An M. Rodriguez (an@preferredframe.com), Anes Palma
31 July 2025
Observers composed entirely of electromagnetic (EM) fields cannot determine the absolute speed of light \(c\). Clocks count field cycles and rulers register field nodes; both rescale with any local change in the vacuum parameters \(\varepsilon(x)\) and \(\mu(x)\). Every laboratory therefore self-calibrates to the value \(c_0 = 1/\sqrt{\mu_0\,\varepsilon_0}\) even if the underlying light speed \(c(x)\) varies from place to place. Using elementary Maxwell theory and a water-wave analogy, we show why present-day cavity experiments can constrain only differences \(\Delta c/c\), not the absolute magnitude of \(c\).
Because laboratory rulers and clocks are built from electromagnetic fields, Maxwell theory allows detection of only spatial or temporal variations in the speed of light—never its absolute value.
Maxwell electrodynamics; metrology; speed of light; Lorentz invariance; Fabry–Pérot cavity.
Michelson–Morley-type experiments and modern optical cavities show no sign of a preferred frame. While special relativity explains this by postulating a universal \(c\), classical Maxwell theory offers a simpler reading: when measuring tools are built from the very field whose speed is sought, only differences in that speed can ever be observed.
In vacuum the fields satisfy
\[
\nabla^{2}\mathbf{E} = \mu_0 \varepsilon_0 \,\partial_t^{2}\mathbf{E},
\]
which fixes the characteristic speed
\[
c_0 = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} .
\]
A Fabry–Pérot cavity of fixed length \(L\) enforces \(n\lambda_n = 2L\). For the fundamental mode (\(n = 1\)) the wavelength is \(\lambda_1 = 2L\). Using \(c = \lambda/T\), the same mode provides the period \(T_1 = \lambda_1/c\). If the local vacuum parameters shift to new values \((\mu,\varepsilon)\), both \(\lambda_1\) and \(T_1\) are multiplied by the same factor \(\sqrt{\mu\varepsilon}\); their ratio \(\lambda_1/T_1\) is unchanged, so any measurement of \(c\) still returns \(c_0\).
A fish tries to measure the speed \(v\) of surface waves. Its rulers (successive crests) and clocks (periods between crests) both stretch if the surface tension varies. Absolute speed is invisible; only relative differences appear. EM observers face the same limitation.
Most laboratory searches for anisotropy compare two orthogonal
high-\(Q\) Fabry–Pérot cavities on a
slowly rotating stage. For the \(m\)-th
longitudinal mode in arm \(i\)
\[
\nu_i \;=\; \frac{m\,c(x_i)}{2L_i},\qquad i = x,y ,
\]
so a directional dependence of \(c(x)\)
would modulate the beat note \(\nu_x-\nu_y\) once per turn. Experiments
find no such modulation down to \(|\Delta c|/c
\le 10^{-18}\). In our framework the setup samples the vacuum
scaling factor differentially: it measures \(c(x)\) relative to \(f(p_x)\) versus \(f(p_y)\) (see next section for \(f(p)\)). If \(f\) were uniform the two arms would differ
only by construction tolerances, and constant-velocity motion would be
completely invisible—precisely the conclusion of special relativity.
Orthogonal cavities are therefore sensitive only to a gradient \(f(p_x) - f(p_y)\); a uniform offset
cancels.
Detecting an absolute shift in \(f(p)\) instead requires a one-arm,
time-of-flight technique—e.g. sending a light pulse through a long fiber
or free-space delay line and comparing the round-trip time against an
independent clock. Such single-path measurements would respond directly
to the optical length
\[
\mathcal{L}(x) = \int n_{\text{eff}}(x)\, \mathrm dx \;=\; \int
f(p)\,\mathrm dx,
\]
and could, in principle, reveal a uniform change in \(f(p)\) rather than just a transverse
gradient.
Let a scalar background \(p(x)\)—a
proxy for local energy density—multiply both permittivity and
permeability by the same positive factor \(f(p)\):
\[
\varepsilon(x) = \varepsilon_0 f(p), \qquad
\mu(x) = \mu_0 f(p).
\]
The local wave speed is
\[
c(x) = \frac{1}{\sqrt{\mu(x)\varepsilon(x)}} = \frac{c_0}{f(p)} .
\]
The cavity’s fundamental frequency becomes
\[
\nu_{\text{ref}}(x) = \frac{c(x)}{2L} = \frac{c_0}{2L\,f(p)} .
\]
The associated wavelength is
\[
\lambda_{\text{ref}}(x) = \frac{c(x)}{\nu_{\text{ref}}(x)} = 2L ,
\]
independent of \(p(x)\). A rod made
from \(N\) such wavelengths has
length
\[
L_{\text{rod}} = N\lambda_{\text{ref}} = 2NL ,
\]
also independent of \(p(x)\).
The period of one oscillation is
\[
T_{\text{ref}}(x) = \frac{1}{\nu_{\text{ref}}(x)} =
\frac{2L\,f(p)}{c_0}.
\]
Combining ruler and clock gives
\[
\frac{L_{\text{rod}}}{T_{\text{ref}}(x)} = c_0 ,
\]
showing that absolute variations in \(c(x)\) remain hidden; only gradients in
\(f(p)\) could, in principle, be
detected.
General relativity expresses the same idea geometrically. A perturbed
metric
\[
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}(p)
\]
gives local null cones identical to those of special relativity, while
global geodesics reproduce light-bending through an effective refractive
index
\[
n_{\text{eff}}(\mathbf{x}) \simeq 1 + \frac{2\Phi(\mathbf{x})}{c_0^{2}}
.
\]
The optical-metric formalism is therefore algebraically equivalent to
Maxwell with variable \((\varepsilon,\mu)\).
Maxwell electrodynamics implies that laboratories built from EM fields cannot measure the absolute speed of light. Their rulers and clocks scale together with any local change in \(\varepsilon\) and \(\mu\), leaving only spatial or temporal differences observable. The water-wave analogy captures the point: one cannot determine the speed of water waves using only water.
The geometrical argument linking Maxwell theory to general relativity was developed entirely by Anes Palma.