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