Mapping Gravity to the Quantum Realm

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Author: Denis Avetisyan

New research bridges the gap between quantum scattering and classical spacetime, offering a powerful framework for understanding black holes and their gravitational signatures.

This review details a multipolar approach reconstructing classical gravity from quantum amplitudes via post-Minkowskian expansion and effective field theory, enabling calculations of black hole properties and the identification of potential mimickers.

Reconciling general relativity with quantum mechanics remains a central challenge in modern physics, yet a direct mapping between quantum processes and classical spacetime geometry has remained elusive. This thesis, ‘From Quantum Amplitudes to Spacetime Geometry: a Multipolar Framework for Black Hole Signatures’, establishes a novel framework reconstructing classical gravity-including metrics and multipole moments-from the analytic structure of quantum scattering amplitudes. By systematically deriving the post-Minkowskian expansion and formulating a momentum-space energy-momentum tensor, this work links internal source distributions to external gravitational fields, even constructing horizon-less black hole mimickers with Kerr-like multipolar structure. Could this amplitude-based approach ultimately provide a pathway towards a complete quantum theory of gravity and a deeper understanding of spacetime itself?

Reconstructing Gravity: A New Paradigm for Spacetime

Historically, determining solutions to Einstein’s equations – the very foundation of general relativity – has presented significant challenges. Conventional methods, often reliant on approximations and iterative techniques, quickly become computationally prohibitive as gravitational fields strengthen or become more complex. These approaches not only demand substantial processing power, but also frequently obscure the underlying physical meaning of the solutions. The resulting mathematical descriptions, while technically correct, can lack intuitive connections to the spacetime geometry they represent, hindering a deeper understanding of gravitational phenomena. This difficulty in both calculation and interpretation motivates the exploration of alternative frameworks, seeking a more efficient and physically transparent pathway to unraveling the mysteries of gravity, particularly in extreme environments where traditional methods falter.

Current research explores a novel approach to understanding gravity by inverting the traditional relationship between spacetime and the events within it. Rather than solving Einstein’s equations to find spacetime, this framework begins with the fundamental description of particle interactions – quantum scattering amplitudes – and reconstructs the geometry of spacetime from them. These amplitudes, which calculate the probabilities of particles scattering, contain hidden information about gravity. By analyzing these probabilities at high energies, physicists can effectively deduce the underlying spacetime geometry, offering a potentially powerful tool for investigating scenarios where traditional methods falter, such as near black holes or during the early universe. This “scattering amplitudes” approach bypasses the need for perturbative expansions – approximations that often become unreliable in strong gravitational fields – and provides a more direct link between quantum mechanics and the classical description of gravity, suggesting a pathway towards a more complete theory.

Conventional calculations in gravity often rely on perturbative methods – approximating solutions as small deviations from a simple background – which struggle when gravity becomes intensely strong, such as near black holes or during the early universe. This new scattering amplitude approach offers a potential solution by reformulating the problem; instead of directly solving Einstein's equations, it focuses on calculating the probabilities of particles scattering off each other. These probabilities, encoded in the amplitudes, contain enough information to reconstruct the underlying spacetime geometry, even in regimes where perturbative methods fail. By sidestepping the need for approximate solutions, this framework promises a more robust and accurate description of strong-field gravity, potentially unlocking insights into phenomena currently beyond the reach of existing theoretical tools and offering a pathway to understanding the universe’s most extreme environments.

The reconstruction of gravitational interactions through the study of scattering amplitudes represents a shift away from traditional methods centered on solving Einstein’s field equations. This approach posits that the fundamental description of gravity resides not in the geometry of spacetime itself, but in the probabilities of particles scattering – the amplitudes. By meticulously analyzing these amplitudes, physicists seek to distill the essential information required to fully characterize gravitational phenomena, effectively ‘reconstructing’ spacetime geometry as an emergent property. This bypasses the complexities of directly tackling the notoriously difficult Einsteinian equations, particularly in regimes of strong gravity where perturbative techniques falter. The focus on amplitudes offers a potentially universal language for describing gravity, potentially unifying it with other fundamental forces and providing new insights into black holes, the early universe, and other extreme astrophysical environments, all derived from the core principles of quantum mechanics and scattering theory.

From Amplitudes to Classical Spacetime: A Derivation

The Energy-Momentum Tensor, T_{\mu\nu}, which describes the density and flux of energy and momentum in spacetime and is fundamental to Einstein’s field equations of General Relativity, can be systematically obtained from the calculation of scattering amplitudes in quantum field theory. This derivation involves expanding the amplitude to higher orders and extracting the classical contribution, which directly corresponds to the components of T_{\mu\nu}. Specifically, the tensor is obtained by taking appropriate derivatives of the amplitude with respect to external momenta and fields. This process demonstrates that the classical description of gravity, as embodied by General Relativity, emerges naturally from the underlying quantum theory, providing a rigorous link between quantum scattering processes and the classical behavior of spacetime.

