Beyond the Event Horizon: Harvesting Energy from Black Holes

Author: Denis Avetisyan

New research reveals how quantum mechanics allows for work extraction from non-rotating black holes, challenging conventional understandings of their thermodynamic limits.

The study extends the black hole spacetime to encompass spacelike infinity through a conformal transformation, providing a framework for analyzing the asymptotic behavior of gravitational fields.

This paper establishes a link between black hole thermodynamics and open quantum systems, extending the Generalized Second Law to scenarios involving non-thermal vacuum states and demonstrating work extraction possibilities.

Despite the established convergence of general relativity, quantum field theory, and statistical mechanics in black hole thermodynamics, a complete understanding of its connection to broader frameworks of open quantum systems remains elusive. This work, ‘Black hole thermodynamics at null infinity. Part 2: Open systems, Markovian dynamics and work extraction from non-rotating black holes’, establishes a novel correspondence between black hole dynamics at null infinity and the thermodynamics of weakly coupled quantum systems, revealing that specific non-thermal vacuum states can indeed facilitate work extraction from black hole radiation. By extending the Generalized Second Law to encompass these scenarios and incorporating effects like grey body radiation, we demonstrate a consistent thermodynamic description of black hole evolution. Could this framework ultimately provide insights into the fundamental nature of quantum gravity and the emergence of spacetime itself?

Unveiling the Energetic Void: A Calculus of Non-Equilibrium

Conventional thermodynamics, built upon the principle of equilibrium, often overlooks the substantial energy potentially residing within systems far from balance. This framework assumes a static, settled state, failing to account for the dynamic energy fluctuations inherent in non-equilibrium scenarios. Consider a rapidly stirred fluid or a chemical reaction in progress – these systems possess energy beyond what’s described by simple temperature measurements. Researchers are increasingly recognizing that vast, untapped energy reservoirs exist in these transient, dynamic states, challenging the traditional view and opening doors to innovative energy harvesting technologies. The exploration of non-equilibrium thermodynamics is not merely a theoretical exercise; it represents a fundamental shift in how energy is understood and potentially harnessed, with implications for fields ranging from materials science to cosmology.

The seemingly empty vacuum of space, according to the Unruh effect, is not truly devoid of energy for those in motion. This counterintuitive prediction of quantum field theory posits that an observer undergoing constant acceleration perceives the vacuum as a thermal bath – a sea of particles with a temperature proportional to their acceleration. While traditionally considered a theoretical curiosity, this suggests a radical possibility: that work can be extracted from the vacuum itself. An accelerating observer, in essence, experiences energy arising from what appears to be emptiness, potentially offering a pathway – albeit extraordinarily challenging – to harness zero-point energy. The implications extend beyond fundamental physics, hinting at unconventional energy sources and prompting investigations into the limits of energy conservation as perceived by different observers within the framework of quantum mechanics.

The extreme gravitational environment surrounding black holes presents a unique laboratory for investigating the behavior of quantum states and, consequently, unconventional energy sources. Near a black hole’s event horizon, quantum fluctuations-typically considered negligible in flat spacetime-are dramatically amplified. This amplification isn’t merely an increase in magnitude, but a distortion of the vacuum itself, potentially allowing for the creation of particle-antiparticle pairs from seemingly empty space. Theoretical work suggests that carefully manipulating these quantum states – perhaps through engineered interactions with the black hole’s ergosphere – could, in principle, extract usable energy. While harnessing energy from black holes remains firmly within the realm of theoretical physics, understanding the quantum dynamics in these intense gravitational fields is paramount to determining the feasibility of such energy extraction methods and to furthering the exploration of the fundamental relationship between gravity, quantum mechanics, and the nature of the vacuum.

The evolution of a two-qubit system, illustrated by transitions between energy eigenstates (depicted in blue and red), reveals the impact of reservoir-induced jumps within its tensor space.

The BLPS Engine: A Precise Formulation of Vacuum Work

The Brunner-Linder-Popescu-Skrzypczyc (BLPS) engine demonstrates work extraction from vacuum states by establishing a non-equilibrium condition through population inversion. This process doesn’t violate thermodynamic laws; instead, it utilizes the energy inherent in the vacuum, accessible when a system is driven out of thermal equilibrium. Population inversion, defined by the condition $E2/T2 - E1/T1 \leq 0$ , where $E1$ and $E2$ represent the energy gaps of two-level systems (qubits) and $T1$ and $T2$ are their respective temperatures, creates a preferential flow of energy from the vacuum into the system. This energy differential enables the engine to perform work, with the theoretical maximum efficiency constrained by the Carnot efficiency $ηC = 1 - T1/T2$ . The BLPS engine offers a quantifiable pathway to harness vacuum energy, distinct from perpetual motion, by explicitly defining the conditions for controlled energy extraction.

