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Ives' Comparison of Voigt and Lorentz Transforms

In my last blog, I stated that light rays have velocities that 1/G times the inverse of the Lobachevsky velocity metrical distance, where G is the Lorentz factor. This implies that the time slippage term is neglected. Since the two metrics must contain the same information, I will now show that this condition is implied in the Wiki article statement that Lorentz time T_L is related to Voigt time, T_V by T_V=G^2T_0=GT_L.


The usual derivation of T_L is to consider a clock placed at x=x_0 which registers two times t_1 and t_2. The time components of the Lorentz transformation are:

t_1'=G(t_1-ux_1/c^2)

t_2'=G(t_2-ux_2/c^2).

Now since x_1=x_2=x_0, subtracting the second from the first gives:

t_1'-t_2'=G(t_1-t_2)

or T_L=GT_0, as above.

However, if we look at the space component

x'=G(x-ut),

things are not a crystal-clear as they may seem. Dividing the above by t'/t=G(1-u/c) since x/t=c, I get

x'/t' G(1-u/c)=G(c-u).

This implies that x'/t'=c also. Then since x and t are not independent variables--according to Einstein--one cannot impose a condition on x without imposing a similar condition on t. The above demonstration of time dilation can be found in any book on relativity, cf. French, "Special Relativity" p. 100.


Consequently, Lorentz time will appear if and only if the time slippage term is neglected in the time component of the Lorentz transformation. According to Ashby, this term is always negligible when applied to GPS.


Since the x' and t' equations do not contain G, it appears that the above won't work for the Voigt transformation. However, let us begin with the pair of equations

x=x'+ut'

t=t'+ux'/c^2.

Eliminate x' in favor of x in the second, and rearrange to obtain

t'=G^2(t-ux/c^2).

Now, neglecting the time-slippage term gives the Voigt time T_V=G^2T_0. The important point is that the time-slippage term must be neglected in both cases.


In fact, Lorentz did not specify to which power of G his equations would take. He wrote them as

x'=L G(x-ut)

y'=L y

z'=L x

t'=L G(t-ux/c^2),

with the caveat that L is a numerical coefficient that can only depend on u and which must reduce to 0 when u=0. Lorentz is associated with 1, while L=1/G for Voigt.


Ives points out that for L=G, the clock period doesn't change since

t'=t-ux'/c^2,

which would "have proved the non-relativity of time." But, this again requires the neglect of the time-slippage term. He concludes that "the shift of the center of gravity of Doppler lines for approaching and receding canal rays...decide unequivocally for L=1."


Ives was also the first to relinquish the constancy of the speed of light. If c_0 and c_b are the "out" and "back" velocities of light, the total time for light to travel a distance d and back is

t=t_0+t_b=d{1/c_0+1/c_b}=2d A/G^2,

where A=1/2(c_0+c_b) the arithmetic mean, and G={c_0c_b}^(1/2) is the geometric mean (not to be confused with the Lorentz factor). Ives acquiesces to the "experimental fact" that the out-and-back measurement would result in c, he writes the total time as t=2d/c. Thus, the universal constant c is identified as the harmonic mean. This leaves open the possibility of velocities greater than c.


Now consider a platform moving with velocity v with length L. The outward signal would travel a distance c_ot_o=L+vt_o in time t_o, and a distance c_bt_b=L-vt_b backward in time t_b. The total time would be:

t=t_o+t_b=L{1/(1-v/c_0)(1+v/c_b)}~(2L/c)/[1-2(v/c)(c_b-c_0)/c_0+c_b)]

retaining only terms of order v/c. The term in the denominator, according to Ives, would represent "contractions of length and clock rate on the 'moving' platform is c_b>c_o and expansions if c_b<c_o."


Since we are considering one-way propagation is it legal to neglect higher order terms in v/c. As c_b->c_0, it would reduce to the usual contraction factor (1-v^2/c^2), a second order term. The difference in propagation times would play the role of an index of refraction, as in the Fresnel modification to the propagation of light through a moving medium. It would be a linear effect, and not a quadratic one as in a two-way speed of light. Setting the square index of refraction, n^2=c_b/c_0, and considering n>>1, the above contraction becomes c-2f v, where f=(1-1/n^2), Fresnel coefficient.

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