zaslavskii, 2024).

The static electric process therefore has a characteristic signature: negative-energy states originate in the electrostatic term, not in $\ell^\mu=\eta^\mu+\omega_H\,\xi^\mu,$4. This is why recent studies emphasize that charged fragments can extract energy from black holes “with no ergosphere,” provided the decay occurs within the generalized negative-energy region defined by the charged-particle effective potential (Alloqulov et al., 2024, Chen et al., 18 Aug 2025).

4. Extensions across geometries and matter sectors

The modern literature uses the electric Penrose process as a diagnostic of how different geometries and matter couplings modify negative-energy regions, escape channels, and efficiencies.

Background	Characteristic condition or formula	Reported effect
Quantum-corrected RN	$\ell^\mu=\eta^\mu+\omega_H\,\xi^\mu,$5, with quantum parameter $\ell^\mu=\eta^\mu+\omega_H\,\xi^\mu,$6 entering $\ell^\mu=\eta^\mu+\omega_H\,\xi^\mu,$7 (Chen et al., 4 Jan 2026)	$\ell^\mu=\eta^\mu+\omega_H\,\xi^\mu,$8 shrinks the generalized ergoregion, raises the barrier, lowers efficiency
4D charged EGB BH	$\ell^\mu=\eta^\mu+\omega_H\,\xi^\mu,$9 (Alloqulov et al., 2024)	Efficiency increases with $\omega_H$0, decreases with $\omega_H$1 relative to RN
ABG(-dS) BH	$\omega_H$2 (Chen et al., 18 Aug 2025)	Larger negative-energy region than RN(-dS); maximum efficiency ratio $\omega_H$3
Magnetized RN	$\omega_H$4 (Baez et al., 19 May 2026)	External $\omega_H$5 induces an ergosphere and controls onset/suppression of extraction
Weakly charged Schwarzschild	$\omega_H$6 (Tursunov et al., 2021)	Ionized particles can reach ultra-high energy; Sgr A* can act as a PeVatron
5D weakly charged Schwarzschild	$\omega_H$7 at $\omega_H$8 (Xamidov et al., 4 Mar 2025)	Slightly better near-horizon acceleration than 4D
Flat space and RN naked singularity	$\omega_H$9 for $T_{\mu\nu}$0 in flat space (Zaslavskii, 2018)	Unbounded efficiency in flat space; bounded-charge super-Penrose near naked RN (Zaslavskii, 2022)
RN–AdS and extremal RN-(A)dS	Recursive decay or BSW-assisted collision (Feiteira et al., 2024, Feiteira et al., 2024)	Energy factories, black-hole bombs with mirrors, and arbitrarily high but finite escaping energy

Several comparative patterns recur. Quantum corrections represented by $T_{\mu\nu}$1 obstruct extraction by shrinking the negative-energy region and can qualitatively change whether an escaping fragment remains unbound; a concrete example in the quantum-corrected Reissner–Nordström paper shows that for $T_{\mu\nu}$2, $T_{\mu\nu}$3 allows escape whereas $T_{\mu\nu}$4 leads to trapping (Chen et al., 4 Jan 2026). Nonlinear electrodynamics can have the opposite effect: Ayón-Beato–García black holes have a larger negative-energy region than RN and a horizon-potential enhancement that yields $T_{\mu\nu}$5 in the small-$T_{\mu\nu}$6 regime (Chen et al., 18 Aug 2025).

External fields also qualitatively reorganize the process. In magnetized Reissner–Nordström, the external $T_{\mu\nu}$7 field induces axisymmetry and an ergosphere in an otherwise static core, produces two disjoint extraction windows in $T_{\mu\nu}$8, and admits analytic critical magnetic fields for onset, reconnection, and suppression of extraction. In RN–AdS, the confining geometry or an imposed mirror permits recursive decay chains; one scenario yields a black-hole energy factory, while another yields a black-hole bomb only when the confinement volume remains fixed, since in pure AdS the effective confining volume can diverge along the decay chain and reduce the extracted energy per unit volume to zero (Baez et al., 19 May 2026, Feiteira et al., 2024).

