This gives c = 2 a.u./22mins = 2.2x10^8m/s .
In 1675, the Danish astronomer Olaus Roemer obtained an approximate value of c using eclipses of the moons of Jupiter. When Earth and Jupiter are on the same side of the sun (E1 and J1 in Figure), eclipses occur ahead of schedule by up to 11 mins; when on opposite sides of sun, they are delayed by up to 11 mins (E2 and J2 in Figure).
This gives c = 2 a.u./22mins = 2.2x10^8m/s .
More accurate values were measured by Fizeau and by Foucault; see Figure below.
Foucault used more precise timing, allowing shorter distance and measurement of v in water and
other transparent materials; v was shown to be less than c, contrary to Newton's view.
We now know that v=c/n, where the refractive index n can be measured by refraction experiments.
Fizeau then (1851) measured the velocity of light in a moving medium (water) see Figure below.
Any effect will be small, because c is so large, but we
need only determine the change in the velocity of light, compared to its value in still water.
If light travels faster, its wavelength must increase, since the frequency is unaltered.
Fizeau used a device called an interferometer, which is very sensitive to small changes in
wavelength.
Consider first the change in properties of light between air and stationary water:
Velocity in still water v = c/n = (3x10^8 m/s)/1.33 = 2.25 x 10^8 m/s
Fizeau used yellow light: measured in air, lambda = 0.5 micrometers = c/f where f = frequency in Hz.
Measured in water, lambda = v/f = c/(nf) = 0.375 micrometers.
Number of wavelengths in a water-filled tube of length L (=1.5m) is N = L/lambda = 1.5 / 0.376 x 10^-6 m = 4.0 x 10^6
Consider now the change in light properties between still and moving water. One possibility is that light velocity is unaffected by the motion of water.
A second possibility is that
light behaves like sound: its velocity with respect to the medium
remains the same. If so, its absolute velocity in water moving at speed u would be: v' = v + u (assuming that
the light is moving in the same direction as the water, otherwise u is preceded by a minus sign)
The new wavelength when the water is moving at velocity u would be: lambda' = (v+u)/f
The number of wavelengths in tube is now N' = L/lambda' = Lf/(v+u) = (Lf/v) (1+u/v)^-1
Using the binomial theorem and neglecting higher-order terms (since u/v << 1) we have: N' = (Lf/v) (1 - u/v )
If we have a second tube in which water is moving in a direction opposite to that of the light,
u is replaced by -u and the number of wavelengths becomes N'' = (Lf/v) (1 + u/v)
The change in the number of wavelengths in the tube is N'' - N' = (Lf/v)(2u/v)
If u = 7 m/s, N'' - N' = (4 x 10^6) [2 x 7/(3 x 10^8/1.33)] = 0.25
So passing light into two tubes with water flowing in opposite directions (see Figure)
provides one quarter--wavelength path difference compared to stationary water. If we now reverse the water flow
through the whole apparatus, the change in optical path length becomes 0.5 wavelength.
For example, if the two light beams combine to give constructive interference with the water flowing in one
direction, the interference would be destructive when the water is moving in the opposite direction and
the interferometer should easily detect this change as a change in intensity when the two light beams are recombined.
The result actually observed was intermediate between these two possibilities ! The interference condition did change, but by an amount which suggested that the velocity in the moving medium must be given by the equation:
v' = c/n + f u
where f(<1) is now the Fresnel coefficient of drag. Experiments with fluids of different refractive index n gave the empirical relation:
f = (1-1/n^2)
The meaning of this drag effect was not understood until Einstein provided an interpretation.
Suppose light from a particular star arrives vertically and that the earth is travelling horizontally with a speed u relative to the ether. During the time interval t which light takes to travel down the telescope, its base moves horizontally by an amount ut (C&N Fig.12-9). Therefore the telescope must be tilted through an angle alpha to enable the light to travel down its axis; a vector triangle of displacements gives: alpha~ tan alpha = u t/(c t) = u/c
If the ether is stationary relative to the sun, u varies between +30km/s and -30km/s, depending on the time of year, giving alpha ~ 10-4 rad (20 arc sec). This is a small tilt: the end of a 10m-long telescope would need to be moved only 1 mm. The observed effect is close to this prediction.
(Note that if the ether is actually moving at some speed V relative to the sun, the earth's velocity relative to the ether would be V +/- 30 km/s, so the prediction is unaltered)
If the telescope is not tilted, light will move down the telescope slightly off-axis; combined with the parallax effect, aberration produces an elliptical apparent motion of the star. [For a star observed at an elevation theta, the angular tilt becomes alpha =(u/c) sintheta, so a smaller effect is observed for stars which are not directly overhead.]
