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I've been given the following scenario:

Observer $B$ is in the center of a train carriage which is moving at velocity $v$ with respect to an observer $A$. Two light signals are emitted from sources L at the left end and R at the right end of the carriage, such that they reach $B$ simultaneously. $B$ observes that the light signals were emitted at the same time. Show that $A$ doesn't agree.

What I've done so far is as follows:

Let the carriage be of length $2d$ from $A$'s point of view. Then the distance that light has travelled from the left is $d+v\Delta t_L=c\Delta t_L$. Similarly, the distance that light has travelled from the right to B is $d-v\Delta t_R=c\Delta t_R$. Can solve these to get $\Delta t_L=\dfrac{d}{c-v}$ and $\Delta t_R=\dfrac{d}{c+v}$. I'm OK with this.

In B's frame of refence, the two times $\Delta t'_L=\Delta t'_R$. Surely two equal time differences will dilate the same and we get that $\gamma \Delta t_L= \gamma \Delta t_R$ and thus $\Delta t_L=\Delta t_R$, which conradicts the above answer? I've been working on this for hours and it's completely inhibiting my progress.

Thanks for any replies!

Lammey
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1 Answers1

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Maybe the following image can help. Hope it's self explanatory.

enter image description here

Gil Bor
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  • Thanks for the answer, but I don't understand what point the diagram is trying to make. At least, it doesn't explain why the time dilation equation I've written doesn't hold here. The thing is with a lot of these problems, I can understand it in some sense, it's just when I try and make it compatible with the rest of my knowledge that it goes pear shaped. – Lammey Feb 14 '14 at 18:24
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    You are trying to show by formulas what is evident in the picture: the coordinate $t_B$ is the same for the events L and R, while the coordinate $t_A$ for these same events is not. The formulas relating the coordinates of the same point in space time (event) in different inertial frames are given by the Lorentz transformations. I thought it may be helpful for you, at least conceptually, to see what's going on in the picture, so you understand why the statement of the problem must be true, then translate this understanding to formulas using analytic geometry. – Gil Bor Feb 14 '14 at 20:48