Abstract:
This paper introduces a novel framework that positions photon mass as the fundamental property from which other physical constants, including Planck’s constant (ℎ), are derived. By analyzing the relationships between photon mass (m), the speed of light (c), and the gravitational constant (G), we reveal the centrality of mass in defining energy, momentum, and other properties of a photon. The convergence of all these properties to photon mass at c = 1 strongly supports mass as the primary physical quantity, challenging traditional views that prioritize energy. This insight provides a unified and causally grounded interpretation of fundamental physics.
1. Introduction
The relationships between fundamental constants—such as Planck’s constant (ℎ), the speed of light (c), and the gravitational constant (G)—have long been foundational in physics. However, traditional frameworks often treat energy as the primary quantity, from which other properties like mass are derived. This paper proposes an alternative perspective: photon mass is the fundamental property, and energy, momentum, ℎ, and other derived quantities emerge from it through powers of c.
2. Theoretical Background Planck’s constant (ℎ) plays a crucial role in quantum mechanics, linking energy (E) and frequency (ν): Simultaneously, mass-energy equivalence from special relativity states: Traditionally, energy has been emphasized as fundamental, with mass often viewed as derived. However, this paper demonstrates that photon mass, coupled with c, is sufficient to define ℎ and other properties, establishing mass as the more fundamental quantity.
3. Derivation of Properties from Photon Mass Using the framework proposed here, we calculate physical properties of photons starting from their mass (m):
Energy (E):
Momentum (p):
Planck’s Constant (ℎ):
Photon Energy-Momentum Relationship:
These relationships converge to the photon’s mass when c = 1. This convergence suggests that ℎ and other constants emerge naturally from mass and c, rather than the reverse.
4. Convergence Analysis: The Centrality of Mass Examining how properties behave as c approaches 1 reveals mass as the central quantity. For instance:
Energy converges to mass.
Momentum converges to mass.
Planck’s constant (ℎ) converges to mass.
The product hc converges to mass.
This behavior is incompatible with frameworks that treat energy as fundamental, as energy would instead converge to itself.
5. Space-Time Ratio and Dimensional Analysis
We first derive the invariant space_time_ratio from Plank mass m_P, this is not the reduced Planck mass, this uses h. In order to isolate the Planck mass we had to cancel a second common factor that I call s_t here because it involves a volume of space and an accleration. So you can cancel m_P and isolate the s_t just as easily.
m = h/c**2 # kg the mass of a photon
m_P = (h*c/G)**(half) # kg the original Planck mass unit using h not hbar
space_time_ratio = m/m_P # dimensionless,
# m_P = m / space_time_ratio #kg this is the new definition for m_P in the framework
In order to isolate the Planck mass we had to cancel a second common factor that I call s_t here because it involves a volume of space and an accleration. So you can cancel m_P and isolate the s_t just as easily.
s_t = (h*c*G)**(half) # m^3/s^2 the s_t is short for space time factor
space_time_ratio = s_t/c**3 # this version has units of second, this cancels an s from denominator in next line
# s_t = space_time_ratio* c**3 # m^3/s^2 - This is the new definition for s_t in the framework
We can then define h, hc and G using these definitions for st and m_P:
hc = s_t * m_P
h = (s_t * m_P)/c
G = s_t / m_P
And then we can expand this out to actual physical values.
# s_t/c^3 = 1.35138507828e-43 # has units of s
# s_t = space_time_ratio* c**3
# m/m_P = 1.35138507828e-43
# m_P = m / space_time_ratio # dimensionless
# G = s_t/m_P
# G = (space_time_ratio* c**3) / (m / space_time_ratio)
# hc = s_t * m_P
# hc = (space_time_ratio* c**3) * (m / space_time_ratio)
# hc = c** 3 * m
p = m * c
h = m * c**2
hc = m * c**3
G = (space_time_ratio**2 * c**3) / m
And now we are defining the properties of the photon directly from the mass and the speed of light. Just like we do at every unit definition but this becomes very clear at c=1.
When we look at the math of how h is defined we can see that it is literally this simple and easy. h is just the mass of the photon times c^2. That is it. h just encodes the mass of the photon and a c squared. It was hidden in plain sight the entire time.
