# Defining Simultaneity

I've been learning about relativity, and I'm just starting to wrap my head around what it means for events to be simultaneous in one reference frame, but not simultaneous in another.

Taking as the only given that the speed of light is constant, I've ended up defining events that happen simultaneously in any given reference frame like so:

Events that are simultaneous in a given reference frame are events such that we could release photons in the direction of both events, so that from the point of view of that reference frame, the photons both pass by the location of those events just as they happen, at the same time.

Nevermind if my definition doesn't really make sense...I know I"m not really wording it well, but that's exactly why I'm asking this question.

I'm looking for great, intuitive (or as intuitive as possible) ways to define what it means for events to be simultaneous in any given reference frame, taking into account the constant speed of light in all reference frames.

“Simultaneous” means “has the same time coordinate value” in that particular frame. For example “they both happened at t=3”

A reference frame can be thought of as having a (hypothetical) clock at each point, all synchronized to the same value. Those are then used to find the time coordinate of each event.

Photons are used to actually (in a hypothetical way) do that synchronization. But it has to be done taking into account the travel time of those photons. A photon can pass by events with different time coordinates if it has to travel a distance between them.

A better definition of events A and B that are simultaneous in a given frame of reference F would use a point P at rest in F. A and B need not be at rest. Two beams of light leave P at the same time, reflect off A and B, and arrive back at P at the same time. The only way this can happen is for the two beams to travel the same distance. At the time of the reflections, A and B must be the same distance from P as seen if F. The time as seen if F to get to A and B must be the same for both beams. The reflections must happen at the same time as seen if F.

There are a few things about this worthy of note. First, an event is a time and place. Two beams leaving P at the same time occur at one event. The two things happen at the same event happen at same time in any frame of reference. The arrival is similar. This definition means that an observer in another frame G will agree that the events were simultaneous in F, even if they are not simultaneous in his own.

G is moving with respect to F. In G, the departure from P and arrival back at P happen at two different places. We are used to the idea that the same place at two different times can be two different places in a different frame. When I drive my car, it stays still and the world moves by. Someone standing by the road sees the car move from place to place.

The most counter-intuitive thing in special relativity is that time behaves this way too. I see two events as simultaneous. If someone is moving, he sees them as happening at different times. It is not an illusion. It is just like seeing them happen in the same place or different places. This idea is very hard to get used to.

I can imagine a procedure for a galactic civilization, for which it is very important.

For the same inertial reference frame, the synchronizer has to go to the midpoint of the two locations to be synchronized, and send a signal.

That signal must inform a $$t = t_0 + (x/2)/c$$ to be set in the clocks, where $$t_0$$ is the time when the signal is sent, and $$x$$ is the distance between the two locations.

An automatic process sets the clocks to that time $$t$$ when the receptors at each location receive the signal.