5. Collisional, repetitive, and super-Penrose regimes

The electric Penrose process naturally extends from single decays to collisions, pair production, and iterative extraction. In a Kerr–Newman collisional model, two neutral particles collide near the horizon and produce oppositely charged particles. The key extraction condition is

$T_{\mu\nu}$9

Because the near-horizon electrostatic increment can dominate the kinematics, a sufficiently charged product can be forced to turn around and escape rather than fall in. In the regime $$P^\mu:=-T^\mu{}_\nu\,\eta^\nu,\qquad M^\mu:=T^\mu{}_\nu\,\xi^\nu.$$0, the escaping energy scales as

$$P^\mu:=-T^\mu{}_\nu\,\eta^\nu,\qquad M^\mu:=T^\mu{}_\nu\,\xi^\nu.$$1

so high efficiency does not require extremality or impact-parameter fine tuning. The paper gives the explicit example of an electron with $$P^\mu:=-T^\mu{}_\nu\,\eta^\nu,\qquad M^\mu:=T^\mu{}_\nu\,\xi^\nu.$$2 near a slightly negatively charged black hole with $$P^\mu:=-T^\mu{}_\nu\,\eta^\nu,\qquad M^\mu:=T^\mu{}_\nu\,\xi^\nu.$$3, for which $$P^\mu:=-T^\mu{}_\nu\,\eta^\nu,\qquad M^\mu:=T^\mu{}_\nu\,\xi^\nu.$$4 and $$P^\mu:=-T^\mu{}_\nu\,\eta^\nu,\qquad M^\mu:=T^\mu{}_\nu\,\xi^\nu.$$5 (Hejda, 26 Nov 2025).

Iterative extraction reveals a second layer of structure. In an initially extremal Kerr–Newman background, a repetitive charged-particle Penrose sequence can be tracked by updating $$P^\mu:=-T^\mu{}_\nu\,\eta^\nu,\qquad M^\mu:=T^\mu{}_\nu\,\xi^\nu.$$6 event by event. Two couplings govern the dynamics: $$P^\mu:=-T^\mu{}_\nu\,\eta^\nu,\qquad M^\mu:=T^\mu{}_\nu\,\xi^\nu.$$7, which controls whether the incident particle can continue to reach the ergoregion, and $$P^\mu:=-T^\mu{}_\nu\,\eta^\nu,\qquad M^\mu:=T^\mu{}_\nu\,\xi^\nu.$$8, which controls how deep the captured negative-energy state can be. Attractive capture, $$P^\mu:=-T^\mu{}_\nu\,\eta^\nu,\qquad M^\mu:=T^\mu{}_\nu\,\xi^\nu.$$9, enhances both the energy return on investment and the energy utilization efficiency; above $$\Delta E_H=\int_{\Delta H}T_{\mu\nu}\,\eta^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2,\qquad \Delta J_H=-\int_{\Delta H}T_{\mu\nu}\,\xi^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2.$$0, the dimensionless spin can even increase transiently despite ongoing angular-momentum loss, because mass decreases faster than $$\Delta E_H=\int_{\Delta H}T_{\mu\nu}\,\eta^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2,\qquad \Delta J_H=-\int_{\Delta H}T_{\mu\nu}\,\xi^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2.$$1. Near the critical charge $$\Delta E_H=\int_{\Delta H}T_{\mu\nu}\,\eta^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2,\qquad \Delta J_H=-\int_{\Delta H}T_{\mu\nu}\,\xi^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2.$$2, the evolution approaches the reversible Christodoulou–Ruffini limit and the energy utilization efficiency approaches unity while the hole remains sub-extremal; beyond that point, the irreducible mass decreases in the formal solution, signaling breakdown of the test-particle approximation (Alipour et al., 29 Jun 2026).

Repetitive extraction in Reissner–Nordström instead reveals a third-law analogue. A recent analysis shows that exact neutrality cannot be reached in any finite number of classical iterative electric Penrose steps. The process stops at a strictly positive residual charge fixed by turning-point and negative-energy constraints, so $$\Delta E_H=\int_{\Delta H}T_{\mu\nu}\,\eta^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2,\qquad \Delta J_H=-\int_{\Delta H}T_{\mu\nu}\,\xi^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2.$$3 is unattainable in finite step number. In that framework the energy return on investment can exceed $$\Delta E_H=\int_{\Delta H}T_{\mu\nu}\,\eta^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2,\qquad \Delta J_H=-\int_{\Delta H}T_{\mu\nu}\,\xi^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2.$$4, but the energy utilization efficiency is typically below about $$\Delta E_H=\int_{\Delta H}T_{\mu\nu}\,\eta^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2,\qquad \Delta J_H=-\int_{\Delta H}T_{\mu\nu}\,\xi^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2.$$5, because an increasing share of the extractable-energy decrease is converted into irreducible mass rather than escaping flux (Hu et al., 30 Oct 2025).

