B Length Contraction Thought Experiment: Spot Mistake/Wrong Assumption

[galm_2727]

TL;DR Summary

I was trying to come up with a thought experiment for showing the length contraction, but I'm not getting the correct expression and I can't see where is the mistake.

Alice travels in a spaceship, which she measures to be L. The spaceship is moving with velocity v relatively to Bob. Alice makes a light beam traveling along the spaceship and measures the time interval it takes to go from one end to another, Δt_A.

So, equation (1): L = c × Δt_A

From Bob's point of view, the light beam will take a time Δt_B and the length of the spaceship will be:

L' = c × Δt_B - v × Δt_B = (1 - β) × c Δt_B

But Δt_B = ϒ Δt_A. Therefore:

L' = (1 - β) ϒ c Δt_A

Using (1), we have:

L' = (1 - β)^1/2 / (1 + β)^1/2 L

Could you help me spot my mistake/wrong assumption?

 

[Dale]

But ΔtB = ϒ ΔtA.

Here is the mistake. This formula does not apply here.

I recommend against using the simplified length contraction and time dilation formulas for exactly this reason. I recommend using the full Lorentz transform instead. That will automatically simplify when appropriate, but will avoid mistakes like this.

 

[Ibix]

As Dale says, you can't naively use the time dilation formula here - doing so neglects the relativity of simultaneity. You need to use the full Lorentz transforms.

 

[galm_2727]

Here is the mistake. This formula does not apply here.

I recommend against using the simplified length contraction and time dilation formulas for exactly this reason. I recommend using the full Lorentz transform instead. That will automatically simplify when appropriate, but will avoid mistakes like this.

Thank you.

 

[David Lewis]

From Bob's point of view, the light beam will take a time ΔtB and the length of the spaceship will be:

If light is traveling in the same direction as the ship (toward the front) then t = L_B/(c-v).
If light is traveling toward the back of the ship, t = L_B/(c+v)