A Worldline-Based Approach to Refraction: Isolating Nonlinear Photon Scaling Factors

J. Rogers, SE Ohio, 02 Mar 2025, 0415

Abstract: Traditional models of refraction rely on material-specific dispersion relations, often requiring extensive empirical fitting for each substance. This paper introduces an alternative approach based on the worldline of the photon, treating refraction as a fundamental scaling effect on the photon's properties due to its interaction with curved spacetime in a medium. By isolating the nonlinear energy-dependent scaling behavior of the photon, we eliminate the need for separate dispersion curves per material and instead require only the material's permittivity, permeability, and a single linear scaling factor. The method achieves sub-1% error across a wide range of materials in the visible spectrum and offers a computationally efficient alternative for applications such as ray tracing and optical simulations.

1. Introduction Refraction has traditionally been modeled using material-specific dispersion equations that fit empirical refractive indices to wavelength-dependent functions, such as Sellmeier or Cauchy equations. However, these models implicitly assume that the refractive index is an emergent property of the material rather than a fundamental interaction between the photon’s worldline and the medium’s properties. By re-examining refraction from a relativistic perspective, we propose a universal nonlinear photon scaling function that applies to all materials, with only a single material-dependent scaling factor required.

2. Theoretical Framework A photon traveling through a medium undergoes a change in velocity due to the medium’s permittivity (ε_r) and permeability (μ_r). The speed of light in the medium is given by: 

        c / sqrt( ε_r * μ_r)

Instead of treating this as an independent material property, we consider this effect as a modification of the photon's worldline, where its energy-dependent scaling follows a universal function. The refractive index is thus decomposed as follows:

        n_predicted = f(photon energy) * s_material

where f() is the universal nonlinear scaling function and s_material is a single empirical scaling factor unique to each material.

3. Computational Model and Implementation To validate this approach, a computational model was developed that:

• Uses known values of permittivity and permeability to determine the baseline refractive index.

• Applies a universal nonlinear function of photon energy to account for wavelength-dependent scaling.

• Uses a single linear material scaling factor  to fit empirical data.

• Compares predicted and empirical refractive indices across visible wavelengths.

The model achieves an accuracy of <0.72% worst-case error, with most values <0.2%, aligning with the limits of accuracy of reference tables.

The model is here: https://github.com/BuckRogers1965/refraction_photon_mass

4. Results and Discussion The results demonstrate that refraction can be accurately modeled using a single universal nonlinear function that scales with photon energy, eliminating the need for per-material dispersion formulas. This suggests that previous dispersion models overcomplicate refraction by implicitly re-deriving photon behavior instead of recognizing its inherent universality.

Furthermore, this method provides an elegant computational advantage: instead of fitting multiple parameters per material, only one material-dependent factor is required. This significantly simplifies calculations for applications like ray tracing and optical design.

5. Future Work and Extensions The model currently applies with high accuracy to the visible spectrum. Preliminary investigations suggest deviations in the infrared and ultraviolet ranges, likely due to additional photon-material interactions such as absorption and scattering. Future work will involve:

• Developing separate scaling functions for IR and UV regimes.

• Investigating the underlying physics of why the universal function holds for visible light but diverges at other wavelengths.

• Exploring the connection between this framework and deeper relativistic effects in materials.

6. Conclusion This paper presents a novel approach to refraction, demonstrating that a photon’s worldline and energy-dependent scaling govern refraction in a way that is largely independent of material-specific dispersion equations. By requiring only permittivity, permeability, and a single material scaling factor, this model simplifies computational optics while maintaining sub-1% error. The results indicate that the behavior of the photon, rather than material-dependent dispersion, is the key driver of refraction, offering new insights into both optics and fundamental physics.