We now rethink the results in Problem 3 in terms of the experiment of Michelson-

Question

We now rethink the results in Problem 3 in terms of the experiment of Michelson-Morley. Instead of an airplane, we consider a beam of light, traveling at speed c. If aether existed. and if it had proper speed v with respect to Earth, what would the times at the point in Problem 3 be? The speed of aether v and the length L are not specified numerically here, so give your results as expressions containing v ,c, L. Take now L = 1 m and c = 3 times 10 8 m / s. Which speed v should the aether possess in order to detect the time difference with a precision of 1%? To answer this question, define first the precision as Delta = where t0 = 2L / c You might want to use the following Taylor expansions for x 1. 1 / 1 + x 1 - x, 1 / 1 -x2 1 + x2 / 2.

Explanation / Answer

distance = L, speed of light = c, speed of aether = v
a) tup-down = L/(c - v) + L/(c + v) = 2Lc/(c2 - v2) = 2L/c * (1 - v2/c2)-1
tcross = 2*L/(c2 + v2) = 2L/c * (1 + v2/c2)-1/2
b) tup-down - tcross = 2L/c * [(1 - v2/c2)-1 - (1 + v2/c2)-1/2] t0 * [(1 + v2/c2) - (1 - 1/2 * v2/c2)]

= t0 * 3v2/(2c2)

= (tup-down - tcross)/t0 = 3v2/(2c2) = 1%

v = c(0.02/3) = 2.45*107 m/s

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