Theoretical proof of the constancy of the speed of light in a vacuum

1. Problem Statement

The paper addresses the lack of a rigorous theoretical proof for the second postulate of Albert Einstein's Special Theory of Relativity (STR): the constancy of the speed of light ($c$) in all inertial frames. While experimental evidence supports this constancy to high precision, the author argues that current theoretical justifications (often relying on group theory and Lorentz transformations) are circular or insufficient. The core question is: Why is the speed of light constant, and is this constancy a fundamental requirement for the existence of quantum mechanics and the observed matter-antimatter asymmetry?

2. Methodology

The author employs a deductive approach based on invariant transformations and Lagrangian mechanics, deliberately relaxing the assumption that $c$ is a constant from the outset.

• Generalized Space-Time Transformations:
  The paper begins by constructing a transformation matrix $U$ between two reference frames moving at relative velocity $V$. Unlike standard STR, the proportionality coefficient between time and space ($x_0 = ct$) is allowed to vary with velocity, denoted as $c(v)$. The transformation matrix includes an odd coefficient $\delta(V)$, leading to a generalized velocity addition formula where $c$ and $c'$ (speeds in different frames) are not initially assumed equal.

• Lagrangian Formulation with Variable $c$:
  An invariant action function $dS = \alpha ds$ is introduced, where the Lagrangian $L$ depends on a velocity-dependent speed of light $c(|v|)$.
  $$L = \alpha c(|v|) \sqrt{1 - \left(\frac{v}{c(|v|)}\right)^2}$$
  From this, the author derives generalized expressions for momentum ($P$) and energy ($E$) that include derivatives of $c$ with respect to velocity ($dc/dv$).

• Four-Vector Invariance Analysis:
  To ensure physical consistency, the author seeks conditions under which the energy-momentum four-vector ($P^i$) remains invariant. By introducing a linear transformation parameter $\beta$, the author derives a relationship between energy and momentum that holds only if the speed of light follows a specific linear law:
  $$c(|v|) = \beta|v| + c_0$$
  Here, $c_0$ is the speed of light at zero velocity, and $\beta$ is a dimensionless parameter ($|\beta| \leq 1$) representing the deviation from constancy.

• Energy Minimization and Quantization:
  The paper analyzes the energy equation derived from the generalized momentum. It investigates the behavior of the system as the parameter $\beta$ approaches zero. Furthermore, the author attempts to quantize the resulting energy-momentum relation to determine if a stable quantum mechanical description (stationary wave functions) is possible for non-zero $\beta$.

3. Key Contributions

A. Derivation of the Speed of Light Constancy

The paper demonstrates that the speed of light is constant ($c = c_0$) only when the parameter $\beta$ approaches zero. This state corresponds to the minimum energy of a particle.

• If $\beta \neq 0$, the speed of light varies with velocity ($c = \beta|v| + c_0$).
• The condition for the existence of a stable, minimum energy state forces $\beta \to 0$, thereby proving that $c$ must be constant in any inertial frame where the system is in its ground state.

B. Emergence of Particles and Antiparticles

The analysis reveals that at the minimum energy limit ($\beta \to 0$), the energy equation yields two distinct, identical solutions:

1. Positive energy ($E > 0$)
2. Negative energy ($E < 0$)
   The author argues that these two roots correspond to the existence of particles and antiparticles. The transition from $\beta \neq 0$ to $\beta = 0$ is the mechanism that allows for the separation of these two states.

C. Necessity of Constant $c$ for Quantum Mechanics

A critical theoretical finding is that quantum mechanics (and quantum field theory) cannot be formulated if $\beta \neq 0$.

• When $\beta \neq 0$, the energy-momentum relation becomes asymmetric and introduces mixed derivative terms ($\frac{\partial^2}{\partial t \partial x}$) upon quantization.
• These mixed derivatives prevent the separation of time and spatial variables in the wave function, meaning no stationary solutions exist.
• Therefore, the constancy of the speed of light is a prerequisite for the existence of the quantum world as currently understood.

D. Explanation of Matter-Antimatter Asymmetry

The paper proposes a mechanism for the observed dominance of matter over antimatter based on fluctuations in the energy minimum.

• The relationship between energy and momentum is asymmetric for non-zero $\beta$.
• The author defines an asymmetry parameter $A_s = N_+ / N_-$ (ratio of positive to negative energy particles).
• Calculations suggest that if the system fluctuates away from the exact minimum energy ($\beta \neq 0$), the probability of generating positive energy particles (matter) is higher than negative energy particles (antimatter).
• Under specific probability distributions, the model predicts a matter-to-antimatter ratio of approximately 4.5:1, offering a theoretical basis for the baryon asymmetry of the universe.

4. Results

1. Proof of Constancy: The speed of light is proven to be constant ($c = c_0$) as a consequence of the system seeking the minimum energy state ($\beta \to 0$).
2. Generalized Transformation: A generalized transformation law is established where $c$ depends on velocity, reducing to the Lorentz transformation only when energy is minimized.
3. Quantum Stability: It is shown that non-constant light speeds ($\beta \neq 0$) render quantum mechanics impossible due to the lack of stationary wave solutions.
4. Asymmetry Quantification: The model provides a mathematical framework where small deviations from the minimum energy state naturally lead to a surplus of matter over antimatter.

5. Significance

• Foundational Physics: The paper attempts to move the constancy of the speed of light from an empirical postulate to a derived consequence of energy minimization and the requirements of quantum mechanics.
• Unification: It links the structure of space-time (Lorentz invariance), the nature of light, and the existence of quantum particles into a single logical framework.
• Cosmological Implication: It offers a novel theoretical explanation for the matter-antimatter asymmetry, suggesting that the universe's current state (dominated by matter) is a result of fluctuations away from the absolute minimum energy state.
• Theoretical Challenge: The work challenges the standard interpretation of Special Relativity by suggesting that the "constant $c$" is a specific solution (minimum energy) rather than an absolute axiom, and that the quantum world is inextricably linked to this specific condition.

Note: As this is a theoretical paper proposing a derivation of fundamental postulates, its claims would require rigorous peer review and experimental validation to be accepted by the broader physics community.