The “super-Penrose” limit also has a purely electric realization. In flat spacetime with a static Coulomb potential, a decaying charged particle can produce one fragment with negative canonical energy and another with arbitrarily large positive energy at infinity, so the efficiency can be unbounded even though the center-of-mass energy remains finite. Near an RN naked singularity the same $$\Delta E_H=\int_{\Delta H}T_{\mu\nu}\,\eta^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2,\qquad \Delta J_H=-\int_{\Delta H}T_{\mu\nu}\,\xi^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2.$$6 Coulomb scaling can drive $$\Delta E_H=\int_{\Delta H}T_{\mu\nu}\,\eta^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2,\qquad \Delta J_H=-\int_{\Delta H}T_{\mu\nu}\,\xi^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2.$$7 as $$\Delta E_H=\int_{\Delta H}T_{\mu\nu}\,\eta^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2,\qquad \Delta J_H=-\int_{\Delta H}T_{\mu\nu}\,\xi^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2.$$8 with bounded charges; this effect persists in the flat-space limit and therefore isolates the electromagnetic, rather than gravitational, core of the mechanism (Zaslavskii, 2018, Zaslavskii, 2022).

6. Astrophysical interpretation, misconceptions, and limitations

In astrophysical applications the electric Penrose process occupies two rather different niches. In the weakly charged, nonrotating picture, it has been proposed as a near-horizon ionization accelerator: a neutral particle falls in, becomes ionized, and the charged fragment escapes with an energy boosted by the Coulomb term. In this toy-model regime, realistic weak charges $$\Delta E_H=\int_{\Delta H}T_{\mu\nu}\,\eta^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2,\qquad \Delta J_H=-\int_{\Delta H}T_{\mu\nu}\,\xi^\mu \ell^\nu\,\sqrt{q}\,dt\,dy^1\,dy^2.$$9 up to $-qA_t$00 are sufficient to make the Galactic-center black hole act as a PeVatron of protons, and the maximum energy can scale as

$-qA_t$01

which becomes approximately mass-independent if $-qA_t$02 (Tursunov et al., 2021).

In rotating magnetized systems, the process merges with relativistic jet phenomenology. The generalized Noether-current treatment identifies Blandford–Znajek and MAD extraction as electromagnetic Penrose processes, while analyses of Kerr black holes in a Wald field show that negative-energy regions need not remain confined to the ergosphere and can form detached toroidal trapping zones. In that setting the effective coupling

$-qA_t$03

sets the scale of negative energies and of the escaping fragment’s boost. As the hole charges toward the Wald value $-qA_t$04, $-qA_t$05, the zero-energy surfaces shrink, the toroidal traps disappear, and efficient extraction is impeded (Lasota et al., 2013, Gupta et al., 2021).

Several misconceptions are clarified by this body of work. First, electric Penrose extraction is not synonymous with rotation: static charged black holes, magnetized static black holes, and even flat spacetime with a Coulomb potential admit electric Penrose analogues. Second, divergent center-of-mass energy does not imply infinite extracted energy at infinity; in the extremal RN-(A)dS BSW-assisted collision problem the escaping particle can be arbitrarily energetic but not infinite, while the partner falls inward with negative energy inside its own electric ergosphere (Feiteira et al., 2024). Third, the mechanism is not a “torque on the horizon”: in the covariant formulation the horizon only absorbs fluxes, and extraction is entirely encoded in negative-energy and negative-angular-momentum absorption (Lasota et al., 2013).

The principal limitations are equally consistent across the literature. Most analyses assume the test-particle or test-field approximation, neglect radiation reaction, self-force, pair production, and plasma screening, and often impose decay at turning points or idealized force-free conditions. Many papers explicitly note that satisfying $-qA_t$06 or $-qA_t$07 is a necessary and sufficient kinematic criterion for extraction, but not by itself a guarantee of astrophysical realizability. Mechanical splitting remains inefficient, while electromagnetic realizations—force-free horizons, pair-production channels, weakly charged ionization, and MAD jets—are the robust regimes in which the electric Penrose process becomes a quantitatively significant channel of black-hole energy extraction (Lasota et al., 2013, Hejda, 26 Nov 2025, Tursunov et al., 2021).