In 1871, Sir George Airy repeated the experiment with a telescope filled with water, refractive index n = 1.33, expecting that the angular tilt might increase to alpha = u/(c/n) = n (u/c). But he observed no difference, compared to the case of a telescope in air.
In fact, this null result had been predicted by Fizeau, assuming that light travelling in water would be dragged horizontally at a velocity f u , where f = 1-1/n^2 is the Fresnel coefficient of drag. The refractive index of the water slows down the light but its coefficient of drag increases the velocity and (because f = 1-1/n^2) the two effects exactly cancel.
So Bradley's original observation and Airy's experiment are both consistent with the earth moving through an ether which is stationary (or moving at constant velocity) with respect to the sun. However, the role of the ether was soon to be challenged.
In 1887, A.W. Michelson built an interferometer to measure the speed u of the earth relative to the ether. The arrangement (C&N Fig. 12-6) is now known as Michelson interferometer. By means of a half-silvered mirror, a light beam is directed in two perpendicular directions, then reflected back towards the same half-silvered mirror, so that some of the light from each path enters a detector (e.g. eyepiece) where it is combined and can exhibit constructive or destructive interference.
As shown in C&N (pp. 301 - 303), light travelling parallel and antiparallel to the"ether wind" would
take a time
tA= L/(c+u) + L/(c-u) = (2L/c) (1 - u^2/c^2)^-1 ,
or (using the binomial theorem) approximately (2L/c) (1 + u^2/c^2)
for its return journey along one arm (length L) of the apparatus.
Light travelling perpendicular to the ether wind would take a time
tB = 2L/(c2-u2)^1/2 = (2L/c)(1-u2/c2)^-1/2, or approximately (2L/c) (1 + u^2/2c^2)
assuming an arm of identical length L. Notice that in both cases, the effect due to drift through the ether is a second-order effect (proportional to u^2) and so is very small, of the order 10^-8 for u = 3x10^4 m/s.
The difference in time should give rise to a difference in phase between the two recombining
waves. It should be visible as a change in intensity at the detector when the whole
apparatus is rotated through 90 degrees, thereby interchanging the ray paths.
In practice, the expected effect is
a shift in a fringe pattern which arises from the fact that the phase (at the observer) changes
over the field of view (with varying displacement from the optic axis).
Michelson first performed the experiment in 1881, expecting to see a fringe shift (upon rotation of the apparatus) of (2L/lambda)(u^2/c^2) = 0.04 fringe, for lambda = 600 nm, L = 1.2 m and u = 30 km/s. He observed no shift. However, there were practical difficulties: rotating the apparatus without introducing distortion, sensitivity to vibration (making the fringes difficult to see, even at 2 a.m. in the city), and temperature changes (the resulting thermal expansion introduced fringe drift).
In 1887, he redesigned the apparatus, in collaboration with E.W. Morley. A system of mirrors was used to increase L by a factor of ten; the apparatus floated on a bed of mercury. They expected to see a fractional fringe shift of 0.4 upon 90-degree rotation of the apparatus but saw nothing.
These null results can't be blamed on incompetence. Michelson was a very gifted experimentalist in the field of optics. For example, he was the first person to measure the diameter of a star, using interferometry.
Michelson and Morley realised that the velocity of the earth relative to the ether might have been zero at the time of measurement, if the earth's orbital velocity around the sun happened to be exactly cancelled by motion of the whole solar system relative to the ether. They planned to repeat the experiment at 3-month intervals.
Although they never carried out these additional measurements, others did so over the next 50 years, with improved apparatus, and also observed no effect. To counteract arguments that the ether might be "entrained" within their basement laboratory, Morley and Miller moved the apparatus to a hill outside Cleveland, in a building of light construction and windows in the direction of the anticipated ether wind.
Miller repeated the experiment on top of Mount Wilson (600 ft above sea level, in California) and in 1925 he announced a small positive effect (u=10 km/s). But results were re-analysed by others, who concluded they were due to temperature variations.
He justified this possibility by arguing that the distance between atoms depends on the strength of electrostatic forces between electrons and atomic nuclei (C&N Fig. 10-9), forces which might be affected by the ether. "We may safely assume that electric and magnetic forces act by means of intervention of the ether".
In 1904, Lorentz pointed out that Maxwell's equations (which govern the behaviour of electric and magnetic fields and predict the properties of electronmagnetic radiation) do not satisfy Galilean relativity. If the Galilean transformation is applied to the Maxwell equations, they take on a different (more complicated) form.
However, the original form of the equations can be preserved if the Galilean transformation is modified to the following:
x' = (x - ut)/(1-u^2/c^2)^1/2
y' = y
z' = z
t' = (t-ux/c^2)/(1-u^2/c^2)^1/2
According to these Lorentz transformations, time is a local rather than universal quantity. Changes in length and time due to motion through the ether always conspire to prevent any measurement of our velocity with respect to it.