Saturday, February 9, 2013

michelson=morley experiment


Observer resting in the aether

Graphical presentation of the expected differential phase shifts in the Michelson–Morley apparatus
Animated presentation of the expected differential phase shifts
Figure 4. Expected differential phase shift between light traveling the longitudinal versus the transverse arms of the Michelson–Morley apparatus
The beam travel time in longitudinal direction can be derived as follows:[A 10] Light is sent from the source and propagates with the speed of light c in the aether. The mirror starts at distance L (the length of the interferometer arm) and is moving with velocity v. The beam hits the mirror at time T_1 and thus travels the distance cT_1. At this time, the mirror has traveled the distance vT_1. Thus cT_1 =L+vT_1 and consequently the travel time T_1=L/(c-v). The same consideration applies to the backward journey, with the sign of v reversed, resulting in cT_2 =L-vT_2 and T_2 =L/(c+v). The total travel time T_{l}=T_{1}+T_{2} is:
T_{l}=\frac{L}{c-v}+\frac{L}{c+v} =\frac{2L}{c}\frac{1}{1-\frac{v^{2}}{c^{2}}} \approx\frac{2L}{c}\left(1+\frac{v^{2}}{c^{2}}\right)
Michelson obtained this expression correctly, however, in transverse direction he obtained the incorrect expression
T_{t}=\frac{2L}{c},
because he overlooked that the aether wind also affects the transverse beam travel time. The result would be a delay in one of the light beams that could be detected when the beams were recombined through interference. Any slight change in the spent time would then be observed as a shift in the positions of the interference fringes. If the aether were stationary relative to the Sun, Michelson expected (using his incorrect formula) that the Earth's motion would produce a fringe shift 4% the size of a single fringe of yellow light. Michelson did not observe the expected 0.04 fringe shift; the maximum possible shift was at most only about 0.02 fringes. He concluded that there is no measurable aether drift.[4]
However, this result has to be modified by considering the correct transverse travel time, first given by Alfred Potier and Lorentz (1886), and which can be derived as follows: The beam is propagating at the speed of light c and hits the mirror at time T_3, travelling the distance cT_3. At the same time, the mirror has travelled the distance vT_3 in x direction. So in order to hit the mirror, the travel path of the beam is L in y direction and vT_3 in x direction. This inclined travel path follows from momentum conservation of the light rays or photons, and is also the consequence of a Galilean transformation from the interferometer rest frame to the aether rest frame. Therefore the Pythagorean theorem gives the actual beam travel distance of \scriptstyle \sqrt{L^{2}+\left(vT_{3}\right)^{2}}. Thus \scriptstyle cT_{3} =\sqrt{L^{2}+\left(vT_{3}\right)^{2}} and consequently the travel time \scriptstyle T_{3} =L/\sqrt{c^{2}-v^{2}}, which is the same for the backward journey. The total travel time T_{t}=2T_{3} is:
T_{t}=\frac{2L}{\sqrt{c^{2}-v^{2}}}=\frac{2L}{c}\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\approx\frac{2L}{c}\left(1+\frac{v^{2}}{2c^{2}}\right)
(This is analoguous to the derivation of time dilation using a light clock.) The fringe shift calculated from the difference between Tl and Tt is given by
\approx\frac{2Lv^{2}}{\lambda c^{2}}
where λ is the wavelength. Lorentz pointed out that when the correct transverse travel time is considered, the expected result in Michelson's experiment is reduced by half to approximately 0.02 fringes, which is nearly equal to the estimated experimental error
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