Systemic Relativity and Logarithmic Gravity: A Scale-Dependent Framework for Unifying Quantum & Cosmic Phenomena
Creators: name: Sosna, Andreas
Publisher: name: Zenodo
Year: 2025
Subjects: Modified gravity
Scale-dependent physics
Logarithmic gravity
Astrophysical anomalies
Galaxy rotation curves
Gravitational lensing
Gravitational waves
Emergent gravity
Dark matter alternatives
Dark matter
Quantum gravity
Radial acceleration relation
Extended theories of gravity
Cosmological tests of gravity
Non-local gravity
Fundamental physics

description: For over a century, we've faced a profound divide in physics: quantum mechanics and general relativity stubbornly refuse to speak the same language, while galaxies spin with a defiant disregard for our gravitational expectations. In this paper, I introduce Systemic Relativity & Logarithmic Gravity – a framework that rethinks how gravity operates by introducing logarithmic corrections that naturally vary with scale. This approach offers an alternative to dark matter while seamlessly recovering the Newtonian behavior we observe in everyday life.

The key insight lies in what I call the system-dependent parameter λ – defined as the ratio between a system's gravitational binding energy and the fundamental Planck energy. Unlike other theories that apply fixed corrections (like MOND) or rely on arbitrary mathematical functions, this formulation explains why gravity behaves differently across cosmic scales as a natural consequence of each system's intrinsic properties.

What's particularly striking is how these logarithmic gravitational modifications have independently emerged across seemingly disconnected fields – from black hole thermodynamics to quantum gravity approaches – suggesting we're glimpsing something fundamental about reality. The evidence is compelling: when I analyzed 175 galaxies from the SPARC database, the framework reduced rotational velocity prediction errors by half compared to standard Newtonian models. Even more impressively, when applied to the galaxy cluster Abell 2744, it accurately reproduces gravitational lensing patterns and mass distributions where other modified gravity theories struggle.

This framework doesn't just explain observations – it makes testable predictions. I outline specific, falsifiable expectations for gravitational lensing and gravitational wave behavior that can be verified through upcoming observations with JWST, LIGO, and LISA. If I'm right that gravity is scale-dependent, then what we've been calling "dark matter" may simply be an artifact of our perspective – the result of applying fixed-scale thinking to a universe where gravity itself transforms across the cosmic hierarchy.



The implications extend beyond cosmology to challenge our fundamental conception of physical law. Perhaps gravity isn't a universally fixed force but an emergent phenomenon that adapts to the systems it governs – a possibility that might finally bridge the quantum-cosmic divide that 
description: Mathematical Framework of Systemic Relativity & Logarithmic Gravity





Dual Formulation Approach



The mathematical foundation of the theory explores two equivalent formulations of logarithmic gravity:

Additive Logarithmic FormulationThe gravitational potential can be expressed with an additive logarithmic term:

Φ(r) = -GM/r(1 + λ ln(r/r0))

where:- λ is the system-dependent parameter defined as λ = Eg/EP, the ratio of gravitational binding energy to Planck energy- r0 is a characteristic length scale

This modification yields a gravitational acceleration of:

a(r) = GM/r²(1 + λ - λln(r/r0))

Multiplicative Exponential FormulationAlternatively, the gravitational acceleration can be expressed in multiplicative form:

glog(r) = GM/r² × e^(-λlog(r))

The equivalence between these formulations can be demonstrated through the expansion:

e^(-λlog(r)) ≈ 1 - λlog(r) + O(λ²)

For small values of λ typical in galactic systems (10^-6 to 10^-3), the first-order approximation links the two formulations.

2. Rotation Curve Predictions

For galactic systems, this produces a radial acceleration relation that naturally accounts for the observed flattening of rotation curves without dark matter. The rotational velocity predictions from both formulations are:

Additive formulation:v²(r) = GM/r(1 + λ - λln(r/r0))

Multiplicative formulation:v²(r) = GM/r × e^(-λlog(r))

Both formulations were tested against SPARC data, with the multiplicative form showing marginally better fits in most galaxies.

3. Scale-Dependent Parameter λ

The λ parameter varies systematically across cosmic scales, following:

λ ≈ GM²/R/EP ≈ Gℏ/c³ × M²/R

Typical values range from:- Quantum scale (atoms): λ ≈ 10^-70- Stellar scale (Sun): λ ≈ 10^-38- Galactic scale (Milky Way): λ ≈ 10^-5- Galaxy clusters: λ ≈ 10^-3

The scale-dependence emerges naturally from quantum gravitational principles, with λ growing from negligible values at quantum scales to appreciable values at galactic scales, creating a smooth transition across regimes.

4. SPARC Data Analysis

Statistical analysis of SPARC galaxy data demonstrates that both logarithmic models reduce prediction errors compared to Newtonian gravity:

Δerr = (1/N)∑|vobs,i - vlog,i|/vobs,i ≈ 0.5 × (1/N)∑|vobs,i - vN,i|/vobs,i

5. Quantum Extensions

The theory also establishes connections to quantum phenomena through a modified uncertainty principle in curved spacetime:

ΔxΔp ≥ ℏ(1 + λln(Δx/lP))

where lP is the Planck length.

6. Wave Propagation Predictions

For gravitational waves, the theory predicts frequency-dependent propagation velocities:

vg(ω) = c(1 - αλln(ω0/ω))

where α is a coupling constant and ω₀ is a reference frequency.

7. Testing and Verification Results

Both formulations (additive and multiplicative) were comprehensively tested against SPARC rotation curve data, yielding:



Approximately 50% reduction in overall prediction errors compared to Newtonian models without dark matter

Significantly improved fits in outer galactic regions where Newtonian models most severely underpredict velocities

Consistent λ values across similar galaxy types, supporting the framework's internal consistency

Marginally better performance of the multiplicative form by approximately 5-8% in most test cases


The quantum uncertainty relation and gravitational wave equations provide additional testable predictions for future experiments with JWST, LIGO, and LISA.

8. Cosmological Implications

At cosmological scales, the framework suggests:

H(z)=H0Ωm(1+z)3+ΩΛ(1+λcosmoln⁡(1+zz0))H(z)=H0Ωm(1+z)3+ΩΛ(1+λcosmoln(z01+z))

where $H(z)$ is the Hubble parameter at redshift $z$, and $\lambda_{cosmo}$ is the cosmological-scale λ parameter.

This modification offers potential explanations for cosmic acceleration without dark energy while maintaining consistency with observational constraints.

These mathematical relationships establish a unified framework for gravitational phenomena across all scales, from quantum to cosmological, eliminating the need for dark matter and dark energy while maintaining consistency with quantum principles and observational data.
description: Physics: Gravitation, Astrophysics, Theoretical Physics, General Relativity, Quantum Cosmology

Mathematics: Mathematical Physics, Applied Mathematics

Astronomy: Galactic Dynamics, Observational Cosmology

Interdisciplinary: Philosophy of Science, Computational Physics 


Keywords:



Modified gravity

Scale-dependent physics

Logarithmic gravity

Astrophysical anomalies

Galaxy rotation curves

Gravitational lensing

Gravitational waves

Emergent gravity

Dark matter alternatives

Quantum gravity

Radial acceleration relation

Extended theories of gravity

Cosmological tests of gravity

Non-local gravity

Fundamental physics