The Mathematical Energy Rest Frame for the Photon

J. Rogers, SE Ohio, 21 Apr 2025, 0201

Abstract

We propose a mathematical "energy rest frame" for the photon: a reference photon with frequency 1 Hz. While this is not a physical rest frame—since photons, being massless, have no inertial rest frame—this convention provides a universal scaling anchor for all photon properties. We demonstrate how this reference works in both SI and natural units, highlighting the arbitrariness of the 1 Hz choice and the universality of the scaling relationships.

1. Introduction

In special relativity, the rest frame of a particle is the frame in which its momentum is zero and its energy is minimized34. For massless particles like photons, no inertial rest frame exists234. However, the mathematical relationships between a photon's energy, frequency, momentum, and (relativistic) mass are linear and universal:

• $E=hf$

• $p=E\mathrm{/}c=hf\mathrm{/}c$

• $m=E\mathrm{/}{c}^2=hf\mathrm{/}{c}^2$

We introduce the concept of a "mathematical energy rest frame"—a reference photon of frequency 1 Hz—serving as a scaling anchor for all photon properties.

2. The 1 Hz Photon as a Reference

2.1. Why 1 Hz?

The choice of 1 Hz is arbitrary and rooted in human conventions for measuring frequency6. We chose this frame because every property scales from here by frequency.

2.2. Properties in SI Units

For a photon with $f_0=1\mathrm{H}\mathrm{z}$:

Property	Formula	Value at 1 Hz
Energy ($E_0$)	$h{f}_0$	$h\approx 6.626\times {10}^{-34}\mathrm{J}$
Wavelength (${\lambda }_0$)	$c\mathrm{/}{f}_0$	$c\approx 3.00\times {10}^8\mathrm{m}$
Momentum ($p_0$)	$h\mathrm{/}c$	$\approx 2.21\times {10}^{-42}\mathrm{k}\mathrm{g}\mathrm{m}\mathrm{/}\mathrm{s}$
Mass ($m_0$)	$h\mathrm{/}{c}^2$	$\approx 7.37\times {10}^{-51}\mathrm{k}\mathrm{g}$
Rotation per second	$2\pi$f0 radians	$2\pi$ radians

For any photon of frequency $f$, all properties scale linearly:

• $E=E_0\cdot f$

• $p=p_0\cdot f$

• $m=m_0\cdot f$

• $\lambda ={\lambda }_0\mathrm{/}f$

3. Natural Units

Keeping the second the same and rescaling the meter and kg so that  $h=c=1$This makes all photon properties dimensionless and reveals their direct proportionality.  This allows all relationships to simplify:

Property	At $f=1$	At general $f$
Frequency ($f$)	$1$	$f$
Energy ($E$)	$1$	$f$
Wavelength ($\lambda$)	$1$	$1\mathrm{/}f$
Momentum ($p$)	$1$	$f$
Mass ($m$)	$1$	$f$
Rotation	$2\pi$	$2\pi f$

All photon properties are dimensionless and scale directly with frequency.

These relationships are always direct proportions, even in SI units. The proportionality constants (h, c) simply reflect our human unit choices. The underlying physics is that energy, frequency, momentum, and mass for a photon are fundamentally and linearly related, independent of the system of units. Allowing h = c = 1 to become true by scaling the kg and meter unit definitions so all the units of measure align in natural units simply reveals the directness of these proportions by removing the conversion factors.

While it is common to view constants like $h$, $c$, and $G$ as fundamental and mysterious, their numerical values are simply a reflection of our chosen units. The real physical content lies in the dimensionless relationships and scaling laws, not in the constants themselves. Natural units make this explicit by aligning the units of measure, showing that the “mystery” of the constants is largely a product of human convention, not a hidden truth of the universe124.

4. Discussion

• Not a Physical Rest Frame:
  The "energy rest frame" is not a real inertial frame—the photon cannot be at rest in any frame234. The reference frequency (1 Hz) is a mathematical convenience, not a physical state6.  

• 1 Hz is a human definition
  This is just a scaling that is decided by 1 / (chosen time unit).  If we made the second
  longer or shorter this would change what point is 1Hz.  This means that all of the values
  for a photon depend on the definition of the second.

• Interpretation:
  This approach clarifies why $E=hf$ is universally valid: Planck's constant is the "energy per Hz," and all photon properties are simple multiples of their values at the reference frequency.

• Valuable for teaching
  By seeing how these properties line up and scale from a starting point it ties all these concepts together.  Also seeing how everything scales by frequency in natural units shows how aligning units of measure is what defines natural units. 

5. Conclusion

The mathematical energy rest frame for the photon—anchored to a 1 Hz reference—provides a transparent, scalable framework for understanding photon properties in any unit system. While not a physical rest frame, it is a powerful conceptual tool for dimensional analysis and for teaching the universality of quantum and relativistic relationships for massless particles.

References:
-3 Rest frame - Wikipedia
-4 Restframe of Photon: Explained - Physics Forums
-6 Do photons with a frequency of less than 1 Hz exist? - Physics Stack Exchange