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Say we have two inertial observers Bob and Alice moving relative to each other at a significant fraction of the speed of light c (say 0.5c). Bob is moving to the left relative to Alice and Alice to the right relative to Bob.

When their origins cross, a pulse of light is emitted to the right. In Bob's frame, he sees Alice moving in the same direction as the light pulse at 0.5c so thinks she must see the light moving away from her at 0.5c. She says she sees light moving at c, and so Bob accuses her of using shorter metre rulers in measuring the distance travelled by light in her frame (I'm considering length contraction and time dilation independently for the moment).

So if she measures some length in her frame that looks the same as what Bob is measuring, she gets a larger numerical value than what Bob is getting.

Now in Alice's frame, she sees Bob moving away from the light pulse at 0.5c so she concludes Bob must see the light moving away from him at 1.5c but he insists the he sees the light pulse moving away at c. So if she accuses him of using shorter rulers to measure the distance travelled by light in his frame. But if he's using shorter metre rulers than shouldn't he measure a larger numerical value for the distance and calculate a larger speed of of light than what Alice gets?

(i.e. he measures the same distance as Alice measures but because his rulers are shorter, he gets a larger number for distance).

What've I misunderstood here? (this is just some preamble before going into the derivation of the Lorentz factor by the way*. An example like the above was given but only for someone going in the same direction as the speed of light as in the second paragraph and I wanted to make sure that it would work the other way around)

* (from 47:50)

When their origins cross, a pulse of light is emitted to the right. In Bob's frame, he sees Alice moving in the same direction as the light pulse at 0.5c so thinks she must see the light moving away from her at 0.5c. She says she sees light moving at c, and so Bob accuses her of using shorter metre rulers in measuring the distance travelled by light in her frame (I'm considering length contraction and time dilation independently for the moment).

So if she measures some length in her frame that looks the same as what Bob is measuring, she gets a larger numerical value than what Bob is getting.

Now in Alice's frame, she sees Bob moving away from the light pulse at 0.5c so she concludes Bob must see the light moving away from him at 1.5c but he insists the he sees the light pulse moving away at c. So if she accuses him of using shorter rulers to measure the distance travelled by light in his frame. But if he's using shorter metre rulers than shouldn't he measure a larger numerical value for the distance and calculate a larger speed of of light than what Alice gets?

(i.e. he measures the same distance as Alice measures but because his rulers are shorter, he gets a larger number for distance).

What've I misunderstood here? (this is just some preamble before going into the derivation of the Lorentz factor by the way*. An example like the above was given but only for someone going in the same direction as the speed of light as in the second paragraph and I wanted to make sure that it would work the other way around)

* (from 47:50)

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