# Maxwellian Energy Transport and the Michelson–Morley Null Result ## 0. Objective We construct, from first principles, a logically complete derivation of: 1. The invariant propagation rate $c$ from source-free Maxwell theory. 2. The unique inertial coordinate transformations preserving Maxwellian propagation. 3. The induced hyperbolic velocity-composition law. 4. The correct computation of Michelson–Morley interferometer timing. 5. The null result as a direct consequence of Maxwell-consistent transport. No geometric assumptions (length contraction, spacetime curvature, etc.) are introduced. All results follow from the structure of Maxwell’s equations. --- # Part I — Maxwell Theory Fixes a Universal Propagation Rate ## 1. Source-Free Maxwell Equations In vacuum, $$ \nabla \cdot \mathbf{E} = 0, \qquad \nabla \cdot \mathbf{B} = 0, $$ $$ \nabla \times \mathbf{E} = -\partial_t \mathbf{B}, \qquad \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \, \partial_t \mathbf{E}. $$ --- ## 2. Derivation of the Wave Equation Take the curl of Faraday’s law: $$ \nabla \times (\nabla \times \mathbf{E}) = -\partial_t (\nabla \times \mathbf{B}). $$ Using $$ \nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}, $$ and $\nabla \cdot \mathbf{E} = 0$: $$ -\nabla^2 \mathbf{E} = -\mu_0 \epsilon_0 \, \partial_t^2 \mathbf{E}. $$ Thus: $$ \partial_t^2 \mathbf{E} = c^2 \nabla^2 \mathbf{E}, \qquad c := \frac{1}{\sqrt{\mu_0 \epsilon_0}}. $$ Similarly, $$ \partial_t^2 \mathbf{B} = c^2 \nabla^2 \mathbf{B}. $$ --- ## 3. The Maxwell Wave Operator Define $$ \square_c := \partial_t^2 - c^2 \nabla^2. $$ Each field component satisfies $$ \square_c \Phi = 0. $$ Therefore: - Maxwell theory fixes a universal transport rate $c$. - Electromagnetic disturbances propagate along the characteristic structure defined by $\square_c$. - $c$ is not an adjustable parameter; it is fixed by field dynamics. --- # Part II — Inertial Re-Description and Operator Invariance ## 4. Minimal Invariance Requirement Let $(t,\mathbf{x})$ and $(t',\mathbf{x}')$ be inertial coordinate systems. Maxwell invariance requires: $$ \square_c \Phi = 0 \quad \Longleftrightarrow \quad \square_c' \Phi = 0. $$ That is, $$ \partial_t^2 - c^2 \nabla^2 \quad \text{and} \quad \partial_{t'}^2 - c^2 \nabla'^2 $$ must represent the same operator up to scale. This ensures that the propagation rate $c$ is not coordinate-dependent. --- ## 5. Linear Structure of Inertial Transformations Homogeneity and inertiality imply linear transformations. Restrict first to motion along $x$: $$ x' = a x + b t, \qquad t' = d x + e t. $$ Wave-operator invariance forces: $$ x' = \gamma(x - v t), $$ $$ t' = \gamma\left(t - \frac{v}{c^2} x \right), $$ where $$ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}. $$ This is uniquely determined by preserving $\square_c$. No geometric assumptions were introduced. --- # Part III — Velocity Composition from Maxwell Invariance ## 6. 1D Velocity Composition Let $u = dx/dt$. Differentiate: $$ dx' = \gamma(dx - v dt), $$ $$ dt' = \gamma\left(dt - \frac{v}{c^2} dx\right). $$ Thus: $$ u' = \frac{dx'}{dt'} = \frac{u - v}{1 - \frac{uv}{c^2}}. $$ This is hyperbolic composition. --- ## 7. 3D Velocity Composition Decompose: $$ \mathbf{u} = \mathbf{u}_\parallel + \mathbf{u}_\perp, $$ with respect to $\mathbf{v}$. Then: $$ \mathbf{u}' = \frac{\mathbf{u}_\perp/\gamma + \mathbf{u}_\parallel - \mathbf{v}} {1 - \frac{\mathbf{v} \cdot \mathbf{u}}{c^2}}. $$ --- ## 8. Critical Identity If $|\mathbf{u}| = c$, then: $$ |\mathbf{u}'| = c. $$ Therefore: Electromagnetic transport speed is invariant under inertial re-description. This directly contradicts Galilean addition: $$ u_{\text{Galilean}} = u \pm v. $$ Galilean addition does not preserve $c$. Maxwell transport forbids it. --- # Part IV — The Logical Error in the Textbook Michelson–Morley Derivation ## 9. The Classical Assumption Textbook reasoning assumes: $$ c_{\text{effective}} = c \pm v. $$ This presumes: - Light behaves as a projectile. - Velocities add algebraically. - The apparatus moves through a medium. But Maxwell theory already forbids additive composition. Thus the $c \pm v$ assumption contradicts the field equations. --- # Part V — Maxwell-Consistent Interferometer Timing Let: - $L$ be arm length in the laboratory, - $v$ lab speed relative to hypothetical medium, - $c$ Maxwell transport rate. The interferometer measures: $$ \Delta T = T_\parallel - T_\perp. $$ --- ## 10. Transport Principle Electromagnetic disturbances propagate at invariant rate $c$. Therefore: - The signal transport rate is $c$ in all inertial descriptions. - Timing must be computed from invariant propagation, not $c \pm v$. --- ## 11. Parallel Arm Distance to mirror (lab frame): $$ L. $$ Transport rate: $$ c. $$ Outbound time: $$ T_{\text{out}} = \frac{L}{c}. $$ Return time: $$ T_{\text{back}} = \frac{L}{c}. $$ Thus: $$ T_\parallel = \frac{2L}{c}. $$ --- ## 12. Perpendicular Arm Same reasoning: $$ T_\perp = \frac{2L}{c}. $$ --- # Part VI — Null Result Therefore: $$ \Delta T = T_\parallel - T_\perp = 0. $$ The Michelson–Morley null result follows directly from: - Maxwell’s wave equation, - invariance of the wave operator, - hyperbolic velocity composition, - invariance of $c$. No length contraction was assumed. No geometric deformation was required. --- # Part VII — Logical Structure of the Argument 1. Maxwell theory fixes a finite propagation rate $c$. 2. Inertial re-description must preserve the Maxwell operator. 3. This forces Lorentz-type transformations. 4. Differentiation yields hyperbolic velocity composition. 5. Hyperbolic composition preserves $c$. 6. Therefore light does not obey $c \pm v$. 7. The Michelson–Morley $c \pm v$ calculation is inconsistent with Maxwell. 8. Correct timing yields equal round-trip times. 9. The null result is expected. --- # Final Statement The Michelson–Morley experiment did not force length contraction. It tested whether electromagnetic propagation obeys Galilean addition. Maxwell theory predicts hyperbolic composition and invariant transport rate $c$. When timing is computed consistently with Maxwell transport, $$ T_\parallel = T_\perp = \frac{2L}{c}, $$ and the null result follows as a direct mathematical consequence of the field equations.