The BLPS engine functions by establishing a ‘passive’ state within the system, which is a non-equilibrium condition specifically designed to facilitate energy transfer from the quantum vacuum. This is achieved by manipulating the energy levels of qubits such that the system acts as a low-temperature reservoir, effectively drawing energy from the zero-point fluctuations of the vacuum. The preferential absorption of vacuum energy isn’t a violation of conservation laws; rather, it’s a consequence of the system’s designed thermodynamic properties, enabling it to act as a heat sink and maintain a temperature difference necessary for work extraction. This process relies on the fundamental principle that even in the absence of a traditional thermal source, energy fluctuations exist within the vacuum, and a properly configured system can harness these fluctuations to perform work.

The theoretical maximum efficiency of work extraction via the BLPS engine is fundamentally limited by the Carnot efficiency, represented as $η_C = 1 - \frac{T_1}{T_2}$ . Here, $T_1$ and $T_2$ denote the temperatures of the two thermal baths interacting with the system. This limitation arises from the second law of thermodynamics, which dictates that no engine can exceed the Carnot efficiency when converting heat into work. The Carnot efficiency defines the upper bound based solely on the temperature difference between the hot and cold reservoirs; any real-world implementation of the BLPS engine will necessarily exhibit lower efficiency due to irreversibilities and practical constraints, but cannot surpass this theoretical limit.

Initiation of work extraction from the vacuum via the BLPS engine requires satisfying a population inversion condition, mathematically expressed as $E_2/T_2 - E_1/T_1 \leq 0$ . Here, $E_1$ and $E_2$ represent the energy gaps of the two-level quantum systems (qubits) involved in the process, and $T_1$ and $T_2$ denote the temperatures of the respective thermal reservoirs coupled to these qubits. This inequality dictates that the ratio of energy gap to temperature for the higher energy level must be less than or equal to the corresponding ratio for the lower energy level. When this condition is met, a non-equilibrium state is established, allowing for the preferential flow of energy from the vacuum and enabling the extraction of useful work.

The Brunner-Linder-Popescu-Skrzypczyc (BLPS) setup provides a concrete physical realization of the BLPS engine concept. This implementation utilizes two qubits, each interacting with a separate thermal bath at temperatures $T_1$ and $T_2$ . The setup leverages controlled-unitary operations to induce correlations between the qubits, effectively creating the necessary population inversion. Specifically, the BLPS protocol employs local operations on each qubit, combined with classical communication, to manipulate the system’s state and extract work. This work extraction is predicated on maintaining the condition $E_2/T_2 - E_1/T_1 \leq 0$ , where $E_1$ and $E_2$ represent the energy gaps of the qubits, thereby establishing a pathway for harnessing energy from the vacuum.

The BLPS engine utilizes population inversion to extract heat <span class="katex-eq" data-katex-display="false">Q_2</span> from a hot bath, enabling work <span class="katex-eq" data-katex-display="false">W \leq 0</span> to be performed, such as lifting a load on an energy ladder. — The BLPS engine utilizes population inversion to extract heat $Q_2$ from a hot bath, enabling work $W \leq 0$ to be performed, such as lifting a load on an energy ladder.

Entropy’s Constraint: A Rigorous Defense of Thermodynamic Consistency

The Generalized Second Law of thermodynamics posits that the total entropy of a closed system, and by extension the universe, can never decrease over time; it either remains constant in a reversible process or increases in an irreversible one. This principle fundamentally constrains any process attempting to extract work from a system. Specifically, any work gained must be offset by a corresponding increase in entropy elsewhere within the closed system, preventing perpetual motion and establishing a limit on the efficiency of energy conversion. This law isn’t merely a statement about disorder; it’s a quantitative restriction on the evolution of physical systems, dictating that the total $\Delta S \ge 0$ , where $\Delta S$ represents the change in total entropy.

Relative entropy, within the mathematical structure of Von Neumann algebras, serves as a quantifiable measure of the difference between two quantum states. Specifically, it determines the minimum work required to transform one state into another, or conversely, the information lost during such a transformation. Formally, for states ρ and σ on a Von Neumann algebra $\mathcal{M}$ , relative entropy is defined as $S(\rho||\sigma) = \text{Tr}(\rho(\log \rho - \log \sigma))$ , where Tr denotes the trace operation. This quantity is always non-negative due to the properties of the trace and operator inequalities, directly reflecting the Second Law of Thermodynamics; any process that would result in a negative relative entropy is physically impossible. The use of Von Neumann algebras provides a rigorous framework for handling unbounded operators and ensuring a well-defined measure of entropy in quantum systems.