The space time ratio came out of the fact that the relationship between
m/m_P = 1.35138507828e-43
and
s_t/c^3 = 1.35138507828e-43
Are the same ratio and this ratio is invariant across all unit definitions. This ratio direct ties h and G together Through the common factors of hc and G: m_P and s_t. This seems significant, the same invariant ratio between two masses and the same exact ratio between two volumes of space time. But this results in the following definitions for hc and G
hc = m * c**3
Lets work backwards from these formulas to show exactly what is happening.
m_P = (hc/G)**2
m_P = ((m * c**3)/((space_time_ratio**2 * c**3) / m))**2
m_P = ((m**2)/space_time_ratio**2)**2
m_P = m/space_time_ratio
m/m_P = space_time_ratio
m/m_P = 1.35138507828e-43
s_t = (hcG)**2
s_t = ((m * c**3)*((space_time_ratio**2 * c**3) / m))**2
s_t = (( c**6)*((space_time_ratio**2)**2
s_t = c**3*space_time_ratio
s_t/c^3 = space_time_ratio s
As you can see, we are getting the right definitions from these formulas.
6. Implications of the Framework
This leads to several important realizations:
- h/c² represents the mass of a 1 Hz photon
- h scales as c² from mass in any definition of mass and c
- h converges to mass when c = 1
- The equation hf = mc² is an identity.
- Redefinition of Fundamental Units: By recognizing photon mass as fundamental, physical constants like ℎ and G can be reinterpreted as derived quantities, simplifying their definitions.
- Unification of Measurements: With photon mass and c, one can calculate ℎ directly, bypassing the need for independent definitions.
- Causal Clarity: Starting with mass aligns physical relationships with causality, clarifying the emergence of energy, momentum, and other properties.
7. Applications and Predictions This framework offers a basis for:
Simplifying unit systems by prioritizing mass and the unit definition for c.
Exploring deeper unifications in physics, particularly in quantum gravity.
Revisiting interpretations of Planck units and their connections to spacetime geometry.
8. Conclusion This paper proposes photon mass as the fundamental property underpinning physical constants and derived quantities. The convergence of energy, momentum, ℎ, and hc to mass at c = 1 highlights mass’s primacy. By reinterpreting ℎ and other constants as emergent from mass and c, this framework provides a unified, causally grounded perspective on fundamental physics.
References
Einstein, A. (1905). "On the Electrodynamics of Moving Bodies." Annalen der Physik.
Planck, M. (1901). "On the Law of Distribution of Energy in the Normal Spectrum." Annalen der Physik.
Dirac, P. A. M. (1938). "The Classical Theory of Radiating Electrons." Proceedings of the Royal Society A.
Appendix A: The code
# Set desired precision
getcontext().prec = 100
c = Decimal(299792458.0) # m/s
h = Decimal(6.62607015e-34) # m^2 kg / s
G = Decimal(6.67430e-11) # m^3 kg / s^2
half = Decimal (1/2)
m = h/c**2 # kg the mass of a photon
m_P = (h*c/G)**(half) # kg the original Planck mass unit using h not hbar
space_time_ratio = m/m_P # dimensionless,
# m_P = m / space_time_ratio #kg this is the new definition for m_P in the framework
s_t = (h*c*G)**(half) # m^3/s^2 the s_t is short for space time factor
space_time_ratio = s_t/c**3 # this version has units of second, this cancels an s from denom in next line
# s_t = space_time_ratio* c**3 # m^3/s^2 - This is the new definition for s_t in the framework
def calculate_and_print_results_new(value_pairs, c_old, m_old):
for pair in value_pairs:
m_scaling, l_scaling = map(Decimal, pair) # Attempt to convert values to decimals
m = m_old * m_scaling
c = c_old * l_scaling # m/s
# s_t/c^3 = 1.35138507828e-43 # has units of s
# s_t = space_time_ratio* c**3
# m/m_P = 1.35138507828e-43
# m_P = m / space_time_ratio # dimensionless
# G = s_t/m_P
# G = (space_time_ratio* c**3) / (m / space_time_ratio)
G = (space_time_ratio**2 * c**3) / m
# hc = s_t * m_P
# hc = (space_time_ratio* c**3) * (m / space_time_ratio)
# hc = c** 3 * m
p = m * c
h = m * c**2
hc = m * c**3
print (f" {1/l_scaling:<10.5e} {m_scaling:<10.5e} {hc:<13.8e} {h:<13.8e} {p:<13.8} {m:<13.8} {G:<13.8}")
x = Decimal(1)
scaling_pairs = [
(x, 100000),
(x, 10000),
(x, 1000),
(x, 100),
(x, 10),
(x, 1),
(x, .1),
(x, .01),
(x, .001),
(x, .0001),
(x, .00001),
(x, 100/c),
(x, 10/c),
(x, 2/c),
(x, 1/c),
]
print()
print()
print("\n ---- Exploring h, hc and G ------- ")
print()
print(" s_length s_mass hc h p m G")
calculate_and_print_results_new(scaling_pairs, c, m)
print()
print()
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