# A Maxwellian, Flat-Space Explanation of the Michelson–Morley Null Result ## 1. Statement of the problem The Michelson–Morley experiment compared round-trip travel times of light along two perpendicular arms of equal length while the apparatus rotated. Under Galilean velocity addition, if light propagated through a stationary medium and the laboratory moved through that medium with velocity \( V \), the two arms should exhibit different travel times depending on orientation. The experiment found no such difference. The usual interpretation invokes length contraction, spacetime geometry, or the absence of an ether. We show here that the null result follows directly from the internal structure of source-free Maxwell transport in flat space, without invoking geometric redefinitions of space itself. --- ## 2. Maxwell transport as intrinsic process In vacuum, Maxwell’s equations read $$ \partial_t \mathbf{B} = -\nabla \times \mathbf{E}, \qquad \partial_t \mathbf{E} = c^2 \nabla \times \mathbf{B}, $$ with constraints $$ \nabla \cdot \mathbf{E} = 0, \qquad \nabla \cdot \mathbf{B} = 0. $$ These equations define a dynamical system for divergence-free vector fields. They do not refer to observers, rods, clocks, or coordinate frames. They describe a physical transport process. Energy density and flux are $$ u = \frac{\epsilon_0}{2} \mathbf{E}^2 + \frac{1}{2\mu_0} \mathbf{B}^2, $$ $$ \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B}. $$ They satisfy the continuity equation $$ \partial_t u + \nabla \cdot \mathbf{S} = 0. $$ This is a description of energy transport through space. The process exists independently of its representation. --- ## 3. Intrinsic propagation rate From the algebra of the fields one obtains the inequality $$ |\mathbf{S}| \le c\, u. $$ Define the local transport velocity $$ \mathbf{v} = \frac{\mathbf{S}}{u}. $$ Then $$ |\mathbf{v}| \le c. $$ For radiation-like configurations, equality holds: $$ |\mathbf{v}| = c. $$ Thus Maxwell transport contains an intrinsic propagation rate \( c \) fixed by the internal rotational coupling of the fields. This rate is not imposed externally. It is not defined relative to a medium. It is a structural property of the curl dynamics. --- ## 4. Why Galilean addition fails Galilean kinematics assumes that if a laboratory moves with velocity \( V \), then signal velocities transform additively: $$ c \mapsto c \pm V. $$ This assumption treats propagation as translation of an object. Maxwell transport is not translation. It is rotation-coupled evolution of divergence-free structure. To impose Galilean addition, one would modify time evolution by introducing a convective term $$ \partial_t \rightarrow \partial_t + \mathbf{V} \cdot \nabla. $$ This alters the curl system itself. The transport law would no longer be identical in the moving laboratory. But the electromagnetic process observed in the laboratory is described by the same Maxwell equations. Therefore no Galilean drift term is present in the evolution. The null result follows immediately: If the transport law is unchanged, then the intrinsic propagation rate remains \( c \) in all uniformly moving laboratories. No anisotropy appears. --- ## 5. Round-trip transport time in flat space Consider an interferometer arm of length \( L \) at rest in the laboratory. The light pulse follows a path determined by the curl transport law. Because the intrinsic propagation rate relative to the apparatus is \( c \), the forward and return travel times are each \( L / c \). Total round-trip time: $$ T = \frac{2L}{c}. $$ Rotation of the apparatus changes neither \( L \) nor \( c \), since both are defined within the same transport law. Therefore $$ \Delta T = 0. $$ This is the Michelson–Morley null result. No appeal to curved space is required. No contraction hypothesis is required. --- ## 6. What the experiment actually constrains The experiment constrains the admissible form of transport laws. It rules out: - mechanical ether models with Galilean addition, - additive velocity composition for curl-based propagation, - external drift terms modifying Maxwell evolution. It supports: - local, divergence-free, curl-based transport, - intrinsic propagation rate determined by field structure, - flat spatial geometry with internal kinematic constraints. --- ## 7. Flat space and transport invariance Nothing in the Maxwell curl system requires curved space. The equations are formulated entirely in flat Euclidean three-space. The isotropy observed in Michelson–Morley is therefore a statement about the internal symmetry of the transport law, not about curvature of space itself. Energy transport happens independently of representation. Observers merely describe it. The null result expresses the fact that the Maxwell transport process is self-contained and does not admit additive drift without altering its defining equations. --- ## 8. Conclusion The Michelson–Morley null result is a direct consequence of source-free Maxwell transport in flat space. Energy transport governed by curl dynamics possesses an intrinsic propagation rate that cannot be modified by Galilean addition without changing the transport law itself. No curved geometry is required. No spacetime reinterpretation is required. The null result follows from the internal kinematics of divergence-free electromagnetic energy transport.