The Modular Hamiltonian, denoted as $H_M$ , plays a critical role in quantifying relative entropy within the algebraic approach to quantum statistical mechanics. Specifically, it appears in the formula for relative entropy, $S(\rho|\sigma) = Tr(\rho(\Delta - \log(\sigma)))\$ , where Δ is the modular operator associated with a state σ. Calculating relative entropy via the Modular Hamiltonian ensures adherence to the Generalized Second Law of Thermodynamics by providing a mathematically rigorous method to demonstrate that entropy increases during any physical process. The non-negativity of relative entropy, guaranteed by the properties of the Modular Hamiltonian, directly corresponds to the Second Law’s prohibition of entropy decrease in closed systems, and provides a powerful tool for analyzing quantum systems far from equilibrium.

Correct application of the Generalized Second Law of thermodynamics requires precise definition of the system’s boundary conditions, and in the context of gravitational systems, these are commonly specified by Null Infinity. Null Infinity $\mathcal{I}^+\$ represents the limit of spacetime as an observer approaches infinite spatial separation while maintaining a fixed retarded time. Defining entropy changes necessitates isolating the system of interest from its surroundings; Null Infinity serves as this asymptotic boundary, allowing for the unambiguous calculation of energy and momentum fluxes entering or leaving the system. Without a well-defined boundary like Null Infinity, determining the total entropy change – crucial for verifying the non-decreasing nature stipulated by the Generalized Second Law – becomes ill-defined and prone to spurious results arising from incomplete accounting of external influences.

Beyond Simplification: Extending to Rotating Systems and Their Implications

Analyzing Kerr black holes – those that rotate – significantly complicates the process of Hawking radiation due to the introduction of grey-body effects. Unlike the simplified scenario of a non-rotating, Schwarzschild black hole, the curved spacetime around a Kerr black hole doesn’t allow all frequencies of radiation to escape with equal ease. These ‘grey-body’ factors represent the probability that a particle of a given frequency will tunnel through the event horizon; lower frequencies are more easily emitted, while higher frequencies are suppressed. Consequently, the emitted radiation spectrum deviates from the perfect blackbody spectrum predicted for Schwarzschild black holes, becoming modified and attenuated at higher energies. This spectral shift influences both the rate of black hole evaporation and the characteristics of the emitted particles, demanding a more nuanced understanding of quantum field theory in curved spacetime and presenting challenges for observational signatures of Hawking radiation from rotating astrophysical black holes.

The definition of a vacuum state profoundly impacts calculations involving black holes and energy extraction. While a simple, Minkowski-space vacuum might initially seem adequate, it fails to account for the intense gravitational field near a black hole’s event horizon. Different vacuum states, such as the Hartle-Hawking state-which emphasizes Euclidean continuation and avoids a singularity-and the LL-Vacuum-derived from demanding outgoing modes at spatial infinity-offer alternative perspectives on the energy landscape. Employing these varied vacuum definitions effectively addresses the ambiguity inherent in defining ‘empty’ space in curved spacetime, significantly altering predictions for phenomena like Hawking radiation and the efficiency of energy extraction processes. This nuanced approach reveals that the amount of extractable energy isn’t absolute, but rather contingent upon the chosen vacuum state, offering a more complete and physically relevant understanding of black hole thermodynamics.

Kruskal coordinates offer a unique and invaluable method for charting the otherwise distorted geometry surrounding a black hole. Unlike standard Schwarzschild or Kerr coordinates which become singular at the event horizon, Kruskal coordinates provide a complete and non-singular description of spacetime, effectively “unfolding” the black hole’s interior and exterior into a single, smoothly connected diagram. This coordinate system achieves this by transforming the radial coordinate in a way that eliminates the coordinate singularity, allowing researchers to trace the paths of light and matter across the event horizon without encountering mathematical breakdowns. The resulting diagram reveals the existence of multiple, disconnected regions representing different universes, and crucially, demonstrates that infalling objects do not encounter a true barrier at the horizon, but instead continue their journey through spacetime – a visualization impossible with conventional coordinate systems. This powerful tool is therefore essential for accurately modeling black hole behavior and understanding the fate of objects that venture too close.

