Implications of the Speed of Light Limit on Maximum Wavelength and Energy Boundaries

James Rogers, SE Ohio, 2024 Nov 08 1434

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Abstract:

This paper explores a novel interpretation of the speed of light as not only an upper limit for velocity but also a boundary condition for wavelength, suggesting that a maximum wavelength exists at a state of minimal energy. By examining the relationship $c = f\lambda$, this interpretation implies that the speed of light constrains both high-energy (short wavelength) and low-energy (long wavelength) extremes, leading to the proposal that wavelength has a finite upper bound. This maximum wavelength at near-zero energy would represent a foundational “rest wavelength,” establishing a new framework where motion, energy, and wavelength are interdependent.

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Introduction:

The speed of light, $c$, is a cornerstone of modern physics, widely recognized as the upper limit for the speed of information and matter. This limit, however, has implications that extend beyond motion, potentially structuring fundamental constraints on wavelength and energy. In this paper, we present two key arguments suggesting that the speed of light implies both a minimum and a maximum wavelength. The resulting framework provides a unified view where the speed of light sets absolute constraints on energy and wavelength, creating a finite range for possible states.

Argument 1: The $c = f\lambda$ Relationship and Maximum Wavelength

In the equation $c = f\lambda$, if we allow frequency $f$ or wavelength $\lambda$ to approach zero, it implies a vanishing speed of light (c = 0)—an apparent contradiction, since $c$ is a constant. Thus, this paradox suggests limits on both frequency and wavelength. If wavelength could be infinite the frequency would go to 0, and visa versa, then $c$ would fail to maintain its constancy. Therefore, we propose that the existence of $c$ as an invariant speed inherently constrains wavelength and frequency, suggesting a maximum wavelength associated with the lowest possible energy state, just as the minimum wavelength is associated with the maximum possible energy state.

Argument 2: Wavelength Compression and Finite Rest Wavelength

As an object approaches the speed of light, its wavelength shortens, which we observe as an increase in energy or frequency. For this process of wavelength shortening to be meaningful, there must be a finite starting point for wavelength at rest. At this lowest energy level, the wavelength reaches its maximum length, implying a “rest wavelength” at minimal energy. Any energy input or time dilation would then shorten this baseline wavelength, indicating that all other wavelengths are compressions of this foundational limit. 

If wavelengths were infinite, then the concept of shortening them would make no sense allowing any speeds without limit.

Implications: A Finite Range for Wavelength and Energy

If the speed of light limits both maximum speed and maximum wavelength, then this reasoning implies a finite range for wavelengths. We propose that:

1. Maximum Wavelength at Rest: At a complete rest state (or minimal energy), the longest possible wavelength would exist, representing a baseline or rest wavelength. This suggests a maximum bound for wavelength, just as $c$ provides a maximum bound for speed.
2. Energy-Wavelength Compression: As energy increases or objects approach the speed of light, wavelengths shorten, compressing from this maximum starting point. Thus, energy states are bounded within a finite spectrum set by the limits of wavelength and frequency as determined by $c$.
3. This idea explains why the speed of light exists. If wavelength has a maximum length then it can only be shortened to just short of zero. So speed of light limit on velocity is directly a result of the geometry of space time that indexed the maximum wavelength of a wavelength. 

This model provides an interdependent structure, where the speed of light defines not only the maximum rate of motion but also the boundary conditions for energy and wavelength.

Discussion: Reconciling with Current Physics

This interpretation diverges from traditional physics, where $c = f\lambda$ is seen as proportional without specific wavelength limits. There is no theory that currently  joins these two different scales together at this time, where frequency and wavelengths switch between maximum and minimum states.

If the same spacetime geometry defines both maximum and minimum speed limits, minimum and maximum speed limits,  and minimum and maximum amounts of time dilation, then this implies that all of these things are intimately related to each other through the geometry of space time.  This would mean that quantum mechanics and relativity are the studying different ends of the exact same geometry.

Conclusion:

By interpreting the speed of light as a constraint on both velocity and wavelength, we propose a framework where the maximum wavelength serves as a starting state from which all energy states derive. This view suggests a structured, finite range for wavelength and energy, bounded by the speed of light as a universal constant. This perspective could add new depth to our understanding of how fundamental limits like $c$ shape the scope of physical phenomena.