Dressed vertices, employed within calculations of scattering amplitudes, represent the interactions of gravitons with matter fields, incorporating self-interactions and loop corrections. These vertices are modified to account for the internal momentum of the graviton and the effects of quantum fluctuations, effectively “dressing” the bare interaction. By analyzing the behavior of these dressed vertices in the low-energy, long-wavelength limit – the classical limit – one can systematically extract the classical gravitational interaction. This process involves identifying the dominant terms in the amplitude as momentum approaches zero, leading to an effective action that describes the classical theory of gravity, specifically General Relativity. The use of dressed vertices therefore provides a pathway to derive classical gravitational dynamics directly from the underlying quantum theory, without the need for semi-classical approximations or assumptions about the nature of spacetime.

The derivation of classical solutions from scattering amplitudes, utilizing the framework of dressed vertices and the energy-momentum tensor, avoids the need for traditional, often empirically-motivated, approximations such as post-Newtonian expansion or weak-field limits. This approach provides a systematic procedure where classical gravity emerges as a well-defined limit of quantum calculations, ensuring consistency between the quantum and classical descriptions. Specifically, the classical Einstein equations are obtained directly from the amplitude calculations without introducing external assumptions about the strength of gravitational fields or the slow-motion of sources. This robustness is due to the framework’s inherent ability to capture all relevant gravitational dynamics within the amplitude structure itself, rather than adding them as corrections.

Classical solutions, derived from scattering amplitudes via the energy-momentum tensor, are fundamental to understanding gravitational phenomena. These solutions represent the spacetime geometry – the metric g_{\mu\nu} – which dictates how matter and energy move within a gravitational field. Specifically, they provide the gravitational potential experienced by test masses and determine the trajectories of objects, including photons. The solutions encompass a broad range of scenarios, from the weak-field approximations relevant to solar system dynamics to the strong-field regimes characterizing black holes and neutron stars. Accurate classical solutions are therefore crucial for predicting and interpreting observational data from gravitational wave detectors, astronomical observations, and cosmological studies, ultimately allowing us to model the universe and its contents.

Approximating and Extending Black Hole Metrics: A Multipole Approach

The multipole expansion is a technique used to approximate gravitational fields by representing the source distribution as a series of nested moments – monopole, dipole, quadrupole, and so on. This approach relies on the energy-momentum tensor, T_{\mu\nu}, which describes the density and flux of energy and momentum, to characterize the source. By expanding the solution to Einstein’s field equations in terms of these multipole moments, the gravitational field can be accurately modeled, particularly when dealing with sources that lack simple symmetries. The accuracy of the approximation improves with each successive term included in the expansion, offering a systematic method for analyzing complex gravitational sources where exact solutions are unavailable. This is especially useful for sources with mass and angular momentum distributions that are not easily described by simpler metrics.

The Hartle-Thorne metric, a well-established solution in general relativity describing the spacetime around slowly rotating axisymmetric sources, serves as a crucial validation point for the multipole expansion method. This metric, commonly used to model astrophysical black holes with moderate angular momentum, is derived from the Kerr metric through a low-rotation approximation. By successfully reproducing the Hartle-Thorne solution using the derived energy-momentum tensor and multipole expansion techniques, the approach demonstrates its accuracy in the limit of known, established physics. This recovery confirms the validity of the mathematical framework and provides confidence in its application to more complex and less understood black hole configurations.

The multipole expansion technique, when combined with a suitable energy-momentum tensor, provides a framework for deriving black hole metrics beyond the Kerr solution. Specifically, it accurately reproduces the Myers-Perry metric, which describes rotating black holes in n dimensions, where n > 4. The Myers-Perry metric is characterized by two parameters: the mass M and the angular momentum parameters a_i, where i ranges from 1 to ⌊n/2⌋. The metric function, analogous to the Δ function in the Kerr metric, determines the event horizon and is dependent on these parameters, enabling the description of rotating black holes in arbitrary higher dimensions.

Generalization of black hole metrics to higher dimensions, beyond the traditional four-dimensional spacetime, permits the theoretical exploration of gravitational configurations not possible in lower dimensions. Specifically, these higher-dimensional frameworks allow for solutions representing black rings – topologically distinct objects characterized by event horizons possessing a toroidal shape rather than a spherical one. These black ring solutions, and other complex configurations arising from higher-dimensional metrics, are not simply extrapolations of four-dimensional black holes but represent fundamentally different classes of gravitational objects with unique properties and stability characteristics. Investigations into these solutions require numerical relativity techniques to determine their existence and assess their physical plausibility, contributing to a broader understanding of gravitational dynamics in diverse spacetime geometries.

