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Lorentz or Voigt? Are Their Transforms Really Equivalent?

Updated: Jun 20, 2019

Woldemar Voigt formulated, a transform from one inertial frame to another, almost two decades prior to Hendrik Lorentz, which the later cedes priority to in his 1909 book, "Theory of Electrons". In the Wikipedia article on Voigt, the claim is that although both transforms predict a null result in the Michelson-Morley experiment, indicating a constant and isotropic speed for the velocity of light, the time dilation in Voigt's case is G times greater than in the Lorentz case, where G=(1-v^2/c^2)^(1/2), and v is the velocity of the ether. This is definitely inaccurate.


We will show that whereas Lorentz's transform leaves the hyperbolic element invariant, Voigt's transform yields the hyperbolic velocity space metric---but for the inverse velocity as calculated from Voigt's metric. The Voigt transform is:


dx'=dx-vdt

dy'=Gdy

dz'=Gdz

dt'=dt-vdx/c^2.


Squaring all and adding the first three, while subtracting c^2 times the last give:


ds^2=dx'^2+dy'^2+dz'^2-c^2 dt'^2=dx^2+G^2(dy^2+dz^2)-c^2 G^2 dt^2.


In contrast, the Lorentz transform,


dx'=(dx-vdt)/G

dy'=dy

dz'=dz

dt'=(dt-vdx/c^2)/G.



would leave the hyperbolic line element invariant,


dx'^2+dy^2+dz^2-c^2dt'^2=dx^2+dy^2+dz^2-c^2dt^2,


making it informationless. The composition law for the difference in two velocities is


u-v={(u-v).(u-v)-(uXv)^2/c^2}^{1/2}/G^2.



Not so with the Voigt transform.


Now transforming the Voigt metric to polar coordinates, the line element becomes:


ds^2={1-(v^2/c^2)sin^2A} dr^2-(c Gdt)^2.


where A is angle between r and dr. Calling G dr/dt=V, we solve for it along the path of a light ray, ds=0, giving


1/V^2=[1-v^2/c^2 sin^2A]/(c G^2)^2.


According to Fock in his book, The Theory of Space, Time and Gravitation, the above is the "square of an infinitesimal relative velocity," or "the square element of length in a certain velocity space." The velocity V is the inverse of the velocity of Lobachevsky space. And if the later satisfies the arithmetic mean, V will satisfy the harmonic mean.


If V_v is Voigt's speed of light, and V_L that of Lorentz, the two satisfy


V_v .V_L = c^2/G > c^2,


so that one velocity must exceed the speed of light.


The velocity of Lobachevsky space is the inverse velocity of the propagation of light in that space, as determined by the Voigt metric. Whereas v is the velocity with respect to the field which creates it, V is the velocity of light in that medium. When v=0, V reduces to the speed of light, c, in vacuum. As Petr Beckmann has pointed out, v cannot be any relative speed with respect to the observer since the velocity is the result of the field that creates it. Otherwise, there would be no unique field, and there would be a multitude of velocities for each and every field. This signifies the demise of observational physics: the observer and the relative velocity he experiences is completely superfluous to the physics at hand.





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