$($e$)$ In the Michelson$-$Morley interferometer experiment, light, which is supposed to have a speed

$c,$ was sent on paths perpendicular and parallel to the direction of the Earth's motion around the

Sun, and the round$-$trip times of the two beams were compared. The round$-$trip distance for each

beam was $22$ meters. If the "ether" is at rest with respect to the Sun, what is the approximate

difference in travel time of the two beams of light?

$($f$)$ This difference in light travel time was impossible to measure directly, so the researchers had to

be clever: if one splits a nearly monochromatic beam of light, sends the two beams on perpendicular

paths, and then recombines the beams, an interference pattern is created that will reveal small

phase shifts in the two beams corresponding to different travel distances. The Michelson$-$Morley

interferometer was mounted on a marble slab and floated on a pool of mercury so that the entire

apparatus could be rotated $90$ degrees to transfer any delay from one beam to the other. Michelson

and Morley used light from a sodium lamp with wavelength $589nm$ and determined that the time

change between the two beams was less than one one$-$hundredth of a wave period. In order to rule

out a coincidental alignment of the ether motion with the Earth's motion around the Sun during

the experiment, they repeated the experiment every three months $($with similar negative results$).$

What is the period of the sodium light? Show that this experiment implies that the motion of the

ether at the surface of the Earth is less than one sixth the orbital speed of the Earth.

$($g$)$ Does this experiment by itself disprove the theory that light propagates through a luminiferous

ether? How could you modify the theory of the ether to agree with the experimental results? What

experiment might you employ to test your new modified theory?

$($h$)$ What can we deduce about a Newtonian absolute standard of rest from this experiment?

$($i$)$ The cosmic microwave background $($CMB$)$ shows a dipole temperature pattern on the sky as

seen from the Earth due to the Earth's motion with respect to the CMB rest frame. Does the

fact that the Universe as a whole has a preferred reference frame invalidate the Michelson$-$Morley

results? Was Newton correct about an absolute standard of rest?