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There are a gazillion tutorials and references about relativity on the web, but sometimes I have a question that all these seem to avoid or gloss over.

Let me introduce Bob and Alice, who are in different inertial frames. Bob sees Alice as moving away from him at ½

*c*. Alice would say the same about Bob. Both Bob and Alice have a light speed measurement device: a track, 1 light second (LS) long, with synchronized clocks on both ends that record when a light pulse passes by. The tracks are aligned along the axis of motion.

Based on the theory of relativity (and a lot of experiments), we know that both devices will time the pulse at exactly 1 LS/sec. This remains true even if Bob originates the pulse and has it travel through his track before reaching Alice and her track.

That’s how each sees their own track measurement. But how does Bob see Alice’s measurement?

My premise, and what I see implied in various articles, is that time dilation and length contraction allow each observer to see their own measurement at 1 LS/sec, while they also see the other’s measurement at 1 LS/sec, even though classical physics would say that the “moving” object should measure light at ½ LS/sec (since the object is moving at ½

*c*).

I thought I would test my understand by checking the numbers. I went to http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/tdil.html to use their handy calculators for time dilation and length contraction. For Bob, Alice’s track is only 0.8660254037844386 LS long. Alice is moving at ½

*c*, so from Bob’s point of view, a light pulse traveling down Alice’s track will take twice the time it would if the track weren’t moving.

Bob will see that the pulse travels 2*0.866… in 2*0.866… seconds, of course. Bob understands that Alice may view herself as the one who is not moving and that she perceives her track as 1 LS long. So, to my naïve way of thinking about it, time dilation will convert Bob’s 2 * 0.866… seconds into Alice’s 1 second and everyone will be happy—no matter how you measure it, light travels at

*c*.

The time dilation is the inverse of the length contraction, but we are going to convert from Bob’s frame to Alices’, so diving by the inverse is the same as multiplying by the original number. Even without doing the math, I can sense trouble. The equation looks like 2 * 0.866… * 0.866… or 3 * 0.866, which is clearly not even close to 1 second.

What factor did I fail to take into account or what concept did I get wrong?