# 13. Two Observers, One Transport Chapter 7 established that source-free energy flow satisfies the wave equation $$ \partial_t^2 \mathbf{F} - c^2 \nabla^2 \mathbf{F} = 0, \qquad \nabla\cdot\mathbf{F} = 0. $$ The same transport process can be described by different observers in relative motion. This chapter asks: what change-of-description is consistent with that same wave equation? The answer forces a particular kinematics — not as a postulate, but as an algebraic consequence of preserving the transport law. ## Isolated observers Define an **isolated observer** as one describing a region in which the net flux of $\mathbf{F}/c^2$ through any large closed surface vanishes. This is the flow-language statement that no external transport organizes the region. Two isolated observers in relative uniform motion, both describing the same transport process, must be able to write the same wave operator with the same constant $c$. This is the only requirement imposed. It is not a statement about geometry or space. It is the statement that two descriptions of the same transport process agree on the transport law. ## The transport bound For propagating solutions of the wave equation — organized wavefronts moving at rate $c$ — energy density and energy flux satisfy $$ |\mathbf{F}| = c\,u. $$ For general configurations (superpositions of modes propagating in different directions), the net flux is bounded: $$ |\mathbf{F}| \leq c\,u. $$ Define the operational transport rate by $$ \mathbf{v} := \frac{\mathbf{F}}{u} \qquad (u > 0). $$ Then $|\mathbf{v}| \leq c$. This bound is not postulated. It is an identity consequence of the wave-equation structure established in Chapter 7. ## Galilean addition violates the bound Galilean composition assigns $$ u \oplus_G v = u + v. $$ For any $0 < u < c$ and $0 < v < c$ with $u + v > c$ — always achievable — Galilean composition gives $u \oplus_G v > c$, violating the bound above. Therefore: **Galilean addition is incompatible with source-free Maxwell transport.** This is a theorem. It follows from the existence of the bound and the requirement that a change-of-description preserve it. We now derive the unique linear re-description that does preserve that same transport law. ## The wave operator forces the unique re-description Let two isolated observers be in relative uniform translation at rate $v$ along the $x$-axis. By homogeneity and straight-line preservation, the change-of-description is linear. By transverse symmetry, $y' = y$, $z' = z$. Write the most general linear mixing: $$ x' = a x + b t, \qquad t' = d x + e t. $$ Derivatives transform by the chain rule: $$ \partial_x = a\,\partial_{x'} + d\,\partial_{t'}, \qquad \partial_t = b\,\partial_{x'} + e\,\partial_{t'}. $$ Impose wave-operator invariance in 1+1 dimensions: $$ \partial_t^2 - c^2 \partial_x^2 = \lambda\!\left(\partial_{t'}^2 - c^2 \partial_{x'}^2\right), \qquad \lambda \neq 0. $$ Expanding the left side and collecting by differential operator: - Cross-term must vanish: $be = c^2 a d$. - Coefficient of $\partial_{t'}^2$: $e^2 - c^2 d^2 = \lambda$. - Coefficient of $\partial_{x'}^2$: $b^2 - c^2 a^2 = -\lambda c^2$. The primed origin $x' = 0$ satisfies $ax + bt = 0$, so the relative rate is $v = -b/a$, giving $b = -av$. From the cross-term condition: $d = -ve/c^2$. Substituting into the coefficient equations: $$ e^2\!\left(1 - \frac{v^2}{c^2}\right) = \lambda, \qquad a^2\!\left(1 - \frac{v^2}{c^2}\right) = \lambda. $$ So $a^2 = e^2$. Choosing the orientation-preserving branch and $\lambda = 1$ so that the inverse transformation takes the same form: $$ a = e = \gamma := \frac{1}{\sqrt{1 - v^2/c^2}}, \qquad b = -\gamma v, \qquad d = -\frac{\gamma v}{c^2}. $$ The unique linear change-of-description consistent with Maxwell transport is therefore $$ x' = \gamma(x - vt), \qquad t' = \gamma\!\left(t - \frac{v}{c^2}\,x\right), \qquad y' = y, \quad z' = z. $$ This is not assumed. It is the only linear map that preserves the wave operator. Note that $\gamma$ requires $|v| < c$: a relative translation rate at or above $c$ would prevent any wavefront emitted by one observer from reaching the other — no closed measurement is possible. ## Composed transport rate Let a transport feature move at rate $u = dx/dt$ in the first description. Differentiating the transformation gives $$ u' = \frac{dx'}{dt'} = \frac{u - v}{1 - \dfrac{uv}{c^2}}. $$ This is the unique composition law consistent with Maxwell transport. It is not postulated; it is a corollary of the operator invariance above. In particular, if $u = c$: $$ u' = \frac{c - v}{1 - v/c} = c. $$ The transport bound is absolute: every isolated observer assigns the same rate $c$ to a propagating wavefront. No composition of rates below $c$ reaches or exceeds $c$. ## Michelson–Morley The 1887 experiment compared round-trip travel times along two perpendicular arms of equal rest length $L$. The classic analysis assumed Galilean composition: outbound rate $c - v$ and return rate $c + v$ along the arm aligned with the laboratory's motion, producing unequal arm times and a predicted fringe shift. That step is not available here. The composition law derived above gives the transport rate as $c$ in all directions in the apparatus description — the one in which the wave operator holds with that same constant. Both arms give $$ T = \frac{2L}{c}, \qquad \Delta T := T_\parallel - T_\perp = 0. $$ The null result is not a surprise requiring additional hypotheses. It is the only answer consistent with Maxwell transport. The $c \pm v$ argument inserts Galilean addition at exactly one step; that step contradicts the operator invariance derived in this chapter. The null result closes the argument.