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The speed of light is independent of the motion of source and it is always constant in vacuum. The earth goes around the sun at speed of 200 kilometers per second. One can ignore the rotational speed which is 460 meters per second at the equator.

Berne and Lucerne lie almost along a straight line where Lucerne is 60 kilometers toward east of Berne.

Light pulses are simultaneously sent to both cities from a tower located in the middle of Berne and Lucerne; the tower is equally distant from both Burne and Lucerne.

As soon as light pulses leave its source, then it's only light and space (ignoring air). Light signal is not co-moving with anything. A person in space would see that light signal has to travel more distance from the tower to Lucerne because Lucerne is moving away from the light signal, and therefore light has to travel through more space. On the other hand, the person would notice that the light pulse has to travel comparatively less distance to get to Berne because Berne is moving toward the pulse.

Now suppose that the tower sends a sound signal to both cities. The sound uses air as the medium which is co-moving with the earth. I believe that the person in space would notice that the sound signals get to both cities at the same time. Do I have it correct?

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It is the whole concept of simultaneity that is affected by relativity. If events A and B are simultaneous in one frame of reference, there is no guarantee that they will (and, in general, they won't) be simultaneous in another frame of reference. For how would the person in space know when the two towns have heard the sound? It would probably be signalled with flashes of light.

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    $\begingroup$ Signalling is the whole root of this problem - many aficionados of relativity don't know how to reason separately about signals. Nor grasp that, to everyone outside the physics community, simultaneity is not in principle affected or dependent upon signalling. $\endgroup$
    – Steve
    Commented Apr 21, 2020 at 10:07
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    $\begingroup$ Nice. I guess they would have to conspire to flash a light to let the space observer know they had received the sound. This is now the experiment to consider. $\endgroup$
    – user196418
    Commented Apr 22, 2020 at 19:35
  • $\begingroup$ @Phoenix87 Part 1: "If events A and B are simultaneous in one frame of reference, there is no guarantee that they will (and, in general, they won't) be simultaneous in another frame of reference."" I had always that if two events are simultaneous in one frame of reference then in another frame of reference they will ALWAYS be non-simultaneous unless the two events happen close to each other, and in such a case observers from different reference frame would agree if the events were simultaneous or not. Please let me know if I have it correct. $\endgroup$
    – PG1995
    Commented Apr 23, 2020 at 3:28
  • $\begingroup$ Part 2: Suppose when the sound signals are received in both cities, light pulses are sent to the space guy and also to the tower which is situated in the middle of both cities. The scenario is more like described in this video: youtube.com/watch?v=wteiuxyqtoM . If the space guy happens to be present right above the tower (just like the guy in the middle of platform), he would receive the light pulses simultaneously therefore he would conclude that sound signals were received simultaneously. Do you agree? $\endgroup$
    – PG1995
    Commented Apr 23, 2020 at 3:30
  • $\begingroup$ Part 3: Moreover, suppose that when sound signals are received in both cities, they in turn activate two clocks. Considering what I said above, the clocks would be synced in the frame of reference of space guy. Suppose there is a guy in Berne. He knows the exact time it takes him to travel to Lucerne (Lucerne is in the east of Berne). He notes the Berne clock time and starts his travel. When he reaches Lucerne, wouldn't he notice that both clocks are in sync? $\endgroup$
    – PG1995
    Commented Apr 23, 2020 at 3:30

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