Griffiths discusses an interesting gedanken experiment to explain the that two events which are simultaneous in one inertial frame are not, in general, simultaneous in another.
Let a freight car is moving along a smooth, straight track with a velocity $v$ as observed by an observer on the ground. Let the length of the car be $2L$ and from the center of the car hangs a light bulb. Let us analyse the motion from the perspective of an observer standing on the ground. At $t=0$, when the bulb is switched on, the light travels from the center to the front-end and the back-end of the car with the same velocity $c$ (postulate of relativity).
What are the times (as measured by the ground observer) at which the light reaches the front end and the back end? Let the light reaches the front end at time $t_1$ and the back end at time $t_2$. Then, $t_1=(L+vt_1)/c$ and $t_2=(L-vt_2)/c$. Is this correct or does one have to take length contraction intro account and change from $L$ to $L/\gamma$? Can we derive properly derive this using the Lorentz transformation formula?