Problem deriving failure of simultaneity at a distance

[Roo2]

Problem deriving "failure of simultaneity at a distance"

Homework Statement

I'm trying to derive the failure of simultaneity at a distance from the spatial part of the Lorentz transformation. In his treatment of the subject, Feynman glosses over the algebra/derivation and produces the formula:

I tried to start with the first equation of the Lorentz transformation and arrive at the difference in simultaneous time. As seen from my work below, I'm not having much luck. Where is my math incorrect?

On a related note, in reading about the Michelson-Morley experiment, it appears that spatial Lorentz contraction completely accounts for the null result. Why, then, is the time dilation necessary? Wouldn't that double-count the difference in perceived time and position between the observer in the stationary and moving reference frames?

Homework Equations

The Lorentz contraction, shown above

The Attempt at a Solution

 

[Chestermiller]

You need to use the second equation of the Lorentz transformation, and recognize that in the S frame of reference, t₁ = t₂. That is, as reckoned from the S frame of reference, two events occur simultaneously (i.e., at t₁=t₂=t) at x₁ and x₂. As reckoned from the S' frame of reference (i.e., using a set of clocks synchronized in the S' frame of reference), the two events are observed to not occur simultaneously.