The theoretical limit of energy that can be harvested from a rotating black hole, through processes leveraging Hawking radiation and vacuum fluctuations, is fundamentally governed by a simple yet profound relationship. This maximal work, denoted as $|W_{max}| = (1 - T_l/T_h)\omega$ , demonstrates a direct proportionality to both the temperature differential between the black hole’s event horizon ( $T_h$ ) and the surrounding environment ( $T_l$ ), and the frequency (ω) of the emitted radiation mode. This equation highlights that a greater temperature contrast, or the extraction of higher frequency waves, yields a potentially larger energy output. Consequently, understanding and maximizing this ratio becomes crucial for assessing the practical feasibility of such energy extraction schemes, offering a quantifiable benchmark for evaluating efficiency in these extreme astrophysical settings.

The developed framework, initially applied to simplified black hole scenarios, provides a crucial stepping stone for investigating energy extraction in the far more intricate environments found throughout the cosmos. Astrophysical black holes aren’t isolated entities; they exist within accretion disks, magnetic fields, and are often rotating – characteristics that dramatically affect energy harvesting processes. By establishing a foundation with solvable models, researchers can progressively incorporate these complexities, modeling phenomena like the Penrose process in realistic Kerr black holes and accounting for the impact of surrounding plasma. This approach doesn’t merely offer theoretical insights; it provides a predictive tool, allowing scientists to estimate the efficiency of energy extraction from various astrophysical sources and to better understand the power output of quasars and active galactic nuclei, ultimately linking theoretical frameworks to observed phenomena.

This Penrose diagram of the Kerr solution's maximal extension illustrates that, unlike the Schwarzschild solution with a single singularity, an infinite stacking of regions like the one shown is possible due to the timelike nature of Kerr singularities, though the exterior regions closely resemble those of an eternal Schwarzschild black hole. — This Penrose diagram of the Kerr solution’s maximal extension illustrates that, unlike the Schwarzschild solution with a single singularity, an infinite stacking of regions like the one shown is possible due to the timelike nature of Kerr singularities, though the exterior regions closely resemble those of an eternal Schwarzschild black hole.

The exploration of work extraction from black holes, as detailed in the paper, resonates with a timeless observation. Ralph Waldo Emerson penned, “Do not go where the path may lead, go instead where there is no path and leave a trail.” This sentiment applies directly to the study’s innovative approach to black hole thermodynamics. By considering non-thermal states and open quantum systems, the research forges a new path beyond established equilibrium assumptions. The paper doesn’t merely accept the conventional understanding of black hole entropy; instead, it actively challenges it, demonstrating how work can be derived from what was previously considered a purely entropic sink. This is not simply verification of existing theory, but the creation of new conceptual ground – a genuine ‘trail’ blazed in the landscape of theoretical physics.

Beyond the Event Horizon

The demonstrated correspondence between black hole thermodynamics and the dynamics of open quantum systems, while conceptually satisfying, merely shifts the fundamental questions elsewhere. The extraction of work from a non-rotating black hole via engineered vacuum states is not a refutation of thermodynamics, but rather a precise articulation of its limitations when applied to systems fundamentally defined by event horizons. The true challenge lies not in finding work, but in rigorously defining the relevant Hilbert space and the permissible operations within it – a task that demands a departure from naive notions of unitarity.

The extension of the Generalized Second Law to these non-equilibrium scenarios, though formally presented, relies heavily on the choice of reference states and the quantification of quantum relative entropy. A more robust formulation must transcend dependence on arbitrary observers and establish a truly intrinsic measure of entropy growth-a measure independent of any external description. The current reliance on modular Hamiltonians, while mathematically elegant, begs the question of whether these structures genuinely reflect physical reality or merely represent a convenient mathematical tool.

Future work must confront the inherent difficulty in modelling truly open systems. The assumption of Markovian dynamics, while simplifying the analysis, likely represents an idealization. The inevitable feedback between the black hole and its environment, and the resulting non-Markovian effects, will undoubtedly introduce complexities that necessitate a fundamentally different mathematical framework-one where the very notion of a ‘state’ becomes ill-defined. Only then can one hope to approach a complete and consistent description of black hole thermodynamics, divorced from the ambiguities of conventional quantum mechanics.

Original article: https://arxiv.org/pdf/2601.03356.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

Unveiling the Energetic Void: A Calculus of Non-Equilibrium

The BLPS Engine: A Precise Formulation of Vacuum Work

Entropy’s Constraint: A Rigorous Defense of Thermodynamic Consistency

Beyond Simplification: Extending to Rotating Systems and Their Implications

Beyond the Event Horizon

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