Simultaneity for stationary and inertially moving frames is different?

Imagine the following scenario: • S, L and R are three stationary observers in stationary, primary inertial reference frame (FS)
• M is an observer moving to the right in a moving, second inertial reference frame (FM)
• S holds a long, rigid rod with bombs at each end - LB and RB
• At t0 (zero), S sends a light pulses in opposite direction along the rod
• At t1, the light pulses reach each of the bombs and they explode - LE and RE
• At t2
• Observer L sees the explosion of the bomb LE
• Observer R sees the explosion of the bomb RE
• Observer M sees the explosion of the bomb RE
• At t3
• S sees both the explosions LE and RE
• At t4
• Observer L sees the explosion of the bomb RE
• Observer R sees the explosion of the bomb LE
• At t5
• Observer M sees the explosion of the bomb LE

Now my question about "relativity of simultaneity"?

I think in the primary frame FS, in which L, S and R are, the special relativity will say that the explosions LE and RE were simultaneous, even though all three see the explosions at different time coordinates t2, t3 andt4. Is that correct? L and R see the explosions at different time because they are not spatially equidistant from the event LE and RE. Therefore we can intuitively and obviously say that the explosions were simultaneous. Of course. And even special relativity agrees with that notion of simultaneity.

Similarly, in the second moving frame FM, in which moving observer M (moving to the right at constant velocity) is, M sees the explosions RE at t2 and LE at t5. But why is it given as an example of RE and LE as not being simultaneous from the point of view of M in frame FM? Of course observer M is not equidistant from LE and RE over the duration of experiment. Then just like observer L and R it sees explosions LE and RE at different time - by the fact of not being equidistant. But knowing that it is moving at constant speed to the right could M not deduce that LE and RE were simultaneous (intrinsically)? Or does the special relativity says also that LE and RE were not simultaneous from the perspective of observers L and R as well? I understand M may not give same value of time coordinate to LE and RE in its frame. But same is true in some sense for time coordinates assigned ot LE and RE by L and R right? Or by definition spatially separated events are simultaneous when their time coordinate is same? That does not sound right from the example of observer L and R. Or does/should simultaneity consider time coordinate and spatial distance to call events simultaneous?

As the events LE and RE are at same distance from the apex of a light cone (which will be the same light cone in every inertial frame) be spatially separated simultaneous events in absolute sense?

I sometimes wonder if some different sense of simultaneity are used in different instances of scenarios. I may be wrong. I guess I am trying to understand and nail down a consistent definition of simultaneity.

Also why is "observation of spatially separated events (reception of photons from the events) by third party" given importance when talking about simultaneity?

I think in the primary frame FS, in which L, S and R are, the special relativity will say that the explosions LE and RE were simultaneous, even though all three see the explosions at different time coordinates t2, t3 andt4. Is that correct?

Yes, that is correct.

Similarly, in the second moving frame FM, in which moving observer M (moving to the right at constant velocity) is, M sees the explosions RE at t2 and LE at t5. But why is it given as an example of RE and LE as not being simultaneous from the point of view of M in frame FM?

This is incorrect. In the frame FM, M does not see the explosions at t2 and t5 . M sees the explosions at t2 and t5 IN FRAME FS . All the events that you mentioned happening at t1, t2, t3 etc. were happening in frame FS. In frame FM, they will happen at different time coordinates , not at t1, t2, t3 etc.

Remember , the spacetime diagram that you have drawn is drawn in the frame of FS. So, the time coordinates of the events are in frame FS. To get the time coordinates of the events in frame FM, you will either have to draw a new spacetime diagram, or draw the space and time axes of frame FM in the same diagram, which will be different than the space and time axes of frame FS.

Once, you do that, you will be able to understand the concept of simultaneity much better

You also have to consider everything from M's point of view, with M stationary and S moving "backwards" (to the left). S saw the light from LE and RE at the same time; this is a physical fact, true in all frames of reference. From M's perspective the light from the left explosion LE to S traveled less distance than the light from RE to S (since S is, in M's frame, moving towards LE). The speed of light is constant, hence in M's frame RE must have happened first in order for the light from it to arrive at S's eye at the same time as light from LE.

The key fact is that the speed of light is the same, c, for all observers regardless of their relative motions.

• Just like S is moving to the left, can we not say the location where LE happened is moving to the left in M frame. Won't the light emitted from LE have leftward component because the source LB is moving to the left? I guess similar to the diagonal path in a light clocks perpendicular to the moving frame, that is used in time dilation explanations? May be I am thinking is wrong in comparing the two scenarios. Mar 16 '21 at 19:54
• Light always travels at the same speed, so there's no additional "leftward component" because LE is moving in M's frame. This is the source of all of the weirdness of relativity: every beam of light always moves at the same speed regardless of who measures it or how the source of the light is moving. (There's a caveat about non-inertial frames of reference, but it's not relevant here.) Mar 16 '21 at 21:06

The astronaut-assignment of t-coordinates can be realized by a radar measurement.
To assign a t-coordinate, the astronaut arranges to send a light-signal to the event and await its echo.
Noting the astronaut-clock reading of Sending and Receiving, the astronaut assigns to event-$$LE$$, for example. $$t_{LE}=t_{midpointEvent}=t_m=\frac{t_r+t_s}{2}$$ $$x_{LE}=t_{halfRoundtripTime}=\frac{t_r-t_s}{2}$$ That is to say, the astronaut says that distant event-LE
is simultaneous with the local event "the astronaut-clock-reads-$$t_m$$".
Thus, the astronaut assigns $$t_{LE}$$ that time on the astronaut-clock $$t_m$$".

On the diagram below,

• $$A_m$$ is the midpoint event for astronaut-L measuring LE.
• $$B_m$$ is the midpoint event for astronaut-L measuring RE

An astronaut determines that two events are simultaneous when their $$t$$-coordinates are equal.
(Receiving light-signals at the same event is not enough information... one needs to know when light-signals from the astronaut would have had to be sent to result in those echo-receptions.)

Since $$A_m$$ and $$B_m$$ coincide, astronaut-L says at LE and RE are simultaneous. Similarly, astronaut-R and astronaut-S say that LE and RE are simultaneous.

However, astronaut-M says that LE and RE are not simultaneous since $$P_m$$ and $$Q_m$$ do not coincide.
(I highlighted the triangle to help emphasize the midpoint-event $$P_m$$.)

(Note: due to time dilation, astronaut-M does not assign $$t_5$$ (as L,S,R would) to event $$Q_r$$.) Abstracting the construction above,
for an inertial astronaut to measure a distant event, locate the intersections of the astronaut-worldline with the light-cone of the distant event.