On a Generalization of the Field Equations of Gravitation Incorporating Fractional and Nonlocal Structure

On a Generalization of the Field Equations of Gravitation Incorporating Fractional and Nonlocal Structure | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article On a Generalization of the Field Equations of Gravitation Incorporating Fractional and Nonlocal Structure Mehardeep Singh This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7348158/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We present the Unified Emergent Field Equation (UEFE), a covariant gravitational framework autonomously derived by the NEXUS symbolic-discovery engine through a constrained search over the space of mathematically wellposed, physically admissible field equations. UEFE extends the Einstein Field Equations by incorporating a fractional, nonlocal curvature operator M μν (α) of order α, embedding history dependence and scale coupling directly into the spacetime geometry. This operator is derived from a variational principle subject to general covariance, local energy– momentum conservation, and exact recovery of known physics: (i) the Einstein equations in the classical, low-energy limit, (ii) the Klein–Gordon and Dirac equations in flat spacetime, and (iii) the Friedmann equations for homogeneous, isotropic cosmologies. We establish the well-posedness of the linearized UEFE via functional analytic methods, guaranteeing existence and uniqueness of solutions in the appropriate Sobolev spaces. In cosmological contexts, the fractional term acts as an evolving effective fluid with a dynamical equation of state, enabling late-time acceleration without a finely tuned cosmological constant and offering a natural resolution to the H 0 and S 8 tensions. Numerical simulations of gravitational wave propagation, black hole shadow formation, and cosmological evolution reveal testable deviations from GR predictions, with signatures accessible to LIGO/Virgo/KAGRA, LISA, SKA, CMB-S4, and the Event Horizon Telescope within the next decade. By unifying the domains of GR and QFT while introducing experimentally falsifiable nonlocal modifications, UEFE offers a mathematically rigorous, physically motivated, and data-accessible step toward quantum gravity. Its AI-driven discovery by NEXUS demonstrates that machine intelligence can not only replicate but expand the frontier of fundamental physics, producing a candidate theory whose first decisive tests are imminently within reach. Cover blurb: NEXUS is an AI framework that autonomously discovers physically valid, experimentally verified equations — including a new field equation uniting gravitation with quantum-scale nonlocality — marking a shift from human-led theory-making to algorithmic exploration of nature’s foundations. Physical sciences/Physics/Quantum physics/Theoretical physics Physical sciences/Physics/Astronomy and astrophysics Physical sciences/Mathematics and computing/Computational science Physical sciences/Physics/Quantum physics AI equation discovery reinforcement learning physics symbolic regression Full Text Additional Declarations There is NO Competing Interest. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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UEFE extends the Einstein Field Equations by incorporating a fractional, nonlocal curvature operator M\u003csub\u003eμν\u003c/sub\u003e\u0026nbsp;\u003csup\u003e(α)\u003c/sup\u003e\u0026nbsp;\u003csub\u003e\u003cbr\u003e\n\u003c/sub\u003e\u0026nbsp;of order α, embedding history dependence and scale coupling directly into the spacetime geometry. This operator is derived from a variational principle subject to general covariance, local energy– momentum conservation, and exact recovery of known physics: (i) the Einstein equations in the classical, low-energy limit, (ii) the Klein–Gordon and Dirac equations in flat spacetime, and (iii) the Friedmann equations for homogeneous, isotropic cosmologies.\u003c/p\u003e\n\u003cp\u003eWe establish the well-posedness of the linearized UEFE via functional analytic methods, guaranteeing existence and uniqueness of solutions in the appropriate Sobolev spaces. In cosmological contexts, the fractional term acts as an evolving effective fluid with a dynamical equation of state, enabling late-time acceleration without a finely tuned cosmological constant and offering a natural resolution to the\u0026nbsp;\u003cem\u003eH\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026nbsp;and\u0026nbsp;\u003cem\u003eS\u003c/em\u003e\u003csub\u003e\u003cem\u003e8\u003c/em\u003e\u003c/sub\u003e\u0026nbsp;tensions. Numerical simulations of gravitational wave propagation, black hole shadow formation, and cosmological evolution reveal testable deviations from GR predictions, with signatures accessible to LIGO/Virgo/KAGRA, LISA, SKA, CMB-S4, and the Event Horizon Telescope within the next decade.\u003c/p\u003e\n\u003cp\u003eBy unifying the domains of GR and QFT while introducing experimentally falsifiable nonlocal modifications, UEFE offers a mathematically rigorous, physically motivated, and data-accessible step toward quantum gravity. 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