Fractional Chrono-Aging Spacetime (FCAS): A Covariant Nonlocal Framework for Quantum Gravity, Cosmology, Temporal Isolation, and Multiscale Dynamics

FCAS Fractional Chrono-Aging Spacetime (FCAS) Theory – Version 5 Abstract The Fractional Chrono-Aging Spacetime (FCAS) theory is a proposed extension of general relativity that incorporates fractional calculus into the fabric of spacetime. This framework treats spacetime as a scale- dependent, fractal-like structure where the effective dimensionality of space and time can vary with scale, leading to what we term “chrono- aging” of time (the flow of time itself can change across different scales or environments). By replacing classical differential operators with fractional-order derivatives, FCAS modifies the Einstein field equations to include non-local, memory-like effects . The theory aims to provide a unified description across quantum and cosmological regimes: at small scales it offers a route toward a renormalizable quantum gravity , and at large scales it reproduces standard general relativity while potentially addressing puzzles like dark energy, dark matter, and the initial conditions of the universe . Key predictions of FCAS include modified gravitational dynamics (fractional field equations and geodesics), possible observational signatures in cosmology (due to varying spacetime dimensions in the early universe), and even the theoretical possibility of temporal isolation (regions where time flows at an anomalously different rate due to fractional structure). This document (version 5) details the theoretical framework of FCAS, discusses major results and implications (quantum gravity unification, cosmology, temporal isolation, and multiscale physics), and provides supporting mathematical formulations in the appendices. Introduction A central challenge in modern physics is reconciling the principles of general relativity (our theory of gravitation and spacetime on cosmic scales) with those of quantum mechanics (the framework governing subatomic particles). Conventional approaches to quantum gravity, such as string theory and loop quantum gravity, attempt this unification but face significant technical and conceptual hurdles. The Fractional Chrono-Aging Spacetime (FCAS) theory is an alternative paradigm that addresses this challenge by introducing fractional (non-integer) dimensions and derivatives into spacetime itself. The core idea is that spacetime may not be a fixed 4-dimensional continuum at all scales, but instead exhibits fractal or fractional structure when examined at extremely small (or extremely high-energy) scales . In this picture, the fabric of spacetime “ages” or changes character with scale – hence chrono-aging, implying that the passage of time and the geometry of space can scale in a way that resembles an aging process. FCAS posits that traditional concepts of distance, duration, and curvature must be generalized. At cosmological scales, the familiar 3+1 dimensional spacetime (three space dimensions and one time dimension) emerges as an average or infrared limit of a more complex, scale- dependent geometry. At microscopic scales, however, the effective dimensionality might differ (for example, one could have $3.5$ spatial dimensions or $0.9$ of a time dimension in a certain sense) . Thisdimensional variation is governed by fractional calculus, which naturally allows interpolation between integer dimensions and introduces non-local behavior. By non-local, we mean that points in spacetime can have interactions or relationships that are spread out – the fractional derivatives integrate information from an extended region, embodying a kind of memory effect . This non-locality could be the key to making gravity more compatible with quantum principles (for instance, smoothing out singularities or improving renormalization behavior). In simpler terms, FCAS suggests: • Fractal Spacetime: Space and time are fractal-like at small scales. The Hausdorff dimension of spacetime is not fixed but changes with scale, and the spacetime metric adapts accordingly . What looks like a point in spacetime at our scale might reveal a complex substructure if zoomed in beyond a certain resolution. • Fractional Field Equations: The laws of gravitation (Einstein’s equations) are generalized by using fractional derivatives. This introduces new terms and scale-dependent effects in the field equations, while reducing to the classic equations in the appropriate limit (when the fractional order $\to 1$) . These modifications encapsulate chrono-aging by allowing time and space to have anomalous scaling. • Chrono-Aging of Time: The flow of time experienced by clocks can depend on the fractional structure. Proper time may accrue in a non- linear way relative to coordinate time – metaphorically, time itself “ages” such that intervals at one scale might contain more or less proper time than expected when viewed from another scale. This concept extends relativity’s idea of time dilation by introducing a scale-dependent component. • Recovery of Standard Physics: In the macroscopic world (large distances, low energy), FCAS must closely approximate normal spacetime physics. Indeed, the theory is constructed so that when fractional effects are tuned off (or become negligible), one recovers ordinary 4D spacetime with Einstein’s general relativity and standard quantum field theory. The departures become significant only as one probes extreme regimes (near the Planck scale, for example). This document provides a comprehensive overview of FCAS theory (version 5). In Section “Theoretical Framework”, we define the mathematical structure of fractional spacetime, derive the modified gravitational field equations, and describe how particle trajectories (geodesics) are affected. In Section “Results and Implications”, we discuss the broader implications of FCAS: how it offers a path toward quantum gravity by taming infinities and introducing a fundamental length scale; how it alters cosmology by possibly explaining inflation or dark energy through geometric means; what is meant by temporal isolation as a potential phenomenon in this theory; and how FCAS acts as a multiscale unification, bridging physics across scales. We then summarize our findings in Conclusions, and provide mathematical details in the Appendices (including derivations related to the fractional parameter – denoted here as α or η in contexts – and its impact on entanglement entropy, an illustration of a WKB approximation in fractional spacetimes, and a table of key parameters and symbols used)