In Part 1,2,3 of this series, we have presented our critical mathematical review of the foundations of SR.

• Part 1:

  • Difficulties and Apparent Incompatibilities in the assumptions (A12) of SR.
  • Einstein proposes Lorentz formulae in attempt to resolve these incompatibilities, requiring two claims.
  • We find the law of propagation of photons particle is underdetermined by (A12) contrary to Riemannian intuition.
  • The geodesic variational equation is trivial on the null cone.
  • Therefore while the speed of light is $c$, the velocity of light is completely unconstrained on the null cone.

• Part 2:

  • We examine Einstein's proof of the compatibility of (A12) and his appeal to spherical light waves.
  • We claim to identify two critical errors in Einstein's classical argument: misidentification of the equation of a sphere, and the erroneous assumption that quadratic summands are numerically Lorentz invariant.
  • We illustrated with trivial computations on ${\bf{R}}^{1,1}$ and the large degrees of freedom of a photon even constrained to a line with respect to Lorentz transformations.

• Part 3:

  • We examined the homogeneous wave equation (HWE) and its properties uner Lorentz transformations.
  • d'Alembert operator is essentially the unique Lorentz invariant second order linear operatort.
  • Radius is not a Lorentz invariant variable.
  • There does not exist a Lorentz invariant set of so-called "radial solutions" to HWE.

As we have discussed in previous Parts, our objections apply simultaneously to the wave and particle models of light. These objections will be elaborated below.

Objections and Responses

Given the critical nature of this series, we here respond to some potential objections.

1.

One might object that all our arguments reduce to the observation that spheres in the frame $K$ are transformed to ellipsoids in $K'$ as is well-known.

But we remind the reader that Lorentz contraction is assumed to affect material objects, even independantly of the nature of the material. So material spheres in $K$ become material ellipsoids in $K'$, where the eccentricity of the $K'$-ellipsoid is nontrivial and independant of the material nature of the sphere. However we respond that light spheres are immaterial and not themselves subject to Lorentz contraction. And indeed if light spheres were subject to the same contraction effects as material spheres, then (A2) would definitely be false.

2.

Critics may object that (A12) only requires the consistent measurement of $c$ in arbitrary reference frames $K, K'$. This would replace the law (A2) with some rule of thumb for measurements.

But this immediately leads to a well-known experimental difficulty at the core of special relativity, namely the impossibility of measuring the one way speed of light. Indeed space and time measurements are always dependant on material objects and often non local, having sources and receivers separated by large distances. The impossibility of synchronizing non local clocks leads to the impossibility of measuring the one-way velocity of light. That all measurements of $c$ only succeed in measuring the two way or "round trip" velocities of light where source and receiver coincide is discussed in [Zhang 1997] and [Perez 2011]. See also Veritasium's video "Why the Speed of Light Can't be Measured".

Moreover in studying the two-way velocity of light, one needs further postulate that the velocity $c$ is constant (uniform) throughout its two-way journey, as Einstein argued. But this assumption is admittedly arbitrary and unverifiable.

3.

Wolfgang Rindler's very interesting textbook attempts "in spite of its historical and heuristic importance, [...] to de-emphasize the logical role of the law of light propagation [(A2)] as a pillar of special relativity."

Rindler claims that "a second axiom [(A2)] is needed only[!] to determine the value of a constant $c$ of the dimensions of a velocity that occurs naturally in the theory. But this could come from any number of branches of physics -- we need only think of the energy formula $E=mc^2$, or de Broglie's velocity relation $u v =c^2$."

Rindler's objection is interesting, and our response is simply that the above quoted formulas are equivalent to (A2), and not independant in any logical or physical sense.

The constant $c$ is central to modern physics. But what is the definition of this constant? Clearly Einstein defines $c$ as the speed of light ("celeritas", or swiftness) en vacuo. But is Einstein silently introducing a new definition of $c$ in his theory? What is the origin of defining $c$ in vacuum?

Let us review history. The constant $c$ was first formulated and estimated by Wilhelm Weber circa 1846, and even before J.C. Maxwell's famous treatise. Weber further studied $c$ with G. Kirchoff in the telegraphy equations. Here $c$ is the speed of propagation of electrical signals in long thin wires of arbitrarily small resistance. But the Weber-Kirchoff definition of $c$ is not equivalent to the $c$ of Einstein's special relativity. Einstein defines $c$ as speed in vacuum, and Weber-Kirchoff define $c$ as speed of signal propagation in a material conductive wire!

There does not appear any independant relation involving $c$ en vacuo apart from Einstein's (A2). Even the incredible Weber-Kohlrausch formula expressing $c$ as ratio of electric and magnetic dielectric constants presumes a material medium, i.e. the ratio is undefined in vacuum. Thus all formulas involving $c$ (where $c$ is Einstein's speed of light in vacuum) are based essentially on some form of (A2), and the logical pillar remains unmoved.

4.

A fourth objection might criticize our argument for not properly accounting for the so-called wave-particle duality of light, e.g. Bohr complementarity.

Our presentation has addressed both corpuscular and undulatory models, showing that (A12) is underdetermined in both cases. The incompatibility of (A12) with both the wave and particle model has been highlighted by A.K.T. Assis [Section 7.2.4, pp.133] from his "Relational Mechanics":

• "we can only conclude that for Einstein the velocity of light is constant not only whatever the state of motion of the emitting body [source], but also whatever the state of motion of the receiving body (detector) and of the observer."

For waves in physical medium, the velocity of emission is independant of the velocity of the source, since waves are transmitted by the medium and their velocities a property of the medium. Furthermore for both particles and waves, it is known that velocity is dependant on the velocity of the receiver.

According to (A2), light is postulated to exhibit properties unlike both waves and particles. Thus we argue that (A2) contradicts the supposed complementarity and wave-particle duality, i.e. (A2) requires light to behave contrary to both the wave and particle interpretations.

5.

[Insert discussion on GPS] Common objection. Details needed.

Sansbury-Fizeau Experiment: Is Light Even Something That Travels At All?

We have discussed this in previous posts at JHM Labs. The purpose of this discussion is to positively demonstrate that there are alternatives beyond SR. Thus we sharply criticize SR, but we also provide a substitute and a new path forward.

From our perspective, there is an extremely important experiment of Fizeau-Sansbury which needs be performed, and which results might be quite shocking to the standard physicist worldview.

Stay Tuned!

-JHM.