On the rarity of rocket-driven Penrose extraction in Kerr spacetime

Abstract

We present a Monte Carlo study of energy extraction from rotating (Kerr) black holes via the Penrose process using rocket propulsion. Through over 250,000 trajectory simulations, we establish sharp constraints on when Penrose extraction with escape to infinity succeeds. The mechanism requires that exhaust ejected inside the ergosphere carries negative Killing energy, which is kinematically accessible only via ultra-relativistic ejection deep within the ergosphere. We find that successful extraction with escape is statistically rare ($\sim$1% in broad parameter scans) and is governed by strict thresholds: it requires high black hole spin (empirically $a/M \gtrsim 0.89$) and ultra-relativistic exhaust velocity (onset at $v_e \approx 0.91c$). When conditions are highly tuned to a specific "sweet spot," success rates can reach 88.5%, representing a narrow extraction window rather than generic behavior. Furthermore, single-impulse thrust at periapsis achieves significantly higher cumulative efficiency ($η_{\rm cum} \approx 19\%$) compared to continuous thrust ($\sim$2--4%) due to path-averaging penalties. These constraints quantify the extreme fine-tuning required for material-based Penrose extraction, consistent with the astrophysical dominance of electromagnetic mechanisms. Simulation code is available at https://github.com/anindex/penrose_process.

Figures (6)

• Figure 1: Orbit classification in $(E_0, L_z)$ parameter space comparing moderate and high spin. (a) $a/M = 0.70$: Six classification regions are present---forbidden (dark gray, inaccessible configurations), bound (blue, $E_0 < 1$), plunge (red, no turning point), outside ergosphere (light blue, $r_{\rm peri} > r_{\rm erg}$), shallow ergosphere (orange, outer 30% of ergosphere), and deep ergosphere (green, extraction zone). The deep ergosphere region is negligible due to the narrow ergosphere width ($0.29M$). (b) $a/M = 0.95$: Higher spin dramatically expands the deep ergosphere region (43.7% of parameter space) at the expense of shallow and outside regions (now absent). The wider ergosphere ($0.69M$) and stronger frame-dragging enable more trajectories to reach the extraction zone. Dashed white contours show constant periapsis $r_p$; vertical dotted line marks $E_0 = 1$ (bound/unbound threshold). Star marker in panel (b) indicates the sweet spot $(E_0, L_z) = (1.22, 3.05)$.
• Figure 2: Penrose success rate versus black hole spin from comprehensive parameter sweep: broad scan ($E_0 \in [0.95, 2.0]$, $L_z \in [-3.0, 6.0]$; blue, $N = 6{,}400$ per spin) and sweet-spot region ($E_0 \in [1.1, 1.4]$, $L_z \in [2.5, 3.8]$; green, $N = 3{,}600$ per spin). Error bars show 95% Clopper-Pearson exact confidence intervals. No successful extractions were observed for $a/M \leq 0.7$ across 12,800 trajectories. Focusing on the sweet spot yields amplification factors of 19$\times$ at $a/M = 0.9$, 12$\times$ at $a/M = 0.95$, and 10$\times$ at $a/M = 0.99$, demonstrating the essential role of precise initial condition tuning.
• Figure 3: Thrust strategy comparison at $a/M = 0.95$, $v_e = 0.95c$. (a) Single-impulse trajectory ($E_0 = 1.18$, $L_z = 2.92$): geodesic infall (blue), impulse at periapsis $r \approx 1.6M$ (orange star), geodesic escape (green); energy gain $\Delta E = +0.0073$. (b) Continuous thrust trajectory ($E_0 = 1.20$, $L_z = 3.0$): coasting infall (blue), sustained thrust with $\sim$34,000 discrete events inside ergosphere (orange), escape (green); energy gain $\Delta E = +0.0025$. (c) Exhaust Killing energy $E_{\rm ex}$ versus radius: large star marks single-impulse event ($E_{\rm ex} \approx -0.12$); small circles show continuous thrust events. For this representative trajectory, all events satisfy $E_{\rm ex} < 0$; ensemble statistics (Table \ref{['tab:thrust-comparison']}) show $P(E_{\rm ex} < 0) > 95\%$. Dashed line marks ergosphere boundary ($r_{\rm erg} = 2M$). (d) Cumulative efficiency: single impulse achieves $\eta_{\rm cum} = 19.2\% \pm 1.5\%$ (Eq. \ref{['eq:eta-cum']}); continuous thrust achieves $\sim$2%. Note: $\eta_{\rm cum}$ is normalized to rocket mass loss; Wald's 20.7% bound is normalized to incident particle energy $E_0$ and is not an upper bound on $\eta_{\rm cum}$.
• Figure 4: Spin dependence of the extraction window. Panels show orbit classification in $(E_0, L_z)$ space for $a/M = 0.99$, $0.95$, $0.9$, and $0.7$ (left to right). Color coding follows Fig. \ref{['fig:orbit-classification']}: green indicates deep ergosphere flyby, orange shallow ergosphere, light blue outside ergosphere, red plunge, blue bound ($E_0 < 1$), and gray forbidden. The deep ergosphere region (extraction zone) expands dramatically with increasing spin: from negligible at $a/M = 0.7$ to dominating the parameter space at $a/M = 0.99$. The bound region (left of vertical dotted line at $E_0 = 1$) is present at all spins but not relevant for extraction in our baseline impulsive scans, since these initial conditions are bound ($E_0<1$) and do not satisfy the escape criterion. High spin ($a/M \gtrsim 0.9$) is essential for practical Penrose extraction.
• Figure 5: Thrust parameter sensitivity for $a/M = 0.95$ at the sweet spot ($E_0 = 1.22$, $L_z = 3.05$), sampled at 0.01$c$ velocity increments. Each configuration's success rate is computed over 1,000 initial conditions in $(E_0, L_z)$ drawn from a Gaussian distribution centered at $(1.22, 3.05)$ with $\sigma_E = 0.03$, $\sigma_L = 0.08$. (a) Penrose success rate versus exhaust velocity $v_e$ for four ejected mass fractions $\delta m \in \{0.1, 0.2, 0.3, 0.4\}$ with 95% Clopper-Pearson confidence intervals. A sharp onset (rapid transition region) occurs around $v_e \approx 0.91$--$0.92c$: at $v_e \leq 0.90c$, success is $\lesssim$1% regardless of $\delta m$; above $v_e = 0.93c$, success increases rapidly with both $v_e$ and $\delta m$. Peak success of 88.5% achieved at $v_e = 0.98c$, $\delta m = 0.4$. (b) Cumulative efficiency $\eta_{\rm cum}$ versus exhaust velocity (Eq. \ref{['eq:eta-cum']}). Values in panel (b) report the sample mean of $\eta_{\rm cum}$ over Penrose-successful escapes (conditioning on $\mathcal{S}$); unsuccessful trajectories are excluded from the efficiency statistic. Cumulative efficiency is inversely related to $\delta m$: smaller mass fractions yield higher per-mass energy gain. In the range $v_e \in [0.90c, 0.99c]$, efficiency increases approximately linearly with $v_e$, reaching $\eta_{\rm cum} = 7.8\%$ at $v_e = 0.99c$, $\delta m = 0.1$. The opposing trends in success rate and efficiency present a design trade-off for practical Penrose propulsion. Figure \ref{['fig:ultrarel-saturation']} extends panel (b) to the ultra-relativistic limit $v_e\to c$.