I'm reading about how the concept of simultaneity breaks down in special relativity. Here's the relevant paragraph:
Consider a photon source $C_0$ in a train car situated equidistant between detectors $C_1$ and $C_2$. The source emits photons back to back. In a reference frame at rest with respect to the train car, photons arrive at the detectors simultaneously. In a reference frame at rest relative to the train station, however, which the train is assumed passing through, the photon source and the detectors are in motion with speed $v$ from left to right. The two frames synchronize their identical clocks when the origins of their coordinate systems coincide, whereupon the photons are emitted. Seen from the frame of the station, event $C_1$ happens before $C_2$; the photon first encounters detector $C_1$ moving toward it.
I'm a bit confused by the part in bold, though I might be wrong. I googled on how clocks are synchronized in the same reference frame, but I've heard that from the perspective of an inertial observer, time goes slower in another inertial reference frame moving w.r.t. them, so isn't it pointless to sync clocks across two reference frames moving w.r.t. each other?
Another aspect that I'm trying to get is exactly how does the platform observer come to "know" of the events in the train. (ref. following poorly drawn diagram)
Let's say in the train's frame, clocks at $C_1$ and $C_2$ are synchronized. They simultaneously (in the train frame) emit photons towards $C_0$. $C_0$ emits red light in all directions when it receives a photon from $C_2$ and emits green light in all directions when it receives one from $C_1$. In the train's frame, both $C_1$ and $C_2$ photons simultaneously reach $C_0$ ==> $C_0$ emits yellow light (red and green simultaneously).
What kind of a setup can a platform observer use to find the time difference between $C_0$ receiving the two photons? Suppose there's an array of clock-detectors lined up on the platform parallel to $v$ and at a distance $D$ from the train - all of them synchronized in the platform's frame.
Each of those clock-detectors are set up in such a way that they record the time when a beam of light perpendicular to the line of detectors (i.e. perpendicular to $v$) in the platform frame reaches them. What should the platform observer notice? I'm confused between two possibilities:
He will notice red light being emitted first from the train: out of that the red ray traveling perpendicular to the platform will set off one of the detectors. After that, he will notice green light coming out from the train which sets off another platform detector further to the right.
He will notice yellow light being emitted from the train, which sets off one of the detectors on the platform. The emission of photons from $C_1$ and $C_2$ was out of sync (in the platform frame) in the first place, and the movement of $C_0$ to the right causes it to receive both simultaneously.
I'm not sure which of these options is correct, or whether both of them are wrong.