Relativity Calculator
"The most incomprehensible thing about the world is that it is comprehensible." - Albert Einstein ( 1879 - 1955 )
During most of pre - Galileo and Newton and for subsequent eras as well, it was supposed that in the interstitial spaces between objects of matter that there existed a "carrying medium" or aether for the transmission of light from source to reflecting object and thence to the human eye for perception. Two French physicists, Jean Bernard Léon Foucault ( 1819 -1868 ) and Armand Hippolyte Louis Fizeau ( 1819 - 1896 ), attempted the determination for the finite speed of light; Fizeau did so singly in 1849 and again in 1850 together with Foucault but thereafter independently sought the speed of light in his famous 1851 Fizeau Water Experiment whenever light was transmitted thru a high velocity flowing medium such as water. In essence, therefore, Fizeau attempted to confirm Augustin - Jean Fresnel ( 1788 - 1827 )'s "velocity drag coefficient"
for light transmitted thru high - velocity ( at least / approx. 30 m/sec ) flowing water. It should be thus noted that Augustin - Jean Fresnel, French mathematical theorist and experimenter in optical wave physics, is the original discoverer of the velocity drag coefficient.
In such a situation, Fizeau discovered that each different flowing liquid acting as a carrying medium or aether for light, in both positive and negative directions of velocity flow, exhibited different refraction indices in the speed of light as demonstrated by measuring the positions of interference fringes for light.
In the 1851 water experiment, Fizeau also discovered that simple Galilean addition of light velocity plus the velocity of the transmitting carrying medium did not fully suffice to explain the resulting interference light fringes. For any explanation of this phenomenon it would take the equations of special relativity ( Albert Einstein, 1905 ) for the addition of relativistic velocities in order to make apparent deeper theoretical understanding.
Fizeau did however successfully confirm that the speed of light, c, is attenuated by a refractive index, n∗, unique to each different carrying medium as well as effected by the velocity, either positive or negative, of the local transmitting medium. Fresnel's convection equation
which Fizeau's 1851 water experiment confirmed, indicates that for increasingly rarified medium such as a vacuum space, n approaches 1 and the observed speed of light, u, becomes c since velocity, v, simply disappears.
Interestingly Fizeau rather accurately determined the speed of light, c, as 3.13 x 108 m/sec and became the first to do so using an earth - based apparatus as opposed to Danish Olaf Roëmer ( 1644-1710 )'s astronomical determination in 1676.
Although no other deeper theoretical explanation existed for the results of the 1851 Fizeau experiment other than the Fresnel factor
representing an "aether drag coefficient" for transmitted light going thru a high velocity moving fluid, it still nevertheless remained a puzzle, especially after the null result of the famed Michelson - Morley experiment ( 1887 ) completely failed to discover an equivalent relationship to the Fizeau - Fresnel equation for light transmitted thru vacuum space.
On the other hand, the FitzGerald - Lorentz transformation equations theoretically resurrected the "existence" of the aether transmitting medium by postulating an "aether drag" in the very instrumentation of the Michelson - Morley experiment resulting in a longitudinal contraction of their equipment apparatus and thereby salvaging the concept of the "aether". In other words, the idea of the "aether" as a transmitting medium for vacuum light similar to the transmitting fluid ( water ) in the Fizeau experiment is thereby sustained, even though undetected, since there became a physical contraction in the length of the Michelson - Morley apparatus in the longitudinal direction of earth's motion in its orbit about the sun due to an "aether drag", thus mathematically shown by the FitzGerald - Lorentz transformation equations.
∗
n = 1.0 for vacuum space >> u = c, speed of light thru vacuum space
n = 1.0002926 for atmosphere at STP ( Standard Temperature and Pressure ) >> u = c/n = 0.9997c, speed of light thru stationary air
n = 1.333 for fresh water >> u = c/n = 0.75c, speed of light thru stationary water
n = 2.419 for diamond >> u = c/n = 0.4134c, speed of light thru diamond
n ≈ 1.5 for glass >> u = c/n = 0.67c, speed of light thru glass
Lorentz Transformation Solution to the Fizeau Experiment
By associating ( relatively stationary ) system S with earth and connecting the high velocity water with ( relatively moving ) inertial frame of reference system S' , the equation for the light will be
and by applying the FitzGerald - Lorentz transformations
we get
.
But when we let time 
, starting time for fluid motion, this then gives
This then is the Fizeau - Fresnel's convection equation by means of FitzGerald - Lorentz transformation equations. Q.E.D.
Relativistic Addition of Velocities as a Solution to the Fizeau Experiment
In several of Einstein's writings this is the proffered solution while still using system S for earth and system S' for the flowing liquid where the relativistic addition of velocities is given by
.
Let 
, phase velocity of light in a stationary transmitting medium ( fluid liquid ), therefore
This again is the Fizeau - Fresnel's convection equation by means of the Einstein relativistic addition of velocities. Q.E.D.!!
Why the historic 1851 Fizeau Experiment is important
The historic 1851 Fizeau Experiment is important because in both employing the Lorentz transformation equations as well as the Einstein relativistic addition of velocities, both of which are at the heart of special relativity mathematics, that the mathematics of special relativity introduced by Einstein in 1905 could explain prior conundrums in 19th century physics such as the Fresnel convection equation and the Fizeau water experiment as well as bringing illumination into 20th century astronomical and atomic physics.
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