DOI: 10.5281/zenodo.17154404
title: Fractional Chrono-Aging Spacetime (FCAS): A Unied Framework for Gravitational-Wave and Collider Constraints, Quantum Entanglement, and Cosmic Expansion.

description: 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)