Relativity Science Calculator - The 1851 Fizeau Water Experiment

The 1851 Fizeau Water Experiment

"The most incomprehensible thing about the world is that it is comprehensible"
- Albert Einstein ( 1879 - 1955 )

Some Common Refractive Indices Before Fresnel's Convection Equation is Applied
u = vp = c/n, phase velocity, in a stationary medium where v = 0
Intervening Medium Physical Material n u
vacuum space pure rarified 1 ( exactly ) 1c
0 °C & 1 atm
hydrogen 1.000132 0.999868c
helium 1.000036 0.999964c
atmosphere at STP
( Standard Temperature and Pressure )
1.0002926 0.9997c
water vapor 1.000256 0.99974c
oxygen 1.000271 0.99973c
Other Optical Materials
room temp
diamond 2.4235 0.41263c
amber 1.55 0.645c
crown glass ( pure ) 1.50 - 1.54 0.66667c - 0.64935c
crown glass ( impure ) 1.485 - 1.755 0.6734c - 0.5698c
silicon 3.4170 0.29265c
fresh water 1.333 0.75018c
benzene 1.501 0.66622c
glycerol 1.47 0.68027c
liquid paraffin 1.48 0.67568c
the velocity by which any one frequency component of the wave propagates in spacetime - i.e., crest - to - crest propagation; this is different from group velocity which is the rate at which the envelope or encompassing amplitude of the waveform is propagating.

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"

Fresnel 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 mathemtical discoverer in 1818 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, , is attenuated by a refractive index, , 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, approaches 1 and the observed speed of light, , becomes   since velocity, , simply disappears as the dependence of on becomes increasingly neglible.

Interestingly Fizeau rather accurately determined the speed of light, , 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.

Lorentz Transformation Solution to the Fizeau Experiment

By associating ( relatively stationary ) system    with earth and connecting the high velocity water with ( relatively moving ) inertial frame of reference system  ,  the equation for the light will be

Fundamental Theorem of The Calculus analogous to the method of exhaustion

and by applying the FitzGerald - Lorentz transformations

definition of the mathematical limit

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 for earth and system 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 it demonstrated experimentally the validity of both the FitzGerald - Lorentz transformation equations
( developed 1889 - 1892 ) and Einstein's relativistic addition of velocities whereby Einstein's special relativity mathematics could explain prior conundrums in 19th century physics as well as bringing illumination into 20th century astronomical and atomic physics.

§ References:

"On the Effect of the Motion of a Body upon the Velocity with which it is traversed by Light", by M.H. Fizeau, published Philosophical Magazine and Journal of Science
[ Fourth Series ], London, Edinburgh, and Dublin, April 1860 for the English translation from Annales de Chimie et de Physique, December 1859, where the original paper was presented to the Parisian Academy of Sciences on September 29, 1851.