Understand Michelson-Morley Experiment from Feynman Lectures

[Bishamonten]

http://www.feynmanlectures.caltech.edu/I_15.html#Ch15-S3

I'm reading through Feynman's chapter on special relativity in the first volume of his books in order to have a more comprehensive idea of it as I go through the special relativity homework given to us as extra credit for our electromagnetism class.

I'm having a tough time however, understanding the logic behind the math in section 3 in the link titled "The Michelson-Morley experiment". When Feynman introduces eq. 15.5(that is, the equation that describes the time it would take for the beam of light to travel to the mirror perpendicular to the direction of the Earth's motion in orbit and back, i.e., the one that Lorentz didn't add the length contraction to later), and then afterwards wants us to compare it to the equation that describes the time the beam would take to the other mirror parallel to the motion and back, he says the following:

"And behold, these modifications are not the same—the time to go to C and back is a little less than the time to E and back, even though the mirrors are equidistant from B, and all we have to do is to measure that difference with precision."

My problem is the equation of time(2t₃) describing the trip to mirror C(the perpendicular-to-motion mirror) and back involves a square root of the same denominator in the equation of time(t₁+t₂) describing the trip to mirror E(the parallel-to-motion mirror) and back. By that logic then, it has a smaller denominator than the equation of the latter, and therefore since the numerators are equivalent(as Feynman himself points out), shouldn't it produce a larger magnitude of time than the latter?(meaning, isn't the math actually saying the opposite: that the beam will take longer to reach C and back than it would to reach E and back?).

Not that I'm getting lost in the math or anything that I forgot the intention of the experiment, but that my misunderstanding of his description of the experiment is bothering me enough to not carry on reading the rest of the chapter until I understand what he was really trying to say.

 

[FactChecker]

When you take the square root of a number in (0,1), it gets larger, not smaller.

 

[Bishamonten]

When you take the square root of a number in (0,1), it gets larger, not smaller.

What do you mean by (0, 1)? In any case, what I meant is that by taking the square root of the term 1 - (u^2/c^2), it will be less than the same denominator in eq. 15.4 in the link that isn't being evaluated the square root of. So therefore, a smaller denominator, and because of a smaller denominator, a larger overall value in eq. 15.5 in the link vs. eq. 15.4.

But according to Feynman, eq. 15.5 is supposed to yield a smaller time, not a bigger one. Unless he was not talking about the entire equations themselves, and just the added denominators in the bottom. In that case, would you be able to explain the logic behind that? I thought the entire equations (t₁ + t₂, 2t₃) are supposed to describe the time it takes to go from one of the respective mirrors and back.

 

[Ibix]

But $u^2 <c^2$, so $0<1-u^2/c^2\leq 1$. And, as @FactChecker noted, the square root of a number in the range 0-1 is larger, not smaller.

 

[Bishamonten]

But $u^2 <c^2$, so $0<1-u^2/c^2\leq 1$. And, as @FactChecker noted, the square root of a number in the range 0-1 is larger, not smaller.

Ah, that's what he was saying, and ultimately that's what Feynman was trying to say to my dumb head too! Now it makes sense to me now that I'm not overlooking this fact. Thank you very much you guys.