A new paradox and the reconciliation of Lorentz and Galilean transformations

One of the most debated problems in the foundations of the special relativity theory is the role of conventionality. A common belief is that the Lorentz transformation is correct but the Galilean transformation is wrong (only approximately correct in low speed limit). It is another common belief that the Galilean transformation is incompatible with Maxwell equations. However, the “principle of general covariance” in general relativity makes any spacetime coordinate transformation equally valid. This includes the Galilean transformation as well. This renders a new paradox. This new paradox is resolved with the argument that the Galilean transformation is equivalent to the Lorentz transformation. The resolution of this new paradox also provides the most straightforward resolution of an older paradox which is due to Selleri in (Found Phys Lett 10:73–83, 1997). I also present a consistent electrodynamics formulation including Maxwell equations and electromagnetic wave equations under the Galilean transformation, in the exact form for any high speed, rather than in low speed approximation. Electrodynamics in rotating reference frames is rarely addressed in textbooks. The presented formulation of electrodynamics under the Galilean transformation even works well in rotating frames if we replace the constant velocity $$\mathbf {v}$$ v with $$\mathbf {v}=\varvec{\omega }\times \mathbf {r}$$ v = ω × r . This provides a practical tool for applications of electrodynamics in rotating frames. When electrodynamics is concerned, between two inertial reference frames, both Galilean and Lorentz transformations are equally valid, but the Lorentz transformation is more convenient. In rotating frames, although the Galilean electrodynamics does not seem convenient, it could be the most convenient formulation compared with other transformations, due to the intrinsic complex nature of the problem.

Asymmetry theory derived from the principle of constant light speed

“The principle of the constancy of the velocity of light” was well established, while the further assumption that the light velocity is independent of the motion of the observer was never directly proven by any experiment. Based solely on this principle without any unproven assumptions, a comprehensive theoretic framework of the electrodynamics of moving bodies, named “Asymmetry Theory”, is derived purely through strict mathematics. A formula of the light velocity was mathematically derived, which is proven by the Sagnac effect and provides mathematical explanations for one-way light speed measurement, stellar aberration, and the M-M experiment. Other mathematically derived results include:1. A formula for observed “time dilation”, which resolves the “twin paradox”.2. Doppler Effect is simply a phenomenon of observed “time dilation” and one general formula covers traditional and transverse Doppler Effects, cosmological redshift, and time-varying velocities.3. Lorentz force law is invariant under Galilean transformation, with the correct definition of velocity following Barnett’s experiment explanation.4. A generalized form of Maxwell wave equations derived from the original equations, which is covariant under Galilean transformation. 5. The electrodynamics including particle acceleration and Mass-Energy relationship. Asymmetry Theory is comprehensive, self-consistent and in harmony with all existing experiments. It provides straightforward and mathematical explanations of key phenomenon without any paradox. Furthermore, Maxwell’s equations provide it the theoretic base and proof. Based on its predictions, two experiment designs are proposed for further conclusive confirmation.

Generalization of Galilean Transformation

In the article, a generalized Galilean transformation was derived. Obtained transformation is the basis for development of new physical theory, which was called the Special Theory of Ether. The generalized Galilean transformation can be expressed by relative speeds (26)-(27) or by the parameter delta(v) (37)-(38). Based on conclusions of the Michelson-Morley’s and Kennedy-Thorndike’s experiments, the parameter delta(v) was determined. This allows the transformation to take a special form (81)-(82), which is consistent with experiments in which velocity of light is measured. On the basis of obtained transformation, the formulas for summing speed and relative speed were also determined. The entire article includes only original research conducted by its author.

Gluing Solutions

This chapter deals with the gluing of solutions and the relevant theorem (Theorem 12.1), which states the condition for a Hölder continuous solution to exist. By taking a Galilean transformation if necessary, the solution can be assumed to have zero total momentum. The cut off velocity and pressure form a smooth solution to the Euler-Reynolds equations with compact support when coupled to a smooth stress tensor. The proof of Theorem (12.1) proceeds by iterating Lemma (10.1) just as in the proof of Theorem (10.1). Applying another Galilean transformation to return to the original frame of reference, the theorem is obtained.