Exploring Cosmological Redshift Through Angular Geometry in a Static Universe

Abstract

We present an alternative geometric model to explain the redshift of light observed from distant celestial bodies, without relying on cosmic expansion or gravitational redshift. By analyzing the spatial arrangement of the light source, the observer, and a fixed reference point positioned relative to the observer, we demonstrate how geometric configurations alone can lead to an apparent elongation of light wavelengths. This model, which utilizes angular variations in triangular formations, suggests that redshift may result from purely geometric factors rather than universal expansion. Our approach provides a fresh perspective on cosmological observations and encourages a re-evaluation of fundamental cosmological assumptions.

1. Introduction

Redshift, the phenomenon where light from distant galaxies shifts toward longer wavelengths, has been a cornerstone of astrophysical research. Traditionally, it has been interpreted as evidence of an expanding universe, forming the basis of the widely accepted Big Bang theory. Hubble’s Law, which establishes a relationship between a galaxy’s redshift and its distance from Earth, has been pivotal in supporting this model.

However, alternative explanations exist that do not require cosmic expansion. By exploring different mechanisms for redshift, we can refine our understanding of the universe. In this study, we introduce a geometric framework based on angular relationships in a static universe, illustrating how spatial configurations can influence observed wavelengths.

2. Geometric Framework

Our model is built upon three key assumptions:

1. Static Universe

  • Assumption: The universe remains structurally unchanged over time.
  • Implication: Any redshift observed must stem from factors other than cosmic expansion.

2. Linear Light Propagation

  • Assumption: Light travels in straight paths unless altered by external influences.
  • Implication: The model employs classical Euclidean geometry for clarity and simplicity.

3. Angular Relationships

  • Assumption: The redshift results from the geometric positioning of the light source, observer, and a fixed reference point.
  • Implication: Changes in these angular configurations with distance influence the observed wavelength.

3. Triangle-Based Redshift Mechanism

Triangle Construction

We define a right-angled triangle to model the geometric interactions involved:

  • Vertices:
    • S (Source): The celestial body emitting light.
    • O (Observer): The point of detection (e.g., Earth).
    • P (Perpendicular Point): A reference point located at a fixed perpendicular distance from the observer.

Effect on Wavelength

As the distance between the source and observer increases, the associated angles in the triangular formation change, leading to an elongation of the effective light path. This elongation results in an apparent stretching of the observed wavelength, producing a redshift effect.

4. Mathematical Representation

4.1 Triangle Relations

For a right-angled triangle with sides h (fixed perpendicular distance), d (horizontal distance), and hypotenuse L:

  • Pythagorean Theorem: L = √(d² + h²)
  • Source Angle: θ = arctan(h/d)

4.2 Wavelength Stretching

The observed wavelength (λ_obs) is expressed as:

λ_obs = λ_emit (1 + ΔL/L_0)

where:

  • λ_emit: The emitted wavelength.
  • ΔL = L - L_0: Change in hypotenuse length relative to a reference point.

4.3 Redshift Expression

The redshift parameter z is given by:

z = (λ_obs - λ_emit) / λ_emit

which simplifies to:

z = ΔL / L_0

5. Interpretation

Angular Influence

As the source distance increases, the triangular formation stretches, elongating the light’s path. This results in an apparent increase in wavelength, mimicking redshift.

Perspective Analogy

This effect is similar to how parallel lines appear to converge at a distance due to perspective distortions. While the physical properties of the light remain unchanged, the geometric configuration alters the observed wavelengths.

6. Compatibility with Observations

  • Relation to Hubble’s Law: The model naturally predicts that redshift increases with distance.
  • Non-expanding Universe: The redshift emerges from geometry rather than space expansion.

7. Challenges and Considerations

7.1 Physical Explanation

Further investigation is required to explain how geometric configurations directly impact light’s measurable properties.

7.2 Energy Conservation

  • Redshift implies changes in photon energy (E = hc/λ), requiring an explanation for energy distribution in a static universe.

7.3 Compatibility with Relativity

Any alternative model must reconcile with the well-established principles of relativity.

7.4 Empirical Validation

  • The model requires parameter tuning to match real-world data accurately.

8. Future Developments

Mathematical Refinement

  • Advanced modeling techniques can enhance predictive accuracy.

Simulation and Data Analysis

  • Comparing predictions with observed astronomical data will help validate or refine the model.

9. Conclusion

This study introduces a geometric explanation for redshift, suggesting that light’s observed elongation may result from spatial configurations rather than universal expansion. While qualitatively aligning with observations, further refinement is needed to address physical mechanisms, energy conservation, and relativistic compatibility.

Future work will focus on enhancing mathematical precision, exploring the wavefront propagation effects, and comparing model predictions with empirical data.

References

  • Milne, E. A. (1935). Relativity, Gravitation and World-Structure. Oxford University Press.
  • Berry, M. V. (1972). "Optical Geometry of Motion." Physics Today, 25(7), 34-40.
  • Schneider, P., Ehlers, J., & Falco, E. E. (1992). Gravitational Lenses. Springer-Verlag.