Refining the Description of Rotating Charges: Towards Greater Precision

The Kerr-Schild gauge offers a significant advancement in the mathematical treatment of rotating black holes, particularly when working with the complex Kerr-Newman metric. This coordinate system skillfully decouples the gravitational field from the black hole’s rotation, effectively ‘shielding’ the flat spacetime behind the event horizon and simplifying the often-intractable equations of general relativity. By transforming the metric into this more manageable form, physicists can more readily analyze the behavior of rotating black holes and their surrounding environments, including the dynamics of accretion disks and the formation of powerful astrophysical jets. This simplification isn’t merely computational; it provides deeper insight into the fundamental properties of these cosmic objects and allows for more accurate modeling of their interactions with the universe.

The Kerr-Newman metric, a cornerstone in the study of rotating charged black holes, provides the foundational mathematics for interpreting numerous high-energy astrophysical observations. These black holes, unlike their static counterparts, possess both angular momentum and electric charge, characteristics believed to be prevalent in the supermassive black holes at the centers of galaxies. The swirling spacetime around these rotating objects directly influences the formation of accretion disks – structures composed of gas and dust spiraling inwards – and is fundamentally linked to the generation of powerful relativistic jets ejected from the poles. These jets, observed across the electromagnetic spectrum, are thought to be powered by the black hole’s rotation and magnetic fields, and understanding the Kerr-Newman metric is therefore essential for modeling their dynamics, luminosity, and overall impact on the surrounding galactic environment. Consequently, precise descriptions of rotating charged black holes are not merely theoretical exercises, but vital tools for unraveling the mysteries of some of the most energetic phenomena in the universe.

The gyromagnetic factor, a fundamental property defining the relationship between an object’s magnetic dipole moment and its angular momentum, plays a critical role in accurately modeling the behavior of rotating black holes. Recent investigations into five-dimensional charged rotating black holes reveal a gyromagnetic factor of 3/2, a result that powerfully validates predictions derived from the CCLP solution – a cornerstone in the study of rotating black hole electrodynamics. This confirmation is significant because it establishes a crucial link between theoretical frameworks and the physical properties of these complex astrophysical objects, furthering the understanding of phenomena like accretion disk dynamics and the formation of relativistic jets. The precise determination of this factor allows for more reliable simulations and predictions regarding the magnetic fields surrounding these black holes and their interaction with surrounding matter.

A comprehensive understanding of rotating black hole behavior necessitates examining the subtle interplay of several key physical effects. Calculations reveal that the gyromagnetic factor – a crucial parameter defining the magnetic properties of these objects – is not a constant, but scales with the dimensionality of spacetime. Specifically, the factor is predicted to be (d-1)/(d-2) for a d+1-dimensional spacetime. Achieving this result requires incorporating both a Chern-Simons interaction, which is constrained to a value of 1 in five dimensions to align with the established CCLP solution, and a Pauli term possessing a coefficient of -1/4 in the same dimensionality. Furthermore, the contribution of electromagnetic multipoles is essential for a complete description. These calculations demonstrate that accurately modeling rotating black holes demands the inclusion of non-minimal couplings – interactions beyond standard electromagnetism – particularly in spacetimes with more than three dimensions, signifying a more complex and nuanced relationship between angular momentum and magnetic moment than previously understood.

The research elegantly demonstrates how a holistic understanding of interconnected components yields profound insights, much like an ecosystem. By reconstructing classical gravity from quantum scattering amplitudes, the framework establishes a clear relationship between quantum processes and classical observables. This mirrors the principle that structure dictates behavior; the underlying quantum structure fundamentally defines the resulting gravitational fields and properties of black holes. As Albert Camus observed, “The struggle itself… is enough to fill a man’s heart. One must imagine Sisyphus happy,” suggesting that the rigorous pursuit of understanding, even within a complex system like spacetime geometry, provides inherent value. The framework’s ability to calculate gravitational multipoles and properties, even for black hole mimickers, highlights the power of clear ideas – what truly scales, not merely computational power.

Beyond the Horizon

The presented framework, reconstructing classical geometries from the ostensibly more fundamental language of scattering amplitudes, offers a compelling, if ambitious, path forward. However, the elegance of the mathematical construction should not obscure the fact that this remains, at present, a reconstruction. The true test lies not in reproducing known solutions-however complex-but in predicting novel phenomena. A complete theory must account for the inevitable imperfections, the ‘grit’ in the machine, that differentiate a true black hole from its many potential mimickers.

Future work will undoubtedly focus on extending these calculations to higher orders in the post-Minkowskian expansion, a task akin to progressively refining the city plan. One does not tear down entire blocks to improve traffic flow; infrastructure should evolve without rebuilding the entire structure. Furthermore, a deeper understanding of the relationship between the effective field theory description and the underlying quantum gravity is paramount.

Ultimately, the goal transcends mere calculation. The framework invites a re-evaluation of the very foundations of gravitational physics, prompting the question: is spacetime a fundamental entity, or merely an emergent property-a useful fiction-arising from the complex interplay of quantum processes? The answer, one suspects, will not be found in increasingly precise solutions, but in a more fundamental shift in perspective.

Original article: https://arxiv.org/pdf/2601.14775.pdf

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

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2026